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Fri, 31 Oct 2014 21:14:39 +0000FeedCreator 1.7.3Comparing the College Football Playoff Top 25 and the Preseason AP Poll
http://blog.minitab.com/blog/the-statistics-game/comparing-the-college-football-playoff-top-25-and-the-preseason-ap-poll
<p>The college football playoff committee waited until the end of October to release their first top 25 rankings. One of the reasons for waiting so far into the season was that the committee would rank the teams off of actual games and wouldn’t be influenced by preseason rankings.</p>
<p>At least, that was the idea.</p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/8ac74acf42052d068b6cd0eeec32f609/cfb_playoff.jpg" style="line-height: 20.7999992370605px; float: right; width: 300px; height: 187px;" /></p>
<p>Earlier this year, I found that the <a href="http://blog.minitab.com/blog/the-statistics-game/has-the-college-football-playoff-already-been-decided">final AP poll was correlated with the preseason AP poll</a>. That is, if team A was ranked ahead of team B in the preseason and they had the same number of losses, team A was still usually ranked ahead of team B. The biggest exception was SEC teams, who were able to regularly jump ahead of teams (with the same number of losses) ranked ahead of them in the preseason.</p>
<p>If the final AP poll can be influenced by preseason expectations, could the college football playoff committee be influenced, too? Let’s compare their first set of rankings to the preseason AP poll to find out.</p>
Comparing the Ranks
<p>There are currently 17 different teams in the committee’s top 25 that have just one loss. I <a href="//cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/File/26e7c8d8d8eee4fe2dfa26dc3d6e3c54/preseason_ap_vs__cfb_playoff_rankings.MTW">recorded the order</a> they are ranked in the committee’s poll and their order in the AP preseason poll. Below is an individual value plot of the data that shows each team’s preseason rank versus their current rank.</p>
<p><img alt="IVP" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/4098bab194a586865d3861f854d65627/ivp.jpg" style="width: 600px; height: 400px;" /></p>
<p>Teams on the diagonal line haven’t moved up or down since the preseason. Although Notre Dame is the only team to fall directly on the line, most teams aren’t too far off.</p>
<p>Teams below the line have jumped teams that were ranked ahead of them in the preseason. The biggest winner is actually not an SEC team, it’s TCU. Before the season, 13 of the current one-loss teams were ranked ahead of TCU, but now there are only 4. On the surface TCU seems to counter the idea that only SEC teams can drastically move up from their preseason ranking. However, of the 9 teams TCU jumped, only one (Georgia) is from the SEC. And the only other team to jump up more than 5 spots is Mississippi—who of course is from the SEC. So I wouldn’t conclude that the CFB playoff committee rankings behave differently than the AP poll quite yet.</p>
<p>Teams below the line have been passed by teams that had been ranked behind them in the preseason. Ohio State is the biggest loser, having had 9 different teams pass over them. Part of this can be explained by the fact that they have the worst loss (a 4-4 Virginia Tech game at home). But another factor is that the preseason AP poll was released before anybody knew Buckeye quarterback Braxton Miller would miss the entire season. Had voters known that, Ohio State probably wouldn’t have been ranked so high to begin with. </p>
<p>Overall, 10 teams have moved up or down from their preseason spot by 3 spots or less. The correlation between the two polls is 0.571, which indicates a positive association between the preseason AP poll and the current CFB playoff rankings. That is, teams ranked higher in the preseason poll tend to be ranked higher in the playoff rankings.</p>
Concordant and Discordant Pairs
<p>We can take this analysis a step further by looking at the concordant and discordant pairs. A pair is concordant if the observations are in the same direction. A pair is discordant if the observations are in opposite directions. This will let us compare teams to each other two at a time.</p>
<p>For example, let’s compare Auburn and Mississippi. In the preseason, Auburn was ranked 3 (out of the 17 one-loss teams) and Mississippi was ranked 10. In the playoff rankings, Auburn is ranked 1 and Mississippi is ranked 2. This pair is concordant, since in both cases Auburn is ranked higher than Mississippi. But if you compare Alabama and Mississippi, you’ll see Alabama was ranked higher in the preseason, but Mississippi is ranked higher in the playoff rankings. That pair is discordant.</p>
<p>When we compare every team, we end up with 136 pairs. How many of those are concordant? Our <a href="http://www.minitab.com/products/minitab">favorite statistical software</a> has the answer: </p>
<p><img alt="Measures of Concordance" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/5f281abfa1e06d5cda492e17b3f9746b/concordance.jpg" style="width: 663px; height: 176px;" /></p>
<p>There are 96 concordant pairs, which is just over 70%. So most of the time, if a team ranked higher in the preseason poll, they are ranked higher in the playoff rankings. And consider this: of the one-loss teams, the top 4 ranked preseason teams were Alabama, Oregon, Auburn, and Michigan St. Currently, the top 4 one loss teams are Auburn, Mississippi, Oregon, and Alabama. That’s only one new team—which just so happens to be from the SEC.</p>
<p>That’s bad news for non-SEC teams that started the season ranked low, like Arizona, Notre Dame, Nebraska, and Kansas State. It's going to be hard for them to jump teams with the same record, especially if those teams are from the SEC. Just look at Alabama’s résumé so far. Their best win is over West Virginia and they lost to #4 Mississippi. Is that <em>really </em>better than Kansas State, who lost to #3 Auburn and beat Oklahoma <em>on the road</em>? If you simply changed the name on Alabama’s uniform to Utah and had them unranked to start the season, would they still be ranked three spots higher than Kansas State? I doubt it.</p>
<p>The good news is that there are still many games left to play. Most of these one-loss teams will lose at least one more game. But with 4 teams making the playoff this year, odds are we'll see multiple teams with the same record vying for the last playoff spot. And if this college football playoff ranking is any indication, if you're not in the SEC, teams who were highly thought of in the preseason will have an edge.</p>
Fun StatisticsHypothesis TestingFri, 31 Oct 2014 13:04:57 +0000http://blog.minitab.com/blog/the-statistics-game/comparing-the-college-football-playoff-top-25-and-the-preseason-ap-pollKevin RudyR-squared Shrinkage and Power and Sample Size Guidelines for Regression Analysis
http://blog.minitab.com/blog/adventures-in-statistics/r-squared-shrinkage-and-power-and-sample-size-guidelines-for-regression-analysis
<p>Using a sample to <a href="http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and-graphs/introductory-concepts/basic-concepts/parameters/" target="_blank">estimate the properties of an entire population</a> is common practice in statistics. For example, the mean from a random sample estimates that <a href="http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and-graphs/introductory-concepts/basic-concepts/parameter-esimates/" target="_blank">parameter</a> for an entire population. In linear regression analysis, we’re used to the idea that the <a href="http://blog.minitab.com/blog/adventures-in-statistics/how-to-interpret-regression-analysis-results-p-values-and-coefficients" target="_blank">regression coefficients</a> are estimates of the true parameters. However, it’s easy to forget that R-squared (R2) is also an estimate. Unfortunately, it has a problem that many other estimates don’t have. R-squared is inherently biased!</p>
<p>In this post, I look at how to obtain an unbiased and reasonably precise estimate of the population R-squared. I also present power and sample size guidelines for regression analysis.</p>
R-squared as a Biased Estimate
<p>R-squared measures the strength of the relationship between the <a href="http://support.minitab.com/en-us/minitab/17/topic-library/modeling-statistics/regression-and-correlation/regression-models/what-are-response-and-predictor-variables/" target="_blank">predictors</a> and <a href="http://support.minitab.com/en-us/minitab/17/topic-library/modeling-statistics/regression-and-correlation/regression-models/what-are-response-and-predictor-variables/" target="_blank">response</a>. The <a href="http://blog.minitab.com/blog/adventures-in-statistics/regression-analysis-how-do-i-interpret-r-squared-and-assess-the-goodness-of-fit" target="_blank">R-squared in your regression output</a> is a biased estimate based on your sample.</p>
<ul>
<li>An unbiased estimate is one that is just as likely to be too high as it is to be too low, and it is correct on average. If you collect a <a href="http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and-graphs/introductory-concepts/data-concepts/why-collect-random-sample/" target="_blank">random sample</a> correctly, the sample mean is an unbiased estimate of the population mean.</li>
<li>A biased estimate is systematically too high or low, and so is the average. It’s like a bathroom scale that always indicates you are heavier than you really are. No one wants that!</li>
</ul>
<p>R-squared is like the broken bathroom scale: it is deceptively large. Researchers have long recognized that regression’s optimization process takes advantage of chance correlations in the sample data and inflates the R-squared.</p>
<p>This bias is a reason why some practitioners don’t use R-squared at all—it tends to be wrong.</p>
R-squared Shrinkage
<p>What should we do about this bias? Fortunately, there is a solution and you’re probably already familiar with it: adjusted R-squared. I’ve written about <a href="http://blog.minitab.com/blog/adventures-in-statistics/multiple-regession-analysis-use-adjusted-r-squared-and-predicted-r-squared-to-include-the-correct-number-of-variables" target="_blank">using the adjusted R-squared</a> to compare regression models with a different number of terms. Another use is that it is an unbiased estimator of the population R-squared.</p>
<p>Adjusted R-squared does what you’d do with that broken bathroom scale. If you knew the scale was consistently too high, you’d reduce it by an appropriate amount to produce an accurate weight. In statistics this is called shrinkage. (You <em>Seinfeld</em> fans are probably giggling now. Yes, George, we’re talking about shrinkage, but here it’s a good thing!)</p>
<p>We need to shrink the R-squared down so that it is not biased. Adjusted R-squared does this by comparing the sample size to the number of terms in your regression model.</p>
<p>Regression models that have many samples per term produce a better R-squared estimate and require less shrinkage. Conversely, models that have few samples per term require more shrinkage to correct the bias.</p>
<p><img alt="Line plot showing R-squared shrinkage by sample size per term" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/c8687540a1adaecc534f746991ce52f0/rsq_shrinkage_w640.png" style="width: 640px; height: 427px;" /></p>
<p>The graph shows greater shrinkage when you have a smaller sample size per term and lower R-squared values.</p>
Precision of the Adjusted R-squared Estimate
<p>Now that we have an unbiased estimator, let's take a look at the precision.</p>
<p>Estimates in statistics have both a point estimate and a <a href="http://blog.minitab.com/blog/adventures-in-statistics/when-should-i-use-confidence-intervals-prediction-intervals-and-tolerance-intervals" target="_blank">confidence interval</a>. For example, the sample mean is the point estimate for the population mean. However, the population mean is unlikely to exactly equal the sample mean. A confidence interval provides a range of values that is likely to contain the population mean. Narrower confidence intervals indicate a more precise estimate of the parameter. Larger sample sizes help produce more precise estimates.</p>
<p>All of this is true with the adjusted R-squared as well because it is just another estimate. The adjusted R-squared value is the point estimate, but how precise is it and what’s a good sample size?</p>
<p>Rob Kelly, a senior statistician at Minitab, was asked to study this issue in order to develop power and sample size guidelines for regression in the <a href="http://www.minitab.com/en-us/products/minitab/assistant/" target="_blank">Assistant menu</a>. He simulated the distribution of adjusted R-squared values around different population values of R-squared for different sample sizes. This histogram shows the distribution of 10,000 simulated adjusted R-squared values for a true population value of 0.6 (rho-sq (adj)) for a simple regression model.</p>
<p><img alt="Histogram showing distribution of adjusted R-squared values around the population value" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/4405515ecfbe8605fdca8347d34dac5d/adjrsqprecision_w640.png" style="width: 640px; height: 427px;" /></p>
<p>With 15 observations, the adjusted R-squared varies widely around the population value. Increasing the sample size from 15 to 40 greatly reduces the likely magnitude of the difference. With a sample size of 40 observations for a simple regression model, the margin of error for a 90% confidence interval is +/- 20%. For multiple regression models, the sample size guidelines increase as you add terms to the model.</p>
Power and Sample Size Guidelines for Regression Analysis
<p>Satisfying these sample size guidelines helps ensure that you have sufficient power to detect a relationship and provides a reasonably precise estimate of the strength of that relationship. Specifically, if you follow these guidelines:</p>
<ul>
<li>The power of the overall F-test ranges from about 0.8 to 0.9 for a moderately weak relationship (0.25). Stronger relationships yield higher power.</li>
<li>You can be 90% confident that the adjusted R-squared in your output is within +/- 20% of the true population R-squared value. Stronger relationships (~0.9) produce more precise estimates.</li>
</ul>
<p style="text-align: center;"><strong>Terms</strong></p>
<p style="text-align: center;"><strong>Total sample size</strong></p>
<p style="text-align: center;">1-3</p>
<p style="text-align: center;">40</p>
<p style="text-align: center;">4-6</p>
<p style="text-align: center;">45</p>
<p style="text-align: center;">7-8</p>
<p style="text-align: center;">50</p>
<p style="text-align: center;">9-11</p>
<p style="text-align: center;">55</p>
<p style="text-align: center;">12-14</p>
<p style="text-align: center;">60</p>
<p style="text-align: center;">15-18</p>
<p style="text-align: center;">65</p>
<p style="text-align: center;">19-21</p>
<p style="text-align: center;">70</p>
<p>In closing, if you want to estimate the strength of the relationship in the population, assess the adjusted R-squared and consider the precision of the estimate.</p>
<p>Even when you meet the sample size guidelines for regression, the adjusted R-squared is a rough estimate. If the adjusted R2 in your output is 60%, you can be 90% confident that the population value is between 40-80%.</p>
<p>If you're learning about regression, read my <a href="http://blog.minitab.com/blog/adventures-in-statistics/regression-analysis-tutorial-and-examples" target="_blank">regression tutorial</a>! For more histograms and the full guidelines table, see the <a href="http://support.minitab.com/en-us/minitab/17/Assistant_Simple_Regression.pdf" target="_blank">simple regression white paper</a> and <a href="http://support.minitab.com/en-us/minitab/17/Assistant_Multiple_Regression.pdf" target="_blank">multiple regression white paper</a>.</p>
Regression AnalysisThu, 30 Oct 2014 12:00:00 +0000http://blog.minitab.com/blog/adventures-in-statistics/r-squared-shrinkage-and-power-and-sample-size-guidelines-for-regression-analysisJim FrostWHO Cares about How Much Sugar You Eat on Halloween?
http://blog.minitab.com/blog/statistics-and-quality-improvement/who-cares-about-how-much-sugar-you-eat-on-halloween
<p>It’s almost Halloween, so there’s lots to do. If you haven’t picked out your costume, get ideas from the National Retail Federation’s list of <a href="https://nrf.com/media/press-releases/disneys-frozen-characters-teenage-mutant-ninja-turtles-top-childrens-costume" target="_blank">the most popular costumes</a> for 2014. Last-minute candy shopping? Check out kidzworld.com’s list of the <a href="http://www.kidzworld.com/article/27503-top-10-halloween-candy" target="_blank">top 10 candies</a> for Halloween. And of course, you have to plan your daily candy consumption to match the <a href="http://www.who.int/nutrition/sugars_public_consultation/en/" target="_blank">limits on free sugar</a> recommended by the World Health Organization (WHO) earlier this year.</p>
<p><img alt="Mixed candy" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/b89ff78e38c6875631f8d86c8eee308e/candy_w640.jpeg" style="line-height: 20.7999992370605px; float: right; width: 200px; height: 133px; border-width: 1px; border-style: solid; margin: 10px 15px;" /></p>
<p>What’s that you say? You didn’t plan your candy consumption yet? Well, the guideline says that no more than 10% of your calories should come from free sugars and that you can achieve increased health benefits by keeping the number below 5%. If you’re a good nutrition tracker, that should be no problem for you. For those of you looking for more general suggestions, we’re going to make a scatterplot in Minitab that should provide a helpful reference.</p>
<p>We like to show some fairly nifty graph features on the Minitab blog. For example, <a href="http://blog.minitab.com/blog/real-world-quality-improvement">Carly Barry</a>’s shown you how to <a href="http://blog.minitab.com/blog/real-world-quality-improvement/making-your-graphs-more-manageable">make your graphs more manageable with paneling</a>, <a href="http://blog.minitab.com/blog/adventures-in-statistics">Jim Frost</a>’s shown you how to <a href="http://blog.minitab.com/blog/adventures-in-statistics/world-travel-bumpy-roads-and-adjusting-your-graph-scales">adjust your scales</a> for travel bumps, and <a href="http://blog.minitab.com/blog/understanding-statistics">Eston Martz</a> adjusted <a href="http://blog.minitab.com/blog/understanding-statistics/studying-old-dogs-with-new-statistical-tricks-part-ii-contour-plots-and-cracking-bones">contour plots</a> while looking at data about hyena skulls. This time though, we’re going to see how our statistical software makes it easy to clarify a graph by taking something away.</p>
<p>The USDA last published their <a href="http://www.cnpp.usda.gov/sites/default/files/dietary_guidelines_for_americans/PolicyDoc.pdf">dietary guidelines</a> in 2010. Appendix 6 contains calorie estimates based on age, gender and activity level, rounded to the nearest 200 calories. Multiply those levels by 0.05 to get an estimate of your recommended sugar limit in calories. To change that into grams that you can find on candy labels, we’ll assume that sugar has 4 calories per gram.</p>
<p>Now, if we create the default graph in Minitab we get something a bit like this. Note the symbols crammed together along each line:</p>
<p><img alt="Crowded symbols make the graph less clear." src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/fe2bb52a63302df85b8b685d22241f54/image1.jpg" style="border-width: 0px; border-style: solid; width: 450px; height: 324px;" /></p>
<p>Let’s be honest, pushing all those symbols together to show a line with no variation looks a bit silly. But select those symbols and a clearer graph is only a right-click away:</p>
<p><img alt="Right-click the symbols and click Delete to make the graph clearer." src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/2271a15debb7aeb4389609b6af61f082/grayend.gif" style="border-width: 0px; border-style: solid; width: 450px; height: 321px;" /></p>
<p>Without the symbols on the graph, the lines and the differences between them are clearer, especially when the lines are closest together during the early phase when people grow rapidly.</p>
<p><img alt="Without the symbols, the graph is clearer." src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/9ba188145759ce1c2720e0e2004b6059/image5.jpg" style="width: 450px; height: 288px;" /></p>
<p>Much has been made of the fact that the 5% WHO guideline is less than the sugar in a can of soda, so Halloween can be a treacherous time for someone who wants to limit their sugar intake. After all, <a href="http://www.popsci.com/article/science/how-new-sugar-stats-will-kill-halloween-save-you" target="_blank">Popular Science</a> reports that the average trick-or-treater begins home over 600 grams. So what do you do if your ghost or goblin brings home more candy than you want? <a href="http://mommypoppins.com/halloween-candy-donation-candy-buy-back-donating-treats-operation-gratitude" target="_blank">Natalie Silverstein</a> offers some suggestions about how to make your candy do some good for others.</p>
<p> </p>
<p style="font-size:8px">The image of mixed candy is by <a href="https://www.flickr.com/photos/stevendepolo/" target="_blank">Steven Depolo</a> and appears under this <a href="https://creativecommons.org/licenses/by/2.0/">Creative Commons</a> license.</p>
Fun StatisticsStatistics in the NewsWed, 29 Oct 2014 12:21:15 +0000http://blog.minitab.com/blog/statistics-and-quality-improvement/who-cares-about-how-much-sugar-you-eat-on-halloweenCody SteeleSimulating Robust Processing with Design of Experiments, part 2
http://blog.minitab.com/blog/statistics-in-the-field/simulating-robust-processing-with-design-of-experiments2c-part-2
<p>by Jasmin Wong, guest blogger</p>
<p> </p>
<p><em><a href="http://blog.minitab.com/blog/statistics-in-the-field/simulating-robust-processing2c-part-1">Part 1</a> of this two-part blog post discusses the issues and challenges in injection moulding and suggests using simulation software and the statistical method called Design of Experiments (DOE) to speed development and boost quality. This part presents a case study that illustrates this approach. </em></p>
Preliminary Fill and Designed Experiment
<p>This case study considers the example of a hand dispensing pump for a sanitiser bottle where the main areas of concern were warpage and the concentricity of the tube, as this had a critical impact on fit and functionality. </p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/f6c68e56710c222c2a20dd002021287f/dispenser_top.png" style="line-height: 20.7999992370605px; margin: 10px 15px; float: right; width: 400px; height: 236px;" /></p>
<div>
<p>In this example, the first step was to carry out a preliminary fill, pack, cool and warp analysis to ensure that the part had no filling difficulties such as short shots or hesitation. DOE was then carried out and, since the areas of concern were warpage and concentricity, these were selected as the quality factor/responses.</p>
<div>
<p>Four control factors that affected warpage and concentricity were used to carry out the DOE: melt temperature, packing pressure, cooling time, and fill time. The factors levels are shown in the table below:</p>
<p><img alt="Taguchi DOE control factors" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/322b2d00c3b22d962ca76ac0485e437b/taguchi_doe_control_factors.png" style="width: 450px; height: 136px;" /></p>
<p>A Taguchi L9 DOE was then created using Minitab Statistical Software. <span style="line-height: 1.6;">It should be noted that a Taguchi DOE assumes no significant interaction between factors, but this may not necessarily be true. In this case, however, it was selected to determine the relationship between the factors and responses in the shortest simulation time.</span></p>
<p>The Minitab worksheet below shows the process settings for the nine runs using the Taguchi L9 Design.</p>
<p><img alt="Taguchi design worksheet" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/7cbc350e2fbe466708f4b5b4a2f58566/taguchi_doe_worksheet.png" style="width: 450px; height: 169px;" /></p>
<p>Moldex3D DOE was then used to perform the mathematical calculations based on the user’s specification (minimum warpage and linear shrinkage between nodes) to determine the optimum process setting.</p>
<p>From the nine different simulated runs, a main effect graph for warpage was plotted. </p>
<p><img alt="Main Effects Plor for Warpage" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/dbec7e75117c7763745e8260d78852fd/main_effects_warpage.png" style="width: 577px; height: 385px;" /></p>
<p><span style="line-height: 1.6;">From this, it could be seen that by increasing the packing pressure and cooling time, warpage was reduced. Increasing melt temperature, on the other hand, lead to higher warpage. Using a filling time of 0.2s or 0.3s seemed to give slightly lesser warpage than 0.1s. Hence, it was determined that to achieve lower warpage, the optimum process setting should be a melt temperature of 225°C, packing pressure of 15MPa, cooling time of 12s and filling time of 0.3s.</span></p>
<p style="line-height: 20.7999992370605px;">Taking the results obtained from Moldex3D, Minitab 17 statistical software was used to determine which of the four factors had the biggest influence on part warpage.</p>
<p style="line-height: 20.7999992370605px;"><img alt="response table for warpage" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/20e65680dd317de7add7a8559b1d50e3/response_table_warpage.png" style="width: 500px; height: 153px;" /></p>
<p style="line-height: 20.7999992370605px;">This data analysis showed that cool time had the biggest impact on part warpage, followed by packing pressure, melt temperature and then filling time. An area graph of warpage (PDF DOWNLOAD CHART 1) showed a quick comparison of the nine different runs, indicating that run 3 gave the least warpage.</p>
<p><img alt="area graph of warpage" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/740d75c1b4424da02ee136a673e43780/area_graph_of_warpage.png" style="width: 500px; height: 333px;" /></p>
<p>Concentricity is difficult to measure, in both real life and in simulation. In real life, the distance between different points is measured using a coordinate-measuring machine (CMM). In the Moldex3D simulation, the linear shrinkage between different nodes was measured. Eight different nodes were identified. The linear shrinkage of the diameter of the tube across was determined and the lower the linear shrinkage, the more circular or better concentricity of the part.</p>
<p>The main effects plot below for shrinkage shows that to get better concentricity/linear shrinkage between the nodes, a lower melt temperature, cooling time and filling time with a high pack pressure was preferable.</p>
<p><img alt="Main Effects Plot for Shrinkage" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/3eb9b51b4bd8caeac5ead713a86ce90b/main_effects_shrinkage.png" style="width: 579px; height: 385px;" /></p>
<p>It had already been established that to achieve lower linear shrinkage, the optimum process setting should be melt temperature of 225°C, packing pressure of 15MPa, cooling time of 8s and filling time of 0.1s. However, a cooling time of 8s may not be practical, as the analysis of warpage shows it would give high warpage.</p>
<p>Minitab was also used to find out which of the four control factors resulted in the greatest impact on linear shrinkage.</p>
<p><img alt="Response Table for Shrinkage" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/9e0e2aca3064320d44a9860223665f48/response_table_shrinkage.png" style="width: 500px; height: 153px;" /></p>
<p>This showed that pack pressure is ranked first, followed by cooling time, melt temperature and lastly the filling time. Since the 8s cooling time would lead to high warpage, a compromise had to be made.</p>
<p>As mentioned earlier, for linear shrinkage the packing pressure was more of a contributing factor than the cooling time, so it makes sense to use 12s cooling time with 15MPa packing pressure. Comparing the nine different runs for linear shrinkage in an area graph showed that run six gave the lowest linear shrinkage.</p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/dfabcb5cb7861c6dc11cc0fdb25c2b2d/area_graph_of_shrinkage.png" style="width: 500px; height: 333px;" /></p>
<p>Based on the user specification, Moldex3D’s mathematical calculations obtained the optimised run<span style="line-height: 1.6;">. For this example, weighting for warpage was the same as for linear shrinkage. However, based on the DOE simulation results obtained, the optimum process setting for the lowest warpage was to have a cooling time of 12s and filling time of 0.3s. The optimum process for the lowest linear shrinkage, on the other hand, required a cooling time of 8s and fill time of 0.1s.</span></p>
Concluding thoughts
<p>Moldex3D simulation resulted in a compromise process setting (melt temperature of 225°C, packing pressure of 15MPa, cooling time of 12s and filling time of 0.1s), which was used as the optimum run. From the area graphs shown below, it can be seen that the optimised run 10 gives the lowest warpage compared to the other nine runs, while having low linear shrinkage.</p>
<p><img alt="optimized run - area chart" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/13c7a74c8d37f74f4acc152b676e53b6/optimized_run_area_graph_w640.png" style="width: 640px; height: 210px;" /></p>
<p>From the simulation in Moldex 3D, shown below, it can be seen that part warpage and concentricity of the tube has been significantly improved (warpage has been improved by 20-30% while linear shrinkage has been kept to 0.6-0.7%).</p>
<p><img alt="Moldex 3D simulation" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/a1b9270c0e645e9db3d7c4f626308aba/moldex_3d_sim.png" style="width: 500px; height: 179px;" /></p>
<p>It is important that designers and moulders understand that numerical results in a simulation such as this provide only a relative comparison and should not be treated as absolute. This is because there are various uncontrollable factors in the actual mould shop environment—‘noise’—which cannot be re-enacted in a simulation. However, running DOE using simulation can give the engineering team a head start on identifying which control factors to focus on and the relationship those factors have with part quality.</p>
<p> </p>
<p><strong>About the guest blogger</strong></p>
<p>Jasmin Wong is project engineer at UK-based <a href="http://www.plazology.co.uk/" target="_blank">Plazology</a>, which provides product design optimisation, injection moulding fl ow simulation, mould design, mould procurement, and moulding process validation services to global manufacturing customers. She is an MSc graduate in polymer composite science and engineering and recently gained Moldex3D Analyst Certification.</p>
<p> </p>
<p> </p>
<p><em>A version of this article originally appeared in the <a href="http://content.yudu.com/htmlReader/A3572w/IWOct14/reader.html?page=26" target="_blank">October 2012 issue of Injection World</a> magazine.</em></p>
</div>
</div>
Design of ExperimentsMon, 27 Oct 2014 12:00:00 +0000http://blog.minitab.com/blog/statistics-in-the-field/simulating-robust-processing-with-design-of-experiments2c-part-2Guest BloggerSimulating Robust Processing with Design of Experiments, part 1
http://blog.minitab.com/blog/statistics-in-the-field/simulating-robust-processing-with-design-of-experiments2c-part-1
<p>by Jasmin Wong, guest blogger</p>
<p><em>The combination of statistical methods and injection moulding simulation software gives manufacturers a powerful way to predict moulding defects and to develop a robust moulding process at the part design phase. </em></p>
<p>CAE (computer-aided engineering) is widely used in the injection moulding industry today to improve product and mould designs as well as to resolve or troubleshoot engineering problems. But CAE can also be used to carry out in-depth processing simulations, allowing the critical process parameters that influence part quality to be identified and to enable determination of an appropriate and achievable process window at the earliest stage of the development process.</p>
<img alt="injection-molded dispenser pump" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/f6c68e56710c222c2a20dd002021287f/dispenser_top.png" style="width: 400px; height: 236px;" />
<p style="text-align: center;">Warpage and tube concentricity were the key<br />
quality criteria in this mold-injected hand dispenser pump.</p>
<p>In order to produce good quality injection mouldings with high consistency, a well-designed part and mould is critical, along with the selection of the right material and processing parameters. Changes to any of these four factors can have a significant effect on the moulded part.</p>
<p>With regard to defining process parameters, the injection moulding industry has been dependent on experienced process engineers using trial-and-error methods. Without the insight into polymer behaviour inside the mould, more often than not engineers would ‘process the part dimensions in.’ Such an approach typically leads to a narrow process window, where just a slight change in processing conditions can cause part dimensions to fall outside of the specification limit. This trial-and-error method is also laborious, expensive, and frequently ineffective, making it unsuitable for today’s fast-moving plastics processing industry.</p>
<p>Plastic injection moulding simulation software such as Moldex 3D from CoreTech System can help in the validation and optimisation of the part and/or mould design by identifying potential moulding defects before the tool is manufactured. The software can reduce the need for expensive prototypes, minimise the cost of tooling (since less rework needs to be done), and shorten validation time. When combined with the Design of Experiments techniques available in <a href="http://www.minitab.com/products/minitab">statistical software such as Minitab</a>, doing simulation <span style="line-height: 20.7999992370605px;">ahead of real world mould trials </span><span style="line-height: 1.6;">can also be used to speed mould approval. </span></p>
The Design of Experiments (DOE) Approach
<p><a href="http://blog.minitab.com/blog/real-world-quality-improvement/leveraging-designed-experiments-doe-for-success">Design of Experiments, or DOE</a>, involves performing a series of carefully planned, systematic tests while controlling the inputs and monitoring the outputs. In the context of injection moulding, the process parameters are usually referred to as the <em>factors </em>or <em>inputs</em>, while the customer requirements (part quality/dimensions or other part specifications) are referred to as <em>responses </em>or <em>outputs</em>. By analysing the results from these tests, moulders can characterise, optimise and/or troubleshoot the injection moulding process effectively and efficiently.</p>
<p>By applying DOE in an injection moulding simulation, designers and/or moulders can study the relationship between the moulding factors (inputs) and response (outputs) prior to the actual trial on the mould floor. This means that they can have a good understanding of which factors will affect the quality or certain part specifications as early as possible in the development process. Optimal moulding process conditions for the specific part design can then be identified so the focus can be directed to the conditions that have the biggest influence on the customer’s requirements. This can save time and increase productivity.</p>
When Should Simulation Be Performed?
<p>Ideally, CAE simulation should be carried out before the actual mould trial so potential mould defects—such as sink marks, weld lines, short shots, etc.—can be predicted and rectified in the original mould design.</p>
<p>The most challenging problem is often warpage. Due to temperature variations and differences in volumetric shrinkage, it is almost impossible to get a part which is exactly the same as the CAD model. It is, therefore, important to conduct a DOE to understand the impact certain processing parameters have and to define the <a href="http://blog.minitab.com/blog/statistics-in-the-field/optimizing-attribute-responses-using-design-of-experiments-doe-part-1">optimum processing settings</a>.</p>
<p>Before the DOE is conducted, however, it is important to carry out a preliminary simulation to understand the root cause of mould defects. Changes to the part are sometimes inevitable to avoid having too narrow a process window to work within. If the fill pattern is not balanced, for example, there is a high possibility of warpage occurring regardless of the process parameters.</p>
<p> </p>
<p><strong>The second half of this two-part post includes a detailed case study illustrating how moulding simulation software and design of experiments can be combined to speed part design and approval. </strong></p>
<p> </p>
<p><strong>About the guest blogger</strong></p>
<p>Jasmin Wong is project engineer at UK-based <a href="http://www.plazology.co.uk/" target="_blank">Plazology</a>, which provides product design optimisation, injection moulding flow simulation, mould design, mould procurement, and moulding process validation services to global manufacturing customers. She is an MSc graduate in polymer composite science and engineering and recently gained Moldex3D Analyst Certification.</p>
<p> </p>
<div> </div>
<div><em>A version of this article originally appeared in the <a href="http://content.yudu.com/htmlReader/A3572w/IWOct14/reader.html?page=26" style="line-height: 20.7999992370605px;" target="_blank">October 2012 issue of Injection World</a> magazine.</em></div>
Design of ExperimentsFri, 24 Oct 2014 17:22:22 +0000http://blog.minitab.com/blog/statistics-in-the-field/simulating-robust-processing-with-design-of-experiments2c-part-1Guest BloggerCan Regression and Statistical Software Help You Find a Great Deal on a Used Car?
http://blog.minitab.com/blog/understanding-statistics/can-regression-and-statistical-software-help-you-find-a-great-deal-on-a-used-car
<p>You need to consider many factors when you’re buying a used car. Once you narrow your choice down to a particular car model, you can get a wealth of information about individual cars on the market through the Internet. How do you navigate through it all to find the best deal? By analyzing the data you have available. </p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/710ce579b4120727bf67e8b48f5965e8/240_used_car_kovacs.jpg" style="line-height: 20.7999992370605px; border-width: 1px; border-style: solid; margin: 10px 15px; float: right; width: 240px; height: 240px;" /></p>
<p>Let's look at how this works using <a href="http://blog.minitab.com/blog/understanding-statistics/we-just-got-rid-of-five-reasons-to-fear-data-analysis">the Assistant</a> in Minitab 17. With the Assistant, you can use regression analysis to calculate the expected price of a vehicle based on variables such as year, mileage, whether or not the technology package is included, and whether or not a free Carfax report is included.</p>
<p>And it's probably a lot easier than you think. </p>
<p>A search of a leading Internet auto sales site yielded data about 988 vehicles of a specific make and model. After putting the data into Minitab, we choose <strong>Assistant > Regression…</strong></p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/9e87de993a0daa39e6643b8c6d3aed9c/regression_dialog.png" style="width: 395px; height: 247px;" /></p>
<p>At this point, if you aren’t very comfortable with regression, <a href="http://www.minitab.com/products/minitab/assistant/">the Assistant makes it easy to select the right option for your analysis</a>.</p>
A Decision Tree for Selecting the Right Analysis
<p>We want to explore the relationships between the price of the vehicle and four factors, or X variables. Since we have more than one X variable, and since we're not looking to optimize a response, we want to choose Multiple Regression.</p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/bc802d35bfb57ca3b86e061da4fa4b09/regression_decision_tree_w640.png" style="width: 640px; height: 502px;" /></p>
<p>This <a href="//cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/File/9ecb2280228deb621ee2db7f6fbe300e/used_cars.MTW">data set</a> includes five columns: mileage, the age of the car in years, whether or not it has a technology package, whether or not it includes a free CARFAX report, and, finally, the price of the car.</p>
<p>We don’t know which of these factors may have significant relationship to the cost of the vehicle, and we don’t know whether there are significant two-way interactions between them, or if there are quadratic (nonlinear) terms we should include—but we don’t need to. Just fill out the dialog box as shown. </p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/b93a0a755e8e73dc7f681ea4b1965749/regression_dialog_box.png" style="width: 532px; height: 382px;" /></p>
<p>Press OK and the Assistant assesses each potential model and selects the best-fitting one. It also provides a comprehensive set of reports, including a Model Building Report that details how the final model was selected and a Report Card that notifies you to potential problems with the analysis, if there are any.</p>
Interpreting Regression Results in Plain Language
<p>The Summary Report tells us in plain language that there is a significant relationship between the Y and X variables in this analysis, and that the factors in the final model explain 91 percent of the observed variation in price. It confirms that all of the variables we looked at are significant, and that there are significant interactions between them. </p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/746574a27bba821ffab4f77ae1a2931b/multiple_regression_summary_report_w640.png" style="width: 640px; height: 480px;" /></p>
<p>The Model Equations Report contains the final regression models, which can be used to predict the price of a used vehicle. The Assistant provides 2 equations, one for vehicles that include a free CARFAX report, and one for vehicles that do not.</p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/58598060212558634d62d75a7045bf0b/regression_equation_w640.png" style="width: 640px; height: 186px;" /></p>
<p>We can tell several interesting things about the price of this vehicle model by reading the equations. First, the average cost for vehicles with a free CARFAX report is about $200 more than the average for vehicles with a paid report ($30,546 vs. $30,354). This could be because these cars probably have a clean report (if not, the sellers probably wouldn’t provide it for free).</p>
<p>Second, each additional mile added to the car decreases its expected price by roughly 8 cents, while each year added to the cars age decreases the expected price by $2,357.</p>
<p>The technology package adds, on average, $1,105 to the price of vehicles that have a free CARFAX report, but the package adds $2,774 to vehicles with a paid CARFAX report. Perhaps the sellers of these vehicles hope to use the appeal of the technology package to compensate for some other influence on the asking price. </p>
Residuals versus Fitted Values
<p>While these findings are interesting, our goal is to find the car that offers the best value. In other words, we want to find the car that has the largest difference between the asking price and the expected asking price predicted by the regression analysis.</p>
<p>For that, we can look at the Assistant’s Diagnostic Report. The report presents a chart of Residuals vs. Fitted Values. If we see obvious patterns in this chart, it can indicate problems with the analysis. In that respect, this chart of Residuals vs. Fitted Values looks fine, but now we’re going to use the chart to identify the best value on the market.</p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/d55ae8720ba281bf37135b68b2069434/multiple_regression_diagnostic_report_w640.png" style="width: 640px; height: 480px;" /></p>
<p>In this analysis, the “Fitted Values” are the prices predicted by the regression model. “Residuals” are what you get when you subtract the actual asking price from the predicted asking price—exactly the information you’re looking for! The Assistant marks large residuals in red, making them very easy to find. And three of those residuals—which appear in light blue above because we’ve selected them—appear to be very far below the asking price predicted by the regression analysis.</p>
<p>Selecting these data points on the graph reveals that these are vehicles whose data appears in rows 357, 359, and 934 of the data sheet. Now we can revisit those vehicles online to see if one of them is the right vehicle to purchase, or if there’s something undesirable that explains the low asking price. </p>
<p>Sure enough, the records for those vehicles reveal that two of them have severe collision damage.</p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/5dbbf5aa405d4b2d53ec720657a09556/vehicles.jpg" style="width: 320px; height: 356px;" /></p>
<p>But the remaining vehicle appears to be in pristine condition, and is several thousand dollars less than the price you’d expect to pay, based on this analysis!</p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/640bd720a3d1f8b04713aa0ec321a570/nice_car.png" style="width: 254px; height: 189px;" /></p>
<p>With the power of regression analysis and the Assistant, we’ve found a great used car—at a price you know is a real bargain.</p>
<p> </p>
Fun StatisticsRegression AnalysisStatisticsStatistics HelpWed, 22 Oct 2014 12:00:00 +0000http://blog.minitab.com/blog/understanding-statistics/can-regression-and-statistical-software-help-you-find-a-great-deal-on-a-used-carEston MartzDoes the Impact of Your Quality Initiative Reach C-Level Executives?
http://blog.minitab.com/blog/understanding-statistics/does-the-impact-of-your-quality-initiative-reach-c-level-executives
<p>Here's a shocking finding from the most recent <a href="http://asq.org/global-state-of-quality/" target="_blank"><em>ASQ Global State of Quality</em></a> report: The higher you rise in your organization's leadership, the less often you receive reports about quality metrics. Only 2% of senior executives get daily quality reports, compared to 33% of front-line staff members. </p>
<p>A quarter of the senior executives reported getting quality metrics on a monthly basis, at least. But just as many reported getting them <em>only on an annual basis</em>. </p>
<p><img alt="reporting on quality initiatives is difficult" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/6424af008c86077421fd08bd84732739/reporting.jpg" style="margin: 10px 15px; width: 202px; height: 202px; float: right;" />This is simultaneously scary and depressing. It's scary because it indicates that company leaders don't have good access to the kind of information they need about their quality improvement initiatives. More than half of the executives are getting updates about quality only once a quarter, or even less. You can bet they're making decisions that impact quality much more frequently than that. </p>
<p>It's depressing because quality practitioners are a dedicated, hard-working lot, and their task is both challenging and often thankless. Their efforts don't appear to be reaching the C-level offices as often as they deserve. </p>
<p><span style="line-height: 20.7999992370605px;">Why <em>do</em> so many leaders get so few reports about their quality programs?</span></p>
Factors that Complicate Reporting on Quality Programs
<p>In fairness to everyone involved, from the practitioner to the executive, piecing together the full picture of quality in a company is daunting. Practitioners tell us that even in organizations with robust, mature quality programs, assessing the cumulative impact of an initiative can be difficult, and sometimes impossible. </p>
<p>The challenges start with the individual project. Teams are very good at capturing and reporting their results, but a large company may have thousands of simultaneous quality projects. Just gathering the critical information from all of those projects and putting it into a form leaders can use is a monumental task. </p>
<p>But there are other obstacles, too. </p>
<ul>
<li>Teams typically use an array of different applications to create charters, process maps, <a href="http://blog.minitab.com/blog/understanding-statistics/four-more-tips-for-making-the-most-of-value-stream-maps">value stream maps</a>, and other documents. So the project record is a hodgepodge of files for different applications. And since the latest versions of some documents may reside on several different computers, project leaders often need to track multiple versions of a document to keep the official project record current. <br />
</li>
<li>Results and metrics aren’t always measured the same way from one team's project to another. If one team measures apples and the next team measures oranges, their results can't be evaluated or aggregated as if they were equivalent. <br />
</li>
<li>Many organizations have tried quality tracking methods ranging from homegrown project databases to full-featured project portfolio management (PPM) systems. But homegrown systems often become a burden to maintain, while off-the-shelf PPM solutions created for IT or other business functions don’t effectively support projects involving methods like Lean and Six Sigma. <br />
</li>
<li>Reporting on projects can be a burden. <span style="line-height: 1.6;">There are only so many hours in the day, and busy team members need to prioritize. Copying and pasting information from project documents into an external system seems like non-value-added time, so it's easy to see why putting the latest information into the system gets low priority—if it happens at all.</span></li>
</ul>
Reporting on Quality Shouldn't Be So Difficult
<p>Given the complexity of the task, and the systemic and human factors involved in improving quality, it's not hard to see why many organizations struggle with knowing how well their initiatives are doing. </p>
<p>But for quality practitioners and leaders, the challenge is to make sure that reporting on results becomes a critical step in every individual project, and that all projects are using consistent metrics. Teams that can do that will find their results getting more attention and more credit for how they affect the bottom line. </p>
<p>This finding in the ASQ report caught my attention because it so dramatically underscores problems we at Minitab have been focusing on recently—in fact, last year we released Qeystone, a <a href="http://www.minitab.com/products/qeystone">product portfolio management system for lean six sigma</a>, to address many of these factors. </p>
<p>Regardless of the tools they use, this issue—how to ensure the results of quality improvement initiatives are understood throughout an organization—is one that every practitioner is likely to grapple with in their career. </p>
<p>How do <em>you </em>make sure the results of your work reach your organization's decision-makers? </p>
<p> </p>
Lean Six SigmaProject ToolsQuality ImprovementMon, 20 Oct 2014 12:00:00 +0000http://blog.minitab.com/blog/understanding-statistics/does-the-impact-of-your-quality-initiative-reach-c-level-executivesEston MartzUsing Data Analysis to Maximize Webinar Attendance
http://blog.minitab.com/blog/michelle-paret/using-data-analysis-to-maximize-webinar-attendance
<p>We like to host webinars, and our customers and prospects like to attend them. But when our webinar vendor moved from a pay-per-person pricing model to a pay-per-webinar pricing model, we wanted to find out how to maximize registrations and thereby minimize our costs.<img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/6060c2db-f5d9-449b-abe2-68eade74814a/Image/8a6733d3b0516b7f1c7ad80ea753d430/mtbnewspromos_w640.jpeg" style="width: 400px; height: 273px; float: right; border-width: 1px; border-style: solid; margin: 10px 15px;" /></p>
<p>We collected webinar data on the following variables:</p>
<ul>
<li>Webinar topic</li>
<li>Day of week</li>
<li>Time of day – 11 a.m. or 2 p.m.</li>
<li>Newsletter promotion – no promotion, newsletter article, newsletter sidebar</li>
<li>Number of registrants</li>
<li>Number of attendees</li>
</ul>
<p>Once we'd collected our data, it was time to analyze it and answer some key questions using <a href="http://www.minitab.com/products/minitab/">Minitab Statistical Software</a>.</p>
Should we use registrant or attendee counts for the analysis?
<strong><span style="line-height: 16.8666667938232px; font-family: Calibri, sans-serif; font-size: 11pt;"><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/6060c2db-f5d9-449b-abe2-68eade74814a/Image/4d9fa1e3c73606627d2ca1ec34b620e2/scatterplot_w640.jpeg" style="width: 300px; height: 197px; margin: 10px 15px; float: left;" /></span></strong>
<p>First we needed to decide what we would use to measure our results: the number of people who signed up, or the number of people who actually attended the webinar. This question really boils down to answering the question, “Can I trust my data?”</p>
<p>Our data collection system for webinar registrants is much more accurate than our data collection system for webinar attendees. This is due to customer behavior and their willingness to share contact information, in addition to the automated database processes that connect our webinar vendor data with our own database. So, for a period of time, I manually collected the attendee data directly from our webinar vendor to see how it correlated with the easily-accessible and accurate registration data. The scatterplot above shows the results.</p>
<p>With a <a href="http://blog.minitab.com/blog/understanding-statistics/no-matter-how-strong-correlation-still-doesnt-imply-causation">correlation coefficient </a>of 0.929 and a p-value of 0.000, there was a strong positive linear relationship between the registrations and attendee counts. If registrations are high, then attendance is also high. If registrations are low, then attendance is also low. I concluded that I could use the registration data—which is both easily accessible and extremely reliable—to conduct my analysis.</p>
Should we consider data for the last 6 years?
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/6060c2db-f5d9-449b-abe2-68eade74814a/Image/5e73f48b852c7afc17762f28bf8887cf/i_mr_chart_of_registrants_w640.jpeg" style="width: 400px; height: 263px; margin: 10px 15px; float: left;" />We’ve been collecting webinar data for 6 years, but that doesn’t mean we can treat the last 6 years of data as one homogeneous population.</p>
<p>A lot can change in a 6-year time period. Perhaps there was a change in the webinar process that affected registrations. To determine whether or not I should use all of the data, I used an Individuals and Moving Range (I-MR, also referred to as X-MR) <a href="http://blog.minitab.com/blog/understanding-statistics/how-create-and-read-an-i-mr-control-chart">control chart</a> to evaluate the process stability of webinar registrations over time.</p>
<p>The graph revealed a single point on the MR chart that flagged as out-of-control. I looked more closely at this point and verified that the data was accurate and that this webinar belonged with the larger population. Based on this information, I decided to proceed with analyzing all 6 years of data together. (Note there is some clustering of points due to promotions, but again the goal here was to determine if we could use data over a 6-year time period.)</p>
What variables impact registrations?
<p>I performed an ANOVA using Minitab's General Linear Model tool to find out which factors—topic, day of week, time of day, or newsletter promotion—significantly affect webinar registrations.<img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/6060c2db-f5d9-449b-abe2-68eade74814a/Image/3758d3d03a604bab9921ad9f94663dc8/main_effects_plot_for_registrants_w640.jpeg" style="width: 400px; height: 263px; float: right; margin: 10px 15px;" /></p>
<p>The ANOVA results revealed that the day of week, time of day, and webinar topic <em>do not</em> affect webinar registrations, but the newsletter promotion type <em>does</em> (p-value = 0.000).</p>
<p>So which webinar promotion type maximizes webinar registrations?</p>
<p>Using Minitab to conduct <a href="http://blog.minitab.com/blog/statistics-and-quality-data-analysis/keep-that-special-someone-happy-when-you-perform-multiple-comparisons">Tukey comparisons</a>, we can see that registrations for webinars promoted in the newsletter sidebar space were not significantly different from webinars that weren't promoted at all.</p>
<p>However, webinars that were promoted in the newsletter <em>article </em>space resulted in significantly more registrations than both the sidebar promotions and no promotions.</p>
<p>From this analysis, we concluded that we still had the flexibility to offer webinars at various times and days of the week, and we could continue to vary webinar topics based on customer demand and other factors. To maximize webinar attendance and minimize webinar cost, we needed to focus our efforts on promoting the webinars in our newsletter, utilizing the article space.</p>
<p>But over the past year, we’ve started to actively promote our webinars via other channels as well, so next up is some more data analysis—using Minitab—to figure out what marketing channels provide the best results…</p>
Data AnalysisHypothesis TestingRegression AnalysisStatisticsFri, 17 Oct 2014 12:00:00 +0000http://blog.minitab.com/blog/michelle-paret/using-data-analysis-to-maximize-webinar-attendanceMichelle ParetHow Important Are Normal Residuals in Regression Analysis?
http://blog.minitab.com/blog/adventures-in-statistics/how-important-are-normal-residuals-in-regression-analysis
<p>I’ve written about the importance of <a href="http://blog.minitab.com/blog/adventures-in-statistics/why-you-need-to-check-your-residual-plots-for-regression-analysis" target="_blank">checking your residual plots</a> when performing linear regression analysis. If you don’t satisfy the assumptions for an analysis, you might not be able to trust the results. One of the assumptions for regression analysis is that the residuals are normally distributed. Typically, you assess this assumption using the normal probability plot of the residuals.</p>
<div style="float: right; width: 250px; margin: 15px 0px 15px 15px;"><img alt="Normal Probability Plot showing residuals that are not distributed normally" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/d84cbe3e157257e1ba07563dacdacbd7/nonnormal_residuals.png" title="Are these nonnormal residuals bad?" width="250" /> <em>Are these nonnormal residuals a problem?</em></div>
<p>If you have nonnormal residuals, can you trust the results of the regression analysis?</p>
<p>Answering this question highlights some of the research that Rob Kelly, a senior statistician here at Minitab, was tasked with in order to guide the development of our <a href="http://www.minitab.com/en-us/products/minitab/" target="_blank">statistical software</a>.</p>
Simulation Study Details
<p>The goals of the simulation study were to:</p>
<ul>
<li>determine whether nonnormal residuals affect the error rate of the F-tests for regression analysis</li>
<li>generate a safe, minimum sample size recommendation for nonnormal residuals</li>
</ul>
<p>For simple regression, the study assessed both the overall F-test (for both linear and quadratic models) and the F-test specifically for the highest-order term.</p>
<p>For multiple regression, the study assessed the overall F-test for three models that involved five continuous predictors:</p>
<ul>
<li>a linear model with all five X variables</li>
<li>all linear and square terms</li>
<li>all linear terms and seven of the 2-way interactions</li>
</ul>
<p>The residual distributions included skewed, heavy-tailed, and light-tailed distributions that depart substantially from the normal distribution.</p>
<p>There were 10,000 tests for each condition. The study determined whether the tests incorrectly rejected the null hypothesis more often or less often than expected for the different nonnormal distributions. If the test performs well, the Type I error rates should be very close to the target significance level.</p>
Results and Sample Size Guideline
<p>The study found that a sample size of at least 15 was important for both simple and multiple regression. If you meet this guideline, the test results are usually reliable for any of the nonnormal distributions.</p>
<p>In simple regression, the observed Type I error rates are all between 0.0380 and 0.0529, very close to the target significance level of 0.05.</p>
<p>In multiple regression, the Type I error rates are all between 0.08820 and 0.11850, close to the target of 0.10.</p>
Closing Thoughts
<p>The good news is that if you have at least 15 samples, the test results are reliable even when the residuals depart substantially from the normal distribution.</p>
<p>However, there is a caveat if you are using regression analysis to generate predictions. <a href="http://blog.minitab.com/blog/adventures-in-statistics/when-should-i-use-confidence-intervals-prediction-intervals-and-tolerance-intervals" target="_blank">Prediction intervals</a> are calculated based on the assumption that the residuals are normally distributed. If the residuals are nonnormal, the prediction intervals may be inaccurate.</p>
<p>This research guided the implementation of regression features in the <a href="http://www.minitab.com/en-us/products/minitab/assistant/" target="_blank">Assistant menu</a>. The Assistant is your interactive guide to choosing the right tool, analyzing data correctly, and interpreting the results. Because the regression tests perform well with relatively small samples, the Assistant does not test the residuals for normality. Instead, the Assistant checks the size of the sample and indicates when the sample is less than 15.</p>
<p><a href="http://blog.minitab.com/blog/adventures-in-statistics/multiple-regression-analysis-and-response-optimization-examples-using-the-assistant-in-minitab-17" target="_blank">See a multiple regression example that uses the Assistant.</a></p>
<p>You can read the full study results in the <a href="http://support.minitab.com/en-us/minitab/17/Assistant_Simple_Regression.pdf" target="_blank">simple regression white paper</a> and the <a href="http://support.minitab.com/en-us/minitab/17/Assistant_Multiple_Regression.pdf" target="_blank">multiple regression white paper</a>. You can also peruse all of our <a href="http://support.minitab.com/en-us/minitab/17/technical-papers/" target="_blank">technical white papers</a> to see the research we conduct to develop methodology throughout the Assistant and Minitab.</p>
Regression AnalysisThu, 16 Oct 2014 12:00:00 +0000http://blog.minitab.com/blog/adventures-in-statistics/how-important-are-normal-residuals-in-regression-analysisJim FrostWith the Assistant, You Won't Have to Stop and Get Directions about Directional Hypotheses
http://blog.minitab.com/blog/statistics-and-quality-improvement/with-the-assistant-you-wont-have-to-stop-and-get-directions-about-directional-hypotheses
<p>I got lost a lot as a child. I got lost at malls, at museums, Christmas markets, and everywhere else you could think of. Had it been in fashion to tether children to their parents at the time, I'm sure my mother would have. As an adult, I've gotten used to using a GPS device to keep me from getting lost.</p>
<p><span style="line-height: 20.7999992370605px;">The Assistant in Minitab is like your GPS for statistics. The Assistant is there to provide you with directions so that you don't get lost. One particular area where it's easy to get lost is with directional hypotheses.</span><img alt="Wait... is my hypothesis the other direction?" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/25dd42362071d2aafc3bfc85f78f5f22/hypothesis_bubble_w640.jpeg" style="line-height: 20.7999992370605px; width: 480px; height: 350px; border-width: 1px; border-style: solid; margin: 10px 15px;" /></p>
What Is a Directional Hypothesis?
<p>When you do a <a href="http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and-graphs/hypothesis-tests/basics/what-is-a-hypothesis-test/">statistical hypothesis test</a>, you have a null hypothesis and an alternative hypothesis. Directional hypotheses refer to two types of alternative hypotheses that you can usually choose. The common alternative hypotheses are these three:</p>
<ul>
<li>The value that you want to test is greater than a target.</li>
<li>The value that you want to test is different from a target.</li>
<li>The value that you want to test is less than a target.</li>
</ul>
<p>If you select an alternative hypothesis with "greater than" or "less than" in it, then you've chosen a directional hypothesis. When you choose a directional hypothesis, you get a one-sided test.</p>
<p>What does it look like to choose a one-sided test, and why would you? Let's consider an example.</p>
Choosing Whether to Use a One-sided Test or a Two-sided Test
<p>Suppose new production equipment is installed at a factory that should increase the rate of production for electrical panels. Concern exists that the change could increase the percentage of electrical panels that require rework before shipping. A quality team prepares to conduct a hypothesis test to determine whether statistical evidence supports this concern. The historical rework rate is 1%.</p>
<p>At this point, you would usually choose an alternative hypothesis. Maybe you remember hearing that you should think about whether to use a one-sided test or a two-sided test, or you may not even know how a test can have a side.</p>
<p>To keep from getting lost, you use your GPS. To keep from getting confused about statistics, you can use the Assistant. The Assistant uses clear and simple language. The Assistant doesn't ask you about "directional hypotheses" or "one-sided tests." Instead, the Assistant asks the question, "What do you want to determine?"</p>
<p><img alt="Is the % defective of Panels greater than .01? Is the % defective of Panels less than .01? Is the % defective of Panels different from .01?" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/b090980e5b08184e7b70b96b9cb05489/test_setup_in_assistant.png" style="width: 573px; height: 198px;" /></p>
<p>In this scenario, it's easy to see why the team would want to determine whether the percent is greater than 1. By performing the one-sided test for whether the percentage is greater than 1, the team can determine if there is enough statistical evidence to conclude that the percentage increased. If the percentage increased, then the concern is justified.</p>
<p>In practical terms, you should consider what it means to limit your decision to whether there is evidence for an increase. A one-sided test of whether the percentage increased will never show a statistically significant decrease in the percentage of boards that require rework. Evidence of a decrease in the number of defectives might guide the quality team to investigate the reasons for the unforeseen benefit.</p>
Why Use a One-sided Test?
<p>Given this possible concern about whether a one-sided test excludes important information from the result, why would you ever use one? The best answer is that you use a one-sided test when the one-sided test tells you everything that you need to know.</p>
<p>In the example about the electrical panels, the quality team might feel completely secure in assuming that the new equipment will not result in a decrease in the percentage of panels that require rework. If so, then a test that checks for a decrease is flawed. The team needs only to determine whether to solve a problem with increased defectives or not.</p>
The Assistant Gets Even Better
<p>While a p-value for a one-sided test can be useful, more analysis can help you make better decisions. For example, in the electrical panel example, if the team finds a statistically significant increase, it will be important to know what the percentage increase is. <a href="http://www.minitab.com/en-us/products/minitab/assistant/">The Assistant</a> produces several reports with your hypothesis tests that help you get as much information as you can from your data. The report card verifies your analysis by providing assumption checks and identifying any concerns that you should be aware of. The diagnostic report helps you further understand your analysis by providing additional detail. The summary report helps you to draw the correct conclusions and explain those conclusions to others. The series of reports includes a variety of other statistics and analyses. That way, you have everything that you need to interpret your results with confidence.</p>
<p><img alt="The % defective of Panels is not significantly greater than the target (p > 0.05)" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/75f280df482574a3aee75ee65741b5c4/1_sample___defective_test_for_panels___summary_report_w640.png" style="width: 480px; height: 360px;" /></p>
<p>The image of the face in the crowd without the thought bubble is by <a href="https://www.flickr.com/photos/akbarsyah/">_Imaji_</a> and is licensed under <a href="https://creativecommons.org/licenses/by/2.0/">this creative commons license</a>.</p>
Hypothesis TestingWed, 15 Oct 2014 18:52:23 +0000http://blog.minitab.com/blog/statistics-and-quality-improvement/with-the-assistant-you-wont-have-to-stop-and-get-directions-about-directional-hypothesesCody SteeleThe Ghost Pattern: A Haunting Cautionary Tale about Moving Averages
http://blog.minitab.com/blog/understanding-statistics/the-ghost-pattern-a-haunting-cautionary-tale-about-moving-averages
<p>Halloween's right around the corner, so here's a scary thought for the statistically minded: That pattern in your time series plot? Maybe it's just a ghost. <em>It might not really be there at all.</em> </p>
<p>That's right. The trend that seems so evident might be a phantom. Or, if you don't believe in that sort of thing, chalk it up to the brain's desire to impose order on what we see, even when it doesn't exit. </p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/336bc5b657c980e1c2769192a4757fa9/ghosts.png" style="line-height: 20.7999992370605px; margin: 10px 15px; float: right; width: 200px; height: 200px;" /></p>
<p>I'm going to demonstrate this with Minitab Statistical Software (get the free 30-day <a href="http://it.minitab.com/products/minitab/free-trial.aspx">trial version</a> and play along, if you don't already use it). And if things get scary, just keep telling yourself "It's only a simulation. It's only a simulation."</p>
<p>But remember the ghost pattern when we're done. It's a great reminder of how important it is to make sure that you've interpreted your data properly, and looked at all the factors that might influence your analysis—including the quirks inherent in the statistical methods you used. </p>
Plotting Random Data from a 20-Sided Die
<p>We're going to need some random data, which we can get Minitab to generate for us. In many role-playing games, players use a 20-sided die to determine the outcome of battles with horrible monsters, so in keeping with the Halloween theme we'll simulate 500 consecutive rolls with a 20-sided die. Choose <strong>Calc > Random Data > Integer...</strong> and have Minitab generate 500 rows of random integers between 1 and 20. </p>
<p>Now go to <strong>Graph > Time Series Plot...</strong> and select the column of random integers. Minitab creates a graph that will look something like this: </p>
<p><img alt="Time Series Plot of 200 Twenty-Sided Die Rolls" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/bc2a4c9bf05e4a61103451fa6e6f8342/20_sided_die_time_series_plot.png" style="width: 577px; height: 386px;" /></p>
<p>It looks like there could be a pattern, one that looks a little bit like a sine wave...but it's hard to see, since there's a lot of variation in consecutive points. In this situation, many analysts will use a technique called the Moving Average to filter the data. The idea is to <span style="line-height: 20.7999992370605px;">smooth out the natural variation in the data </span><span style="line-height: 1.6;">by looking at the <em>average </em>of several consecutive data points, thus enabling a pattern to reveal itself. It's the statistical equivalent of applying a noise filter to eliminate hiss on an audio recording. </span></p>
<p>A moving average can be calculated based on the average of as few as 2 data points, but this depends on the size and nature of your data set. We're going to calculate the moving average of every 5 numbers. Choose <strong>Stat > Time Series > Moving Average...</strong> Enter the column of integers as the Variable, and enter 5 as the MA length. Then click "Storage" and have Minitab store the calculated averages in a new data column. </p>
<p>Now create a new time series plot using the moving averages:</p>
<p><img alt="moving average time series plot" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/f93eff7bceb62bd5da113de356afcd8e/moving_average_time_series_plot.png" style="width: 576px; height: 384px;" /></p>
<p>You can see how some of the "noise" from point-to-point variation has been reduced, and it does look like there could, just possibly, be a pattern there.</p>
Can Moving Averages Predict the Future?
<p>Of course, a primary reason for doing a time series analysis is to forecast the next item (or several) in the series. Let's see if we might predict the next moving average of the die by knowing the current moving average. </p>
<p>Select <strong>Stat > Time Series > Lag</strong>. In the dialog box, choose the "moving averages" column as the series to lag. We'll use this dialog to create a new column of data that places each moving average down 1 row in the column and inserts missing value symbols, *, at the top of the column.</p>
<p>Now we can create a <a href="http://blog.minitab.com/blog/understanding-statistics/using-statistics-software-and-graphs-to-quickly-explore-relationships-between-variables">simple scatterplot</a> that will show if there's a correlation between the observed moving average and the next one. </p>
<p><img alt="Scatterplot of Current and Next Moving Averages" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/78607f90333600cdeb6eeba721c62ee7/scatterplot_of_moving_averages.png" style="width: 578px; height: 386px;" /></p>
<p>Clearly, there's a positive correlation between the current moving average and the next, which means we <em>can </em>use the current moving average to predict the next one. </p>
<p><span style="line-height: 1.6;">But wait a minute...this is </span><em style="line-height: 1.6;">random data!</em><span style="line-height: 1.6;"> </span><span style="line-height: 20.7999992370605px;">By definition, you <em>can't </em>predict random</span><span style="line-height: 1.6;">, so how can there be a correlation? This is getting kind of creepy...it's like there's some kind of ghost in this data. </span></p>
<p>Zoinks! What would Scooby Doo make of all this? </p>
Debunking the "Ghost" with the Slutsky-Yule Effect
<p>Don't panic—there's a perfectly rational explanation for what we're seeing here. It's called the Slutsky-Yule Effect, which simply says an autoregressive time series (like a moving average) can <em>look like </em>patterned data, even if there's no relationship among the data points. </p>
<p>So there's no ghost in our random data; instead, we're seeing a sort of statistical illusion. Using the moving average can make it seem like a pattern or relationship exists, but that apparent pattern could be a side effect of the tool, and not an indication of a real pattern. </p>
<p>Does this mean you shouldn't use moving averages to look at your data? No! It's a very valuable and useful technique. However, using it carelessly could get you into trouble. And if you're basing a major decision solely on moving averages, you might want to try some alternate approaches, too. Mikel Harry, one of the originators of Six Sigma, has a <a href="http://drmikelharry.wordpress.com/2014/04/08/beware-the-moving-average/">great blog post</a> that presents a workplace example of how far apart reality and moving averages can be. </p>
<p>So just remember the Slutsky-Yule Effect when you're analyzing data in the dead of night, and your moving average chart shows something frightening. <span style="line-height: 20.7999992370605px;">Shed some more light on the subject with follow-up analysis and you might find there's nothing to fear at </span><span style="line-height: 1.6;">all. </span></p>
Data AnalysisFun StatisticsStatisticsStatsMon, 13 Oct 2014 12:00:00 +0000http://blog.minitab.com/blog/understanding-statistics/the-ghost-pattern-a-haunting-cautionary-tale-about-moving-averagesEston MartzThe Influence of the AP Preseason College Football Poll
http://blog.minitab.com/blog/the-statistics-game/the-influence-of-the-ap-preseason-college-football-poll
<p><img alt="AP Poll" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/516de4c3fce2a1810010ceef3afedbae/ap_top_25_college_football.jpg" style="float: right; width: 248px; height: 140px; margin: 10px 15px;" />A few weeks ago I looked at how the preseason college football poll <a href="http://blog.minitab.com/blog/the-statistics-game/has-the-college-football-playoff-already-been-decided">influences the rankings at the end of the year</a>. I found that for the most part, the teams that ranked higher in the preseason tend to be ranked higher going into the postseason. So if Team A and Team B both finish the regular season undefeated, the team that was ranked higher in the preseason tends to be the one ranked higher going into the postseason.</p>
<p>The biggest exception was, and I hope you’re sitting down for this, SEC teams. SEC teams that finished the regular season with 0 or 1 loss tended to be ranked higher than non-SEC teams with the same amount of losses, regardless of the preseason poll. On the other side, Boise State (a team from a smaller conference) tended to get jumped by teams with the same number of losses that were ranked lower than them in the preseason poll. And the oddball was USC, who in both 2007 and 2008 was jumped by 4 different teams despite starting the season ranked lower than the Trojans and finishing the regular season with the same number of losses.</p>
Looking at 2014
<p>Now let’s turn our attention to this season. Are the same trends happening? There are currently 10 undefeated teams. I looked at the order those 10 teams were ranked in the preseason AP Poll and the order they are currently ranked in the AP Poll. Below is an individual value plot of <a href="//cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/File/a782180e097ef631f2b2119c7d04e5db/2014_ap_poll_rankings.MTW">the data</a> that shows each team's preseason rank versus their current rank.</p>
<p><img alt="Individual Value Plot" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/b40e9e5aff71dd7c6d3722093680c66a/ivp_w640.jpeg" style="width: 640px; height: 427px;" /></p>
<p>Teams on the diagonal line haven’t moved up or down since the preseason. So of the 10 undefeated teams, Florida St and Auburn were ranked #1 and #2 respectively in the preseason. And they remain that way currently.</p>
<p>Teams above the line have jumped teams that were ranked ahead of them in the preseason. Just as we saw before, it’s the SEC teams that rise to the top regardless of where they were ranked in the preseason. Mississippi State had 22 points in the AP preseason poll, which is actually one less than TCU received, and weren’t even close to being in the top 25. But now? They have 1,320 points, are ranked 3rd, and jumped 5 other undefeated teams that were ranked higher than them in the preseason. It’s good to win games in the SEC.</p>
<p>Teams below the line have been jumped by teams ranked lower than them in the preseason. Baylor and Notre Dame are simply casualties of the Mississippi teams. If somebody moves up, somebody else has to move down. But Marshall appears to fit our “Boise State” profile, as they are a team from a small conference. They received 41 points in the preseason AP Poll, putting them just outside the top 25. And now, despite being 5-0, they have only 78 points and are the only undefeated team still outside the top 25.</p>
Measuring Correlation and Concordance
<p>It appears that the preseason poll is influencing the current rankings, but we can get some statistics to confirm our hunches. The first is a <a href="http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and-graphs/tables/other-statistics-and-tests/what-are-spearman-s-rho-and-pearson-s-r-for-ordinal-categories/">correlation coefficient</a>. The correlation coefficient can range in value from -1 to +1. The larger the absolute value of the coefficient, the stronger the relationship between the variables. An absolute value of 1 indicates a perfect relationship, and a value of zero indicates the absence of relationship. </p>
<p><img alt="Correlation" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/da99536fcee47f439b9a99a5408cbc9b/correlation.jpg" style="width: 249px; height: 51px;" /></p>
<p>For these data, both values equal about 0.685, indicating a positive association between preseason rankings and the current rankings. Teams ranked higher in the preseason are also currently ranked higher.</p>
<p>We can also look at the <a href="http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and-graphs/tables/data-and-table-layouts/what-are-concordant-and-discordant-pairs/">concordant and discordant pairs</a>. A pair is concordant if the observations are in the same direction. A pair is discordant if the observations are in opposite directions. More concordant pairs means voters are keeping the same order as the preseason, while more discordant pairs means they’re switching teams.</p>
<p><img alt="Concordance" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/05decbf8033686d033bf18999833e6f8/concordance_w640.jpeg" style="width: 640px; height: 162px;" /></p>
<p>Of the 45 pairs of teams, 35 of them are concordant, meaning most of the time voters are keeping teams in the same order as the preseason. The 10 discordant pairs all involve either Mississippi, Mississippi State, or Marshall. So it’s pretty clear when voters are switching teams from the preseason.</p>
Implications for the College Football Playoff
<p>So far we’ve shown that voters in the AP Poll prefer to keep the same order teams are ranked in the preseason, minus SEC teams and teams from smaller conferences. But the voters of the AP Poll won’t decide the teams in the college football playoff, a separate college football playoff committee will. Luckily for us, that committee will start releasing their own set of rankings on October 28. So we’ll come back then and see if the committee is also influenced by preseason rankings the same way the AP voters are.</p>
Fun StatisticsFri, 10 Oct 2014 12:00:00 +0000http://blog.minitab.com/blog/the-statistics-game/the-influence-of-the-ap-preseason-college-football-pollKevin RudyAttribute Acceptance Sampling for an Acceptance Number of 0
http://blog.minitab.com/blog/applying-statistics-in-quality-projects/attribute-acceptance-sampling-for-an-acceptance-number-of-0
<p>Suppose that you plan to source a substantial amount of parts or subcomponents from a new supplier. To ensure that their quality level is acceptable to you, you might want to assess the capability levels (<a href="http://blog.minitab.com/blog/michelle-paret/process-capability-statistics-cpk-vs-ppk">Ppk and Cpk indices</a>) of their manufacturing processes and check whether their critical process parameters are fully under control (using <a href="http://blog.minitab.com/blog/understanding-statistics/control-chart-tutorials-and-examples">control charts</a>). If you are not sure about the efficiency of the supplier quality system or if you cannot get reliable estimates of their capability indices, you will probably need to actually inspect the incoming parts from this vendor.</p>
<p><img alt="Parts for visual inspection" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/fcc024f038edabcaae5c4fee9476e7c7/parts.jpg" style="margin: 10px 15px; float: right; width: 202px; height: 202px;" />However, checking all parts is expensive and time consuming. In addition to that, visually inspecting 100% of all parts will not necessarily ensure that all defective parts are detected (operators will eventually get tired performing repetitive visual inspections).</p>
<p><a href="http://www.minitab.com/Support/Tutorials/Smart,-Inexpensive-Inspections-with-Acceptance-Sampling/">Acceptance sampling</a> is a more efficient approach: to reduce costs, a smaller sample of parts is selected (in a random way to avoid any systematic bias) from a larger batch of incoming products, these sampled parts are then inspected.</p>
Attribute Acceptance Sampling
<p>The Acceptable Quality Level (AQL) of your supplier is the quality level that you expect from them (a proportion of defectives that is still considered acceptable). If the proportion of defectives is larger than that, the whole batch should get rejected (with a financial penalty for the supplier). The RQL is the Rejectable Quality Level (a proportion of defectives that is not considered acceptable, in which case the whole batch should be rejected).</p>
<p>The graph below represents the probability to accept a batch for a given proportion of defectives. The probability to accept the whole batch when the actual percentage of defectives in the batch is 1% (1% is the AQL in this case) is 98.5%, but if the true percentage of defectives increases to 10% (10% is the RQL), the probability to accept the whole batch will be 9.7%.</p>
<p>The inspection criterion, in this case, should be the following: check 52 parts, and if there are more than 2 defective parts, then reject the whole batch. If there are two defective parts or less, then do not reject. The AQL and the RQL need to be negotiated with your supplier, whereas the acceptance criteria are calculated by Minitab.</p>
<p><img alt="" spellcheck="true" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/31b80fb2-db66-4edf-a753-74d4c9804ab8/Image/7b7ae253a774a961819ff72c20186f3e/oc_1.jpg" style="width: 577px; height: 385px;" /></p>
<p>This graph represents the probability to accept a batch for a given proportion of defectives.</p>
<p>In Minitab, go to <strong>Stat > Quality Tools > Acceptance Sampling by Attributes...</strong> and enter your AQL and RQL as displayed in the dialogue box below to obtain the acceptance criteria.</p>
<p><img alt="" spellcheck="true" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/31b80fb2-db66-4edf-a753-74d4c9804ab8/Image/686400ca3090fa2c5d03a469efa7aa8c/dialogue_box_1.jpg" style="width: 493px; height: 403px;" /></p>
C = 0 Inspection Plans (Acceptance Number of 0):
<p>From a quality assurance point of view, however, in many industries the only acceptable publicized quality level is 0% defective parts. Obviously, the ideal AQL should be 0. You may have a difficult time explaining your final customers that a small proportion of defectives is still acceptable. So let's focus on 0 defective control plans, when the acceptance number is 0 and a batch is rejected as soon as a single defective is identified in the sample.</p>
<p>Note that Minitab will not allow you to enter an AQL of exactly 0 (it should always be larger than 0).</p>
The Producer’s Risk
<p>If the acceptance number is set to 0, the conditions for accepting a lot become considerably more restrictive. One consequence of setting very strict standards for accepting a batch is that if quality is not 100% perfect, and even with a very small proportion of defectives, the probability of rejecting a batch will increase very rapidly.</p>
<p>The Alpha risk (the Producer’s risk) is the probability to reject a batch even though the proportion of defectives is very small. This impacts the producer since many of the batches they deliver will get rejected if the true proportion of defectives is not exactly 0. </p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/31b80fb2-db66-4edf-a753-74d4c9804ab8/Image/eee9a8b20fe157a07bc4a2559301e6f8/capture_dialogue_box.JPG" style="width: 492px; height: 400px;" /></p>
<p>In the graph below the probability to accept a batch with a 1% defective rate is now 80% (so that nearly 20% of the batches will get rejected if the true proportion of defectives is 1%)! This high rejection rate is the price we need to pay for the very strict 0 acceptance number.</p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/31b80fb2-db66-4edf-a753-74d4c9804ab8/Image/325ba3e06d463cc4cf30b45ec14889bb/oc2_w640.jpeg" style="width: 640px; height: 410px;" /></p>
Conclusion
<p>The sample size to inspect is smaller with an acceptance number of 0 (22 parts are inspected in the second graph vs. 52 in the first graph). However, this is a very ambitious objective. If the true percentage of defectives is, say, 0.5% in the batches (if the AQL is set at 0.5%), then 10,4% of all batches will get rejected.</p>
<p>To obtain a lower and more realistic proportion of rejected batches, the level of quality from your supplier should be <em>nearly </em>100% perfect (almost 100% good parts).</p>
Quality ImprovementSix SigmaStatisticsWed, 08 Oct 2014 12:00:00 +0000http://blog.minitab.com/blog/applying-statistics-in-quality-projects/attribute-acceptance-sampling-for-an-acceptance-number-of-0Bruno ScibiliaCapability Snapshot in the Minitab 17 Assistant
http://blog.minitab.com/blog/marilyn-wheatleys-blog/capability-snapshot-in-the-minitab-17-assistant
<span style="font-size: 13px; line-height: 1.6;">In quality initiatives such as Six Sigma, practitioners often need to assess the capability of a process to make sure it can meet specifications, or to verify that it produces good parts. </span><span style="font-size: 13px; line-height: 1.6;">While many Minitab users are familiar with the capability analysis tools in the Stat menu and in </span><a href="http://www.minitab.com/products/minitab/assistant/" style="font-size: 13px; line-height: 1.6;">Minitab’s Assistant</a><span style="font-size: 13px; line-height: 1.6;">, the Assistant includes a less-frequently used feature</span><span style="font-size: 13px; line-height: 1.6;">—</span><span style="font-size: 13px; line-height: 1.6;">the Capability Snapshot.</span>
<strong>What Is the Capability Snapshot, and When Is It Useful?</strong>
<p>The Shapshot can give you a capability estimate when data have not been collected over a long enough period of time to validate process stability (a key assumption for a true process capability analysis) or to capture the different sources of process variation.</p>
<p>The Capability Snapshot provides the most information possible from the limited data provided.</p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/f6d0da32-ba1d-41d4-ace1-af34dcb51351/Image/b02a6122e95cc4debf1d10fe6836ef0f/pic1.png" style="border-width: 2px; border-style: solid; width: 611px; height: 508px;" /></p>
<p>The Assistant's Snapshot mentions using this option only with data that are not in time order, but remember that even if the measurements being used are stored in the worksheet in the order in which they were collected, data collected over a short period of time won't capture the true long-term variation in the process. The time order here refers to data that has been collected over a long period of time, not data that is merely sequential.</p>
<p>According to the guidelines set forth in the Assistant’s Capability Analysis menu for a standard Capability Analysis, sufficient data must be collected over a long-enough period of time to obtain reliable estimates of process capability:</p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/f6d0da32-ba1d-41d4-ace1-af34dcb51351/Image/8d742e13794277c6adc9edec4467c200/pic2_w640.png" style="border-width: 2px; border-style: solid; width: 640px; height: 485px;" /></p>
<p>In some situations, it may not be possible to collect enough data over a sufficiently long period of time to obtain precise estimates or capture the different sources of variation. </p>
<p>For example, parts produced in an R&D laboratory are not likely produced in the same environment—by the same operators, using the same equipment, at the same temperature, or more generally under the same operating conditions—as they would be in a manufacturing plant making parts for customers. Similarly, it may not be possible to produce enough parts in an R&D laboratory. In some situations only a few parts can be produced and it may be necessary to estimate the capability of the process based on this limited information.</p>
<strong>Capability Snapshot to the Rescue!</strong>
<p>If you choose the Capability Snapshot, Minitab only estimates the overall standard deviation. Because the analysis makes no assumption about the amount of time over which you collected the data, it is not possible to determine whether the variation in the data represents the inherent variation (typically captured by within-subgroup variation) or the overall variation of the process (which can only be estimated over the long-term). </p>
<p>In other words no subgroups are assumed, and only the overall variation in the sample is used.</p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/f6d0da32-ba1d-41d4-ace1-af34dcb51351/Image/62a54ccbc50f92d56a3bd6c52eada5ff/pic3_w640.png" style="border-width: 2px; border-style: solid; width: 640px; height: 477px;" /></p>
<p><span style="line-height: 1.6;">Because the calculations use the overall standard deviation of the sample, the resulting output is </span><span style="line-height: 115%; font-family: Calibri, sans-serif; font-size: 11pt;">labeled <a href="http://blog.minitab.com/blog/statistics-and-quality-data-analysis/mayberry-ppk-making-intuitive-sense-of-capability-output-episode-1">Ppk</a></span><span style="line-height: 1.6;">…. However that number should be interpreted with caution! The overall standard deviation for a snapshot of the process may not capture all sources of process variation that would be observed if the data were collected over the long-term. Because of that, the Ppk in this output cannot be interpreted in the usual way.</span></p>
<strong>So Do the Capability Snapshot Statistics Represent Cpk or Ppk?</strong>
<p>As one of Minitab’s fantastic (and hilariously entertaining) trainers, <a href="http://www.minitab.com/en-us/training/trainers/">Paul Sheehy</a>, likes to say, “No mother, no father—it’s neither!” Some like to argue that it represents Cpk, or within-subgroup variation. But to truly capture within-subgroup variation, we calculate the variation averaged across many subgroups of data, and in this case, we only have one ‘subgroup.’</p>
<p>Others prefer to think of the resulting capability estimate as Ppk, or overall variation. And even though that is how the output is labeled in the Capability Snapshot, the data that is typically used for this type of analysis does not capture the variation from the process over the long term, so it's not quite the same metric as traditional Ppk.</p>
<p>If that seems like a lot to remember, don't worry about it! T<span style="line-height: 1.6;">he output from the Assistant is packed full of guidance and helpful tips to guide your interpretation of the results:</span></p>
<p><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/f6d0da32-ba1d-41d4-ace1-af34dcb51351/Image/78eca99eea4c32cc20e37c6c0fe14e0b/pic4_w640.png" style="border-width: 2px; border-style: solid; width: 640px; height: 281px;" /></p>
<p><span style="line-height: 1.6;">To read more about Capability Analysis, check out </span><span style="line-height: 115%; font-family: Calibri, sans-serif; font-size: 11pt;"><a href="http://blog.minitab.com/blog/starting-out-with-statistical-software/the-beginners-curve">Eric Heckman’s</a></span><span style="line-height: 1.6;"> blog post (which you’ll really like if you’re a Star Wars fan!): </span><a href="http://blog.minitab.com/blog/starting-out-with-statistical-software/starting-out-with-capability-analysis" style="line-height: 1.6;">Starting Out with Capability Analysis</a><span style="line-height: 1.6;">.</span></p>
Data AnalysisQuality ImprovementMon, 06 Oct 2014 12:00:00 +0000http://blog.minitab.com/blog/marilyn-wheatleys-blog/capability-snapshot-in-the-minitab-17-assistantMarilyn WheatleyExoplanet Statistics and the Search for Earth Twins
http://blog.minitab.com/blog/adventures-in-statistics/exoplanet-statistics-and-the-search-for-earth-twins
<p>Astronomy is cool! And, it’s gotten even more exciting with the search for exoplanets. You’ve probably heard about newly discovered exoplanets that are extremely different from Earth. These include hot Jupiters, super-cold iceballs, super-heated hellholes, very-low-density puffballs, and ultra-speedy planets that orbit their star in just hours. And then there is PSR J1719-1438 which has the mass of Jupiter, orbits a pulsar, and is probably one giant diamond!</p>
<p>In this post, I'll use statistics to look at the overall planetary output from the Milky Way’s planet-making process. Where does Earth fit in the overall distribution of planets? In light of the extreme exoplanets, is Earth actually the oddball? I’ll also look into the search for an Earth twin, and highlight data that suggest exciting finds down the road.</p>
Our Sample of Exoplanets
<p>We have 1,826 confirmed exoplanets! That’s a very good sample size, but is this sample representative of all planets? To obtain a representative sample, you need to collect a <a href="http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and-graphs/introductory-concepts/data-concepts/why-collect-random-sample/" target="_blank">random sample</a>. It’s easy to point our instruments at random stars, but that doesn’t guarantee a representative sample. Equipment characteristics may make certain types of exoplanets easier to detect than others, thus biasing the sample.</p>
<p>Let’s look at the two methods that have been used to discover the most exoplanets to see how the sample might be biased. These are the radial-velocity and transit methods of planet discovery.</p>
<p><img alt="illustration of the radial velocity method for detecting exoplanets" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/76f572f722a95e6ec19887cb2cb798a0/radial_velocity.gif" style="float: right; width: 200px; height: 200px; margin: 10px 15px;" />There’s nothing better than working with data yourself to truly understand it! These data are from the Planetary Habitability Laboratory’s <a href="http://phl.upr.edu/projects/habitable-exoplanets-catalog/data/database" target="_blank">exoplanet catalog</a>, and I encourage you get the <a href="http://it.minitab.com/products/minitab/free-trial.aspx">free trial version of our statistical software</a> and explore the data yourself. .</p>
Radial-velocity method
<p>Astronomers who use the radial-velocity method look for the wobble that an orbiting exoplanet causes in its star. The bigger the wobble, the easier the exoplanet is to detect. Large exoplanets that are close to small stars cause the largest wobbles and are, therefore, easier to detect by this method. Hot Jupiters are easiest to detect with this method because of their large size and close proximity to the star.</p>
Transit method
<p>The transit, or photometry, method measures the decrease in brightness as an exoplanet passes in front of the parent star. The Kepler space telescope uses the transit method and it was specifically designed to be able to detect Earth-size planets. For this to work, the orbits of the exoplanets have to be perfectly aligned from the astronomers' vantage point.</p>
<p><img alt="illustration of the transit method of detecting exoplanets" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/df3ab82a41c6fc5c7b14296de208f1b7/320px_planetary_transit_svg.png" style="float: right; width: 200px; height: 63px; margin: 15px 10px;" />Kepler must observe at least three transits to flag a candidate exoplanet. The multiple transits help rule out other potential reasons for a light decrease but doesn’t prove it’s a planet. These candidates need to be confirmed via other methods, such as direct imaging. Unfortunately, Kepler was crippled by a failure after four years of data collection. Consequently, detecting exoplanets much more than one AU out from its star is not expected with the Kepler data.</p>
Method comparison
<p>The histograms below show the distribution in mass and distance by detection method for all confirmed exoplanets.</p>
<p><img alt="Mass of exoplanet grouped by detection method" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/4734daf08f590db172cb75a3663b5581/massbymethod.png" style="width: 576px; height: 384px;" /></p>
<p><img alt="Exoplanet distance from parent star by detection method" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/29c6a33c3f11c3ccaecad888e5211911/distancebymethod.png" style="width: 576px; height: 384px;" /></p>
<p>Both methods found a large proportion of planets that are both close to the star and not particularly massive. Even the radial-velocity method, which favors massive planets, found more smaller planets. As expected, the Kepler stopped finding planets that are further than one astronomical unit (AU) away from their star, but the detections by radial-velocity continue out to greater distances.</p>
<p>So, while we probably don’t have a completely representative sample, the two methods agree on the general shape of the distribution: smaller, closer planets far out number massive, more distant planets.</p>
Overall Distribution of Exoplanets
<p>Next, we’ll look at the overall distribution of all confirmed exoplanets by several key variables. The green bar in each graph shows where the Earth fits in.</p>
<p><img alt="Histogram of mass of confirmed exoplanets" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/398260dfb4cdb08089994ee591c9ae6b/massconfirmed.png" style="width: 576px; height: 384px;" /></p>
<p>We can see that the range in exoplanet mass varies greatly, from very small to thousands of Earth masses. For comparison, Jupiter is 317 Earth masses. There are many more small planets, like Earth, than large planets! Let's zoom in.</p>
<p><img alt="Zoomed in look at mass of Earth-sized planets" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/3930142790647e095af79b50c6fa48af/masszoomconfirmed.png" style="width: 576px; height: 384px;" /></p>
<p>In this graph, I've truncated the X-axis. Here we can see that Earth is actually smaller than the peak value. Among rocky planets, it turns out that super-Earths are more common than Earth-sized planets. Super-Earths are rocky like Earth, but have 2 to 5 times the mass of Earth. So, Earth might be slightly unusual for rocky planets by being on the small side. </p>
<p><img alt="Average distance of exoplanet from parent star" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/63c52f6f1b30ab52db287e8f641969a1/distanceconfirmed.png" style="width: 576px; height: 384px;" /></p>
<p>Earth is a bit further from the sun (1 AU) than the more frequent distances on the graph. This reflects the fact that red dwarf stars are by far the most common type of star (80%). These smaller, cooler stars have planetary systems that are much more compact than those of stars like our sun.</p>
<p><img alt="Orbital eccentricity of confirmed exoplantes" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/49e98696fb0e7e6a7e74e08c873e77c0/eccentricityconfirmed.png" style="width: 576px; height: 384px;" /></p>
<p>Orbital eccentricity measures whether an object's orbit is close to circular or more elliptical (oval shaped). Highly eccentric orbits cause extreme climate changes because there is a greater difference between the minimum and maximum distances from the parent star.</p>
<p>Zero is a perfect circle while just less than one is the most elliptical an orbit can be. The orbits of the planets in the solar system are very circular. We can see on the graph that Earth's very low eccentricity is not unusual among confirmed exoplanets.</p>
<p>These graphs show that Earth really isn’t such an oddball—there’s a wide range of planets, and Earth falls near the more common values in each graph.</p>
The Search for an Earth Twin
<p>There’s more to our search than just looking at the distributions by mass, length of year, and orbital eccentricity. We want to know about specific cases where everything lines up just right to produce exoplanets that are habitable Earth twins.</p>
<p>Let me introduce you to the <a href="http://en.wikipedia.org/wiki/Earth_Similarity_Index" target="_blank">Earth similarity index</a> (ESI). This measure indicates how similar an exoplanet is to Earth. Values range from 0 to 1, where Earth has a value of one. ESI is based on <span style="line-height: 20.7999992370605px;">estimated</span><span style="line-height: 20.7999992370605px;"> </span><span style="line-height: 1.6;">parameters for each exoplanet, such as radius, density, surface temperature, and escape velocity. In our solar system, Mars has an ESI of 0.64 and Venus is 0.78.</span></p>
<p>The <a href="http://blog.minitab.com/blog/statistics-and-quality-data-analysis/opening-ceremonies-for-bubble-plots-and-poisson-regression" target="_blank">bubbleplot</a> below shows all of the confirmed exoplanets <strong>and</strong> unconfirmed Kepler candidates that have ESI values greater than 0.80 and are in the habitable zone. For comparison, the blue bubble is Earth.</p>
<p><img alt="Bubbleplot of Earth-like exoplanets by mass, length of year, and star mass" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/20c1c2091bc58c422475502d6753028a/bubbleplot_earthlike_planets_w640.png" style="width: 640px; height: 427px;" /></p>
<p>There are 23 exoplanets and candidates that have an ESI greater than 0.80. In fact, five are greater than or equal to 0.90, with the highest being 0.93. The blue Earth bubble is smaller than most other bubbles on the plot, which again indicates that Earth is on the smaller side for rocky planets. On the graph, 18 out of 23 (78%) are super-Earths, four are classified as Earth-sized, and one is smaller than Earth.</p>
<p>Even though a majority are super-Earths, that’s fine because, super-Earths might be even <a href="http://news.nationalgeographic.com/news/2014/01/140117-exoplanets-superhabitable-planets-space-astronomy-science/">more habitable</a> than Earth-sized planets!</p>
<p>There are two groups of Earth-like planets on the graph. Let’s call them Earth cousins and Earth twins.</p>
<p>The Earth cousins are on the bottom-left. These exoplanets are similar to Earth, but they orbit red dwarf stars that are much cooler and less massive than our sun. These exoplanets need to orbit much closer to be in the habitable zone, which produces the short years.</p>
<p>The Earth twins are on the top right. These exoplanets are like Earth and orbit stars that are like our sun. Consequently, they have years that are more similar to our own.</p>
<p>The bubbleplot contains both confirmed planets and unconfirmed Kepler candidates. The green bubbles indicate confirmed planets, but they’re all in the Earth cousin group. So far, all Earth twins are unconfirmed by other methods. Kepler has detected three transits for each candidate, but some of the candidates may be false positives. The false positive rate for Kepler candidates varies by planet size, and for Earth-sized planets it is 12.3%.</p>
<p>While it is reasonable to expect that some of the 14 Earth twins are false positives, we can also expect that 88% (12) will eventually be confirmed. That will be exciting news! And those are just the twins that we currently have data for.</p>
<p>Given the context about the distribution of planets, it’s not surprising that scientists estimate there are 40 billion Earth-sized planets in the habitable zones of their stars in the Milky Way!</p>
<p><em>The image of the radial velocity method is by Rnt20 and the image of the planet transit method is by Nikola Smolenski. Both images are used under this <a href="http://creativecommons.org/licenses/by-sa/3.0/deed.en" target="_blank">Creative Commons license</a>.</em></p>
Fun StatisticsStatistics in the NewsThu, 02 Oct 2014 12:00:00 +0000http://blog.minitab.com/blog/adventures-in-statistics/exoplanet-statistics-and-the-search-for-earth-twinsJim Frost