*This is part two in a series where we assess what information we can obtain from the various estimates of quarterly GDP growth using statistical analysis and a control chart. Read part one **here**.* *You can download the Minitab Statistical Software data file used in this series here. *

## Understanding the I-MR Chart

An I-MR chart comprises two plots, the individuals (I) chart on the top and the moving range (MR) chart on the bottom. The I chart displays the measurements and provides a means to assess the process center. The MR chart displays the absolute change between successive measurements and provides a means to assess process variation. The I chart is only valid if the MR chart is in control. For charts where variability is in control but the I chart is out of control, this scenario indicates that there are non-random, unexpected patterns in the series of measurements.

So, for the GDP data, the I chart displays each quarter’s GDP growth estimate that BEA reports. The MR chart plots the absolute change in the quarterly growth. In this case, the MR chart reflects a change in the change. This may sound complicated, but it’s really not. We already think about these data this way. For example, if the GDP growth was 2.5% for last quarter and 3.5% this quarter, we say the economy is heating up, or that growth has increased by 1.0%. However, keep in mind that the MR chart just graphs the absolute changes, not the direction.

## I-MR Chart of the Latest GDP Estimates

In this I-MR chart*, we are looking at the gold standard values for the quarterly GDP growth:

Looking at the MR chart, we see that the average for the moving ranges is 2.170. Each moving range value is the absolute change from one quarterly estimate to the next. If you average all of those absolute changes, you get 2.170. In other words, you can expect a change of 2.170% between consecutive quarterly estimates. This is not a small amount when you consider that the average GDP growth estimate is 2.81%.

On the MR chart, we see that the upper control limit is at 7.089%. This value represents the magnitude of the change that goes beyond expected changes based on the observed variability. In other words, points beyond the control limit represent a signal, not noise. It really means something! Only one value, which happens in Q3 2000, exceeds this control limit and appears as a red point. We’ll come back to this after looking at the I chart, but this point does represent a meaningful change in GDP growth.

Because the MR chart is generally in control, we can interpret the I chart, which displays BEA’s quarterly growth estimates. The I chart includes tests for special causes, which detect points beyond the control limits and specific patterns in the data. Failed points are either too far from the mean or exhibit a pattern that is unlikely given the observed variation in the MR chart. In other words, failed tests represent patterns or values that stand out from the noise. Points that have failed a test for special causes are marked in red with a number that indicates the meaning of the failed test.

There seem to be 4 identifiable patterns, which are circled and labeled above:

1) **Moderately above-average GDP growth for an extended time.** See how all the points are above the green center line (average) for an extended time? There is not one extreme value that trips the alarm, but a series of values that collectively trip the alarm. Test 2 indicates that there are 9 points in a row on the same side of the center line. The first data point that fails this test is the 9th, so it was detected as soon as possible for this data set. The "5" label for the last data point in this pattern indicates that 2 out of 3 points are more than two standard deviations from the center line on the same side (higher). This extended run above the average is unlikely given the variability in the MR chart.

2) **Moderately below-average GDP growth for an extended time.** Again, there is not one point that trips the alarm. Test 5 indicates that 2 of 3 points are more than two standard deviations away from the center line, this time on the lower side. Look at both the I and MR charts, and you'll notice the one failed point in the MR chart corresponds to the drop from pattern 1 to pattern 2.

3) **Average GDP growth.** This is a time of average growth that falls randomly around the mean. There are no test failures because this is what you expect to see given the observed variability.

4) **Economic downturn.** Not surprisingly, the recent economic turmoil shows up as 3 extreme points that fall below the lower control limit. Additionally, the 4th point fails test 6, which indicates that 4 out of 5 points are more than 1 standard deviation from the center line and on the same side (low).

## Observations

For me, at least, the control chart makes it easier to see the patterns in this particular type of data than the time series plot. Additionally, the test results confirm the visibly apparent patterns. So, it appears to be a good way to assess these data.

In the MR chart, the change between consecutive Latest estimates generally does not provide information by itself. It did this only once during the 15 years that this data set covers. The typical variability (noise) is moderately large in this context and it is hard for a signal in the MR chart to stand out from it. In short, large changes during normal times are not unusual. However, the one failed point in the MR chart did correctly signify the shift from the long standing pattern 1 to pattern 2.

In the I chart, it is generally not one GDP growth estimate that exceeds a control limit and tips you off that the economy is in an unusual pattern. More often, it is a *pattern *of growth estimates that is unlikely to occur given the observed variability and mean. This pattern can be an extended run that is moderately, but consistently, above or below the mean.

These observations are all based on the Latest, or gold standard, estimates. Tomorrow, in part three, we'll look at the early estimates and see if they might—just possibly—paint a different picture!

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** In creating all I-MR charts in this blog series, I told Minitab to ignore 2008 Q4 and 2009 Q1 when calculating the control limits because these are extreme values that have not been seen since the Great Depression. Also, I told Minitab to run all of the available tests on the data. The effect of both actions is to make the chart more sensitive to special-cause variation. If this had produced unusual results, I could have dialed down the sensitivity. But, the results look reasonable.*