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## Reassessing GDP Growth with Data and Statistics, Part 2

### Understanding the IM-R chart

This is part two in a three-part 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, and part three here.

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 nonrandom, unexpected patterns in the series of measurements.

So, for the gross domestic product (GDP) data, the I chart displays each quarter's GDP growth estimate that the U.S. Bureau of Economic Analysis (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 percent for last quarter and 3.5 percent this quarter, we say the economy is heating up, or that growth has increased by 1.0 percent. However, keep in mind that the MR chart just graphs the absolute changes, not the direction.

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

Figure 1: Individuals moving range (I-MR) chart of the latest gross domestic product (GDP) estimates

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 percent between consecutive quarterly estimates. This is not a small amount when you consider that the average GDP growth estimate is 2.81 percent.

On the MR chart, we see that the upper control limit is at 7.089 percent. 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 the 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 four identifiable patterns, which are circled and labeled in figure 1:

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 nine points in a row on the same side of the center line. The first data point that fails this test is the ninth, 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 two out of three 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 two of three 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 one to pattern two.

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 surprising, the recent economic turmoil shows up as three extreme points that fall below the lower control limit. Additionally, the fourth point fails test 6, which indicates that four out of five points are more than one 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 longstanding 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 we'll look at the early estimates and see if they might—just possibly—paint a different picture.

### Jim Frost

Jim Frost is a statistical technical communications specialist at Minitab Inc. He has a background in a wide variety of academic research and became known as the “data/stat guy” on research projects that ranged from osteoporosis prevention to quantitative studies of online user behavior. At Minitab, he is a technical writer who helps people use Minitab software to gain insights from their own data, whether they’re working in quality improvement, academic research, or another field entirely. He also writes in The Minitab Blog about various experiences and practical knowledge he’s learned along the way that may help others’ research endeavors.

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