Handling Non-Normal Data

Segmenting Data

Transforming Data

Using Different Distributions

Non-Parametric Statistics

When performing statistical tests on data, it is important to realize that many statistical tests assume normality. If you have non-normal data, there are parametric equivalent statistical tests that should be employed. Table 2 below summarizes the statistical tests to use with normal process data, as well as the and non-parametric statistical test equivalents.

Table 2: Common Statistical Tests For Normal & Non Parametric Data
Assumes Normality	No Assumption Required
One sample Z test	One sample Sign
One sample t test	One sample Wilcoxon
Two sample t test	Mann - Whitney
One way ANOVA	Kruskal - Wallis Moods Median
Randomized Block (Two way ANOVA Analysis)	The Friedman Test

If we look back at our Table 1 data set where zero was the hard limit, we can illustrate what tests might be employed when dealing with non-normal data.

Setting A Hard Limit
If we set a hard limit at zero and re-run the process capability, the results are presented in Figure 4.

Figure 4 now indicates that the long term PPM is 249535.66, as opposed to 286011.34 in Figure 2. This illustrates that quantification can be more accurate by first understanding whether the distribution is normal or non-normal.

Weibull Distribution
If we take this analysis a step further, we can determine which non-normal distribution is a best fit. Figure 5 displays various distributions overlaid on the data. We can see that the Weibull distribution is the best fit for the data.

Knowing that the Weibull distribution is a good fit for the data, we can then recalculate the process capability. Figure 6 shows that a Wiebull model with the lower bound at zero would produce a PPM of 233244.81. This estimate is far more accurate than the earlier estimate of the bounded normal distribution.

Box-Cox Transformation
The other method that we discussed earlier was transformation of data. The Box-Cox transformation can be used for converting the data to a normal distribution, which then allows the process capability to be easily determined. Figure 7 indicates that a Lambda of 0.5 is most appropriate. This Lambda is equivalent to the square root of the data.

After using this data transformation, the process capability is presented in Figure 8. This transformation of data estimates the PPM to be 227113.29, which is very close to the estimate provided by the Weibull modeling.

We have seen three different methods for estimating the appropriate process capability of the process in case the data is from a non-normal source: Setting a hard limit on a normal distribution, using a Weibull distribution, and using the Box-Cox transformation.

Sub-Group Averaging
Now let us assume that the data is collected in time sequence with a subgroup of one. The X bar R chart with subgroups cannot be used. Had the data been collected in subgroups the Central limit theorem would come in handy and the data would have exhibited normality. If we use the Individual Moving Range chart -- which is the more appropriate chart to use -- Table 3 displays the results.

Table 3: I/MR for Cycle Time

Test Results for I Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 12 36 55 61 103 132 TEST 2. 9 points in a row on same side of center line. Test Failed at points: 28 TEST 3. 6 points in a row all increasing or all decreasing. Test Failed at points: 49 50 TEST 5. 2 out of 3 points more than 2 sigmas from center line (on one side of CL). Test Failed at points: 23 24 25 61 80 104 203 TEST 6. 4 out of 5 points more than 1 sigma from center line (on one side of CL). Test Failed at points: 15 22 23 24 25 26 27 45 82 159 160 TEST 8. 8 points in a row more than 1 sigma from center line (above and below CL). Test Failed at points: 27 Test Results for MR Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 12 55 62 69 70 78 127 132 133 TEST 2. 9 points in a row on same side of center line. Test Failed at points: 52 53 54 200 201 202 203

Figure 9 indicates that the process is plagued by special causes. If we focus only on those points that are beyond the three Sigma limits on the I chart, we find the following data points as special causes.

TEST 1. One point more than 3.00 sigmas from centerline.
Test Failed at points: 12 36 55 61 103 132.

The primary assumption in the Figure 9 control chart is that the data is normal. If we plot the I-MR chart applying the Box-Cox transformation as above, the looks much different (see Figure 10).

Table 4: I/MR for Cycle Time

Test Results for I Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 80 TEST 2. 9 points in a row on same side of center line. Test Failed at points: 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 110 111 TEST 3. 6 points in a row all increasing or all decreasing. Test Failed at points: 49 50 TEST 5. 2 out of 3 points more than 2 sigmas from center line (on one side of CL). Test Failed at points: 61 80 92 104 165 203 TEST 6. 4 out of 5 points more than 1 sigma from center line (on one side of CL). Test Failed at points: 15 22 23 24 25 26 27 45 82 113 159 160 TEST 8. 8 points in a row more than 1 sigma from center line (above and below CL). Test Failed at points: 27 Test Results for MR Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 55 69 78 80 132 133 140 TEST 2. 9 points in a row on same side of center line. Test Failed at points: 29 30 31 32 33 52 53 54 200 201

If we attempt to study the number of points outside the three sigma limits on the I chart, we note that the test fails at only one point -- number 80 -- and not at points 12 36 55 61 103 132 as indicated by the Figure 8 chart earlier. In fact, if you study the results of other failure points one would realize the serious consequences of assuming normality: doing so might cause you to react to common causes as special causes, which would lead to tampering of the process. It is important to note that for X bar R chart the problem with normality is not serious due to the central limit theorem, but in the case of Individual Moving Range chart there could be serious consequences.

Now let's consider a case where we would like to study whether the mean cycle time of this process is at 20 days. If we assume data normality and run a one sample t test to confirm this hypothesis, Table 5 displays the results.

Table 5: One-Sample T - Cycle Time

Test of mu = 20 vs mu not = 20 Variable N Mean StDev SE Mean turn around 207 21.787 12.135 0.843 Variable 95.0% CI T P turn around ( 20.125, 23.450) 2.12 0.035

Based on the above statistics, one would pronounce at an alpha risk of 5% that the mean of the data set is different than 20. If we were to verify the fact that the data is not normal, we would have run the one sample Wilcoxon test which is based on the medians rather than the means, and we would have obtained the results found in Table 6.

Table 6: Wilcoxon Signed Rank Test: Cycle Time

Test of median = 20.00 versus median not = 20.00 N for Wilcoxon Estimated N Test Statistic P Median turn aro 207 202 11163.5 0.273 21.00

The Wilcoxon test indicates that the null hypothesis (test median is equal to 20) is accepted, and there is no statistical evidence that the median is different than 20.

The above example illustrates the fact that assuming that the data is normal and applying statistical tests is dangerous. As a better strategy in data analysis, it is better to verify the normality assumption and then -- based on the results -- use an appropriate data analysis method.

About The Author
Shree Phadnis is a Master Black Belt at KPMG India. Mr. Phadnis is an ASQ certified Quality Manager and ASQ Certified Quality Engineer. Mr. Phadnis can be reached at shreephadnis@usa.net.