17.1.7.3 Choosing Normality Tests and Interpreting ResultsNormalityTestEX
Summary
Suppose you want to investigate the health status of some students. You collect 40 students and record their names, genders, ages, heights and weights. After collecting your data, you use a Normality Test procedure to examine whether the weights of the students follow a normal distribution.
Example
 Import body.dat in the \Samples\Statistics folder.
 Highlight Column E
 Select Statistics: Descriptive Statistics: Normality Test.
 Check all tests under the Quantities to Compute branch.
 Select Histograms and Box Chart under the Plots branch.
 Click OK.
Interpreting Results
Statistical models usually depend on some underlying assumptions. One common assumption is that of a normallydistributed population. Unfortunately, many analysts assume normality without any empirical evidence or test. If the assumption of normality is violated, then what we infer might not be reliable.
It is difficult to define a standard for interpreting the results of normality tests because needs vary by discipline and by analyst. Some test methods may be satisfactory for one field but unsatisfactory in another field.
There are two primary approaches to normality testing: graphical methods and numerical methods. Graphical methods tend to be intuitive and easy to interpret. Numerical methods are more precise and hence, more objective.
Graphical Methods
Stemandleaf plots, (skeletal) box plots, dot plots, histograms, and PP or QQ plots, are useful for visualizing the difference between an empirical distribution and a theoretical normal distribution. Origin's Normality Test tool offers histograms and box charts, but it should be mentioned that Origin also offers PP and QQ plots from the Plot menu.
One very straightforward way to "test" for normality is to create a histogram. It is well known that the shape of a normal distribution is symmetrical and classically "bellshaped." Looking at a histogram, we can we can obtain a rough idea as to the nature of the population distribution. The box chart is something of a complement to the histogram. A box chart effectively summarizes major percentiles, such as minimum, 25th percentile (1st quartile), 50th percentile (median), 75th percentile (3rd quartile) and maximum, using a box and lines. If the 25th and 75th percentiles are symmetrical with respect to the median, and median and mean values are seen to be located at roughly the same position near the center of the box, then we have reason to believe that the variable of interest may be normally distributed. In the body.dat example above, the shape of histogram is not exactly symmetrical, but it is near to a "bell" shape. The box chart of the same data also indicates rough symmetry.
Choosing Normality Test Methods
It is well known that measures of skewness and kurtosis can be applied to normality testing. Skewness is generally defined to be a third standardized moment, a measure of the degree of symmetry. If skewness is greater than zero, the distribution is rightskewed and we count more observations on the left side of the distribution curve; conversely, when skewness is less than 0, observations will be distributed to the right side of the curve. Kurtosis, a fourth standardized moment, measures peak expression or thinness of tails. Note that the standard normal distribution has kurtosis = 0 (in "excess kutosis" definition convention). So, if a calculated kurtosis > 0, the distribution has thinner tails and a higher peak as compared with the standard normal distribution. Origin calculates both skewness and kurtosis. See Statistics on Columns for details.
The KolmogorovSmirnov, KolmogorovSmirnovLilliefors, AndersonDarling and Cramervon Mises tests are empirical distribution function (EDF) based methods, while Jarque–Bera and SkewnessKurtosis (aka D'Agostino KSquared) tests are Chisquared distribution based. The ChenShapiro test is a normalized spacingbased method found to be both powerful and simple. ShapiroWilk, RyanJoiner and ShapiroFrancia tests, like ChenShapiro, are regression and correlationbased methods.
 KolmogorovSmirnov: The KS test, though known to be less powerful, is widely used. Generally, it requires large sample sizes.
 KolmogorovSmirnovLilliefors: An adaptation of the KS test. More complicated than KS, since it must be established whether the maximum discrepancy between empirical distribution function and the cumulative distribution function is large enough to be statistically significant. KSL is generally recommended over KS. Some analysts recommend that the sample size of KSL be larger than 2000.
 AndersonDarling: One of the best EDFbased statistics for normality testing. Sample size of less than 26 is recommended, but industrial data with 200 and more might pass AD. The pvalue of the AD test depends on simulation algorithms. The AD test can be used to test for other distributions with other specified simulation plans. See D’Agostino and Stephens (1986) for details.
 D'Agostino KSquared: Based on skewness and kurtosis measures. See D’Agostino, Belanger, and D’Agostino, Jr. (1990) and Royston (1991) for details. It is worthwhile mentioning that skewness and kurtosis are also affected by sample size.
 ShapiroWilk: The recommended sample size for this test ranges from 7 to 2000. Origin allows sample sizes from 3 to 5000. However, when sample size is relatively large, D'Agostino Ksquared or Lilliefors are generally preferred over ShapiroWilk.
 ChenShapiro: The CS test extends the SW test without loss of power. The motivation for CS is based on the fact that the ratios of the sample spacing to their expected spacing would converge to one due to the consistency of sample quantiles. From the standpoint of power, CS performs more like the SW test rather than the SF test.
Origin provides users with commonly used methods like SW, KS, Lilliefors, AD, D'AgostinoK Squared and CS tests. There are six additional normality tests in Origin. You can use the following table to guide your choice of tests. Note that in Origin, we report critical values rather than pvalues for ChenShapiro test. Critical values can also be used for testing. If the specified statistical value is less than or equal to the 5% critical value, then the pvalue should be greater than or equal to 0.05. Hence, we would not reject the null hypothesis for alpha = 0.05.
Summary of Normality tests in Origin
Test Method 
Statistic 
N Range 
Distribution Based

KolmogorovSmirnov 
D 
3 <= N 
EDF

Lilliefors 
L 
4 <= N 
EDF

AndersonDarling 
Asquare 
8 <= N 
EDF

D'Agostino KSquared 
Chisquare 
4 <= N 

ShapiroWilk 
W 
3 <= N <= 5000 


ChenShapiro 
QH 
10 <= N <= 2000 


Note: Just because you meet sample size requirements (N in the above table), this does not guarantee that the test result is efficient and powerful. Almost all normality test methods perform poorly for small sample sizes (less than or equal to 30).
