Data Normality Tests using p value and critical values

Jay Arthur
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KnowWare International, Inc.
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Is Your Data Normal?

Statistical analysis may rely on your data being "normal" (i.e., bell-shaped), so how can you tell if it really is normal? The two tests most commonly used are:

• Normal Probability Plot
• p-value or Critical Value method

Normal Probability Plot Method

If you've used any of the QI Macros X Chart templates, you know that the normal probability plot is part of the XmR, XbarR and XbarS templates:

Just by looking at the histogram (bell shaped) and probability plot, you can see that this data is fairly normal.

The probability plot transforms the data into a normal distribution and plots it as a scatter diagram.

• Normal data will follow the trend line.
• Non-normal data will have more points farther from the trend line.

p-value and Critical Value Method

The Descriptive Statistics or Normality Test in the QIMacros Anova Tools uses the Anderson-Darling method to analyze normality more rigorously. The output includes the Anderson-Darling statistic, A-squared, and both a p-value and critical values for A-squared. Using Cells A1:A26 from the XbarR.xls in c:\qimacros\testdata, you would get:

The Anderson-Darling values shown are:

• A-squared = 0.270
• p value= 0.648
• 95% Critical Value = 0.787
• 99% Critical Value = 1.092

In this case, the "null hypothesis" is that the data is normal.
The "alternative hypothesis" is that the data is non-normal.

Reject the Null hypothesis (i.e., accept the alternative) when p<=alpha or A-squared>critical value.

Using the p value p = 0.648 which is greater than alpha (level of significance) of 0.01. So we cannot reject the null hypothesis (i.e., the data is normal).

Using the critical values, you would only reject this "null hypothesis" (i.e., data is non-normal) if A-squared is greater than either of the two critical values. Since 0.270 < 0.787 and 0.270 < 1.092, you can be at least 99% confident that the data is normal.

Another Example

Using Cells D1:D41 (after deleting the blank row) from the XbarR.xls in c:\qimacros\testdata, you would get the following result. Notice how the normality plot curves at the right so that some of the points are farther from the line. Using Anderson-Darling we discover that the data is considered normal at one level (99%), but not at another (95%).

Using the p value p = 0.016 which is greater than alpha of 0.01 (0.01 < 0.016 < 0.05), we can reject the null hypothesis (i.e., the data is normal) at alpha = 0.05, but not at alpha = 0.01.

Using the critical values, . Since 0.787 <0.932 < 1.092, you would reject the null hypothesis at 95% but not reject it at 99%.

Frankly, the double negatives of "not rejecting the null hypothesis" makes my brain tired. All I really want to know is: "Is my data normal?" So, in summary:

• if the dots fit the trend line on the normal probability plot, then the data is normal
• if p > alpha then the data is normal
• if A-squared < Critical Value, then the data is normal.

The data normality test is just one of the functions in the QI Macros for Excel.

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It's easy to evaluate data normality with the QI Macros

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