# Data Normality Tests Using p and Critical Values in QI Macros

### Is Your Data Normal?

Statistical analysis (e.g., ANOVA) 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:

**Note: Excel does not do normality tests; QI Macros adds this functionality.**

### Anderson Darling p-value and Critical Value Method

The Normality Test and Descriptive Statistics in the QI Macros uses the Anderson-Darling method to analyze normality. The output includes the Anderson-Darling statistic, A-squared, and both a p-value and critical values for A-squared.

**To run a normality test in the QI Macros**, just select your data then select Statistical Tools, Normality Test from the QI Macros menu.

The QI Macros will run an Anderson-Darling Normality Test and other descriptive statistics giving both numerical and graphical representations of the data:

### Interpreting the Normality test results

#### The Anderson-Darling values shown are:

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

If |
Then |

p value <= a | Reject the null hypothesis |

p value > a | Cannot reject the null hypothesis - the data is normal. |

A-squared > critical value |
Reject the null hypothesis |

A-squared <= critical value |
Cannot reject the null hypothesis - the data is normal. |

"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.

### Normal Probability Plot Method

The QI Macros Normal Probability Test will give you a Probability Plot. 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.

Hypothesis Testing Quick Reference Card