Statistics Definitions > Assumption of Normality / Normality Test

## What is Assumption of Normality?

Assumption of normality means that you should make sure your data roughly fits a bell curve shape before running certain statistical tests or regression. The tests that require normally distributed data include:

## How to Test for Normality

You’ve got two main ways to test for normality: eyeball a graph, or run a test that’s specifically designed to test for normality. The data doesn’t have to be *perfectly* normal. However, **data that definitely does not meet the assumption of normality is going to give you poor results** for certain types of test (i.e. ones that state that the assumption must be met!). How closely does your data have to meet the test for normality? This is a judgment call.

If you’re uncomfortable with making that judgement (which is usually based on experience with statistics), then you may be better off running a statistical test (scroll down for the test options). That said, some of the tests can be cumbersome to use and involve finding test statistics and critical values.

**Stuck on which option to choose? **If you’re new to statistics, the easiest graph to decipher is the histogram. The easiest test to run is probably the Jarque-Bera Test.

## Using a Graph for a Normality Test.

**1. Q Q Plot**.

A Q Q plot compares two different distributions. If the two sets of data came from the same distribution, the points will fall on a 45 degree reference line. To use this type of graph for the assumption of normality, compare your data to data from a distribution with

*known*normality.

**2. Boxplot**.

Draw a boxplot of your data. If your data comes from a normal distribution, the box will be symmetrical with the mean and median in the center. If the data meets the assumption of normality, there should also be few outliers.

**3. Normal Probability Plot.**

The normal probability plot was designed specifically to test for the assumption of normality. If your data comes from a normal distribution, the points on the graph will form a line.

**4. Histogram.**

The popular histogram can give you a good idea about whether your data meets the assumption. If your data looks like a bell curve: then it’s probably normal.

## Statistical Tests for Normality

You’ve got *lots* of options to test for normality. Most of these are included with statistical packages like SPSS.

**Chi-square normality test**. You can use a chi square test for normality. The advantage is that it’s relatively easy to use, but it isn’t a very strong test. If you have a small sample (under 20), it may be the*only*test you can use. For larger samples, you’re much better off choosing another option.**D’Agostino-Pearson Test**. This uses skewness and kurtosis to see if your data matches normal data. It requires your sample size to be over 20.**Jarque-Bera Test**. This common test is also relatively straightforward. Like the D’Agostino-Pearson, the basic idea is that it tests the skew and kurtosis of your data to see if it matches what you would expect from a normal distribution. The larger the JB statistic, the more the data deviates from the normal.**Kolmogorov-Smirnov Goodness of Fit Test**. This compares your data with a known distribution (i.e. a normal distribution).**Lilliefors Test**. The Lilliefors test calculates a test statistic T which you can compare to a critical value. If the test statistic is bigger than the critical value, it’s a sign that your data isn’t normal. It also computes a p-value for your distribution, which you compare to a significance level.**Shapiro-Wilk Test**This test will tell you if a random sample came from a normal distribution. The test gives you a W value; small values indicate your sample is*not*normally distributed.

## References

Klein, G. (2013). The Cartoon Introduction to Statistics. Hill & Wamg.

Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences, Wiley.

Vogt, W.P. (2005). Dictionary of Statistics & Methodology: A Nontechnical Guide for the Social Sciences. SAGE.

Wheelan, C. (2014). Naked Statistics. W. W. Norton & Company