How to test normality statistically?

The previous article explained the importance of testing normality t for a dataset before performing regression. It also explained the various ways to test normality graphically using the SPSS software. However, graphical normality test has several shortcomings, the biggest one being lack of reliability due to the probability of inaccurate results.

For this purpose, statistical or empirical normality tests are conducted. This article explains three such tests using SPSS and E-Views software:

  1. Kolmogorov-Smirnov Goodness of Fit (K-S) test,
  2. Jarque-Bera test and,
  3. Shapiro-Wilk test.

Normal distribution of data is also called ‘Gaussian distribution’. The below equation shows the mathematical formula for normal or gaussian distribution.

Importance of testing normality of a dataset

Normality tests help in checking whether the data is normally distributed or not. Statistical tests such as regression assume the existence of normally distributed data. For example, simple linear regression analysis for determining the impact of social factors on women’s empowerment does not include the normality test of the dataset. However, this assumption is not always accepted.

Data scientists strictly prefer to test normality and work on normally distributed data because of its benefits (Parbhakar, 2018). Some of the important characteristics of a normal distribution are –

  • Provide a high confidence level in the analysis.
  • Better model fit for nature and social science-based studies.

Thus, considering the characteristics of normally distributed data, a normality test needs to be performed for generating more effective results.

Methods to test normality

A normality test is typically represented by the below hypothesis.

H0: Sample is not derived from a normally distributed population.

Ha: Sample is derived from a normally distributed population.

Statistical tests of checking normality of a dataset

Statistical test of normality calculates the probability of deriving sample from the normally distributed population. The empirical methods of normality test are classified as under.

Tests Purpose Benefit Disadvantage Normal Distribution Criteria
Kolmogorov-Smirnov Goodness of Fit (K-S) Test. Derive the deviation of the cumulative frequency distribution of the variable with the expected normally distributed data. Information on the normally distributed data not required.
Easy to compute.
Helps in testing normality and goodness of fit.
Suitable for small sample size.
Modified to Lilliefors test for more accurate results.
Requires more specification of data.
Not applicable for discrete distributions.
Centre values distribution is more sensitive.
Kolmogorov-Smirnov Table yield conservative results.
Historical concept.
Test statistic value < critical Value Or P-Value > α value.
Shapiro-Wilk Test
Identifies the probability of having data from normally distributed population

Reliable test.
High power of the test.
Determine the correlation between the observations.
Not suitable for large sample size*.
Biased by sample size.

Test statistic value < critical Value Or P-Value > α value.
Jarque-Bera Test Check the joint probability of skewness and kurtosis from the normal distribution values. Suitable for large sample size.
Very Effective.
Not suitable for a heteroscedastic and autocorrelated sample.
Low power of the test for a finite sample.
Not suitable for small sample size.
Test statistic value > critical Value Or P-Value < α value.

* Best-suited for the sample between 3 and 2000 but can work till 5000. However, work best for dataset < 50.

Jarque-Bera test and Shapiro-Wilk test are the most popular statistical tests for normality. Shapiro-Wilk test can be performed in SPSS and Stata. EViews and Stata support the Jarque-Bera test. However, K-S Test can only be applied in SPSS.

Case example of statistical tests of normality

This case example involves the representation of empirical or statistical tests of normality using data of FDI inflows of India from 1994-2015. The results are represented below.

K-S test and Shapiro-Wilk test of normality in SPSS

K-S test and Shapiro Wilk test results of normality in SPSS
Figure 1: K-S test and Shapiro Wilk test results of normality in SPSS

The table shows that the significance or p-value of the K-S test (0.000) is less than the tolerable significance level of 5% i.e. 0.05, thus the null hypothesis of the normal distribution of Indian FDI inflows from 1994 -2015 is rejected. Shapiro-Wilk test results are similar to K-S test results i.e. the p-value of 0.001 < 0.05, hence, the null hypothesis is rejected. Hence, the FDI Inflows sample is not derived from the normally distributed population.

Jarque-Bera test of normality in E-Views

Jarque-Bera test of normality in E-Views
Figure 2: Jarque-Bera test of normality in E-Views

The table shows that the p-value (0.277740) is greater than the significance level of 5% i.e. 0.277740 > 0.05. Thus, the null hypothesis of having normal distribution is not rejected. Hence, FDI Inflows for a period of 1994-2015, is normally distributed. Results of the Jarque-Bera test are not aligned with other statistical results thus depicting that it is not suitable for a small sample size.

Jarque-Bera test and Shapiro-Wilk test are the most effective normality tests but the difference is that the former is suitable for large sample size, whereas the latter is applicable in case of a small sample size.

Riya Jain

Research analyst at Project Guru
Riya is a master in Economics from Amity University. She has a keen interest in econometrics and data analysis. She was a part of the Innovation Project of Daulat Ram College, Delhi University. Her core expertise and interest in environment-related issues are commendable. Apart from academics, she loves music and travelling new places.
Riya Jain