Statistical Tests for Checking Normality

Introduction

Testing for normality is a critical step in data analysis, especially when using statistical methods that assume a normal distribution. This blog provides a comprehensive overview of several statistical tests used to assess the normality of your data, helping you ensure the validity of your analyses.

What is a Normal Distribution?

A normal distribution, or Gaussian distribution, is a bell-shaped curve that is symmetrical around the mean. Key characteristics include:

• Symmetry: The left and right sides of the curve are mirror images.
• Central peak: Most of the data points are concentrated around the mean.
• Tails: The tails approach, but never touch, the horizontal axis.
• Empirical Rule: Approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.

Why Test for Normality?

Many statistical methods, such as t-tests, ANOVA, and linear regression, assume that the data follow a normal distribution. Using these methods on non-normal data can lead to inaccurate results and misleading conclusions. Therefore, it is essential to verify the normality of your data before proceeding with such analyses.

Common Normality Tests

1. Graphical Methods
   • Histogram
   • Q-Q Plot (Quantile-Quantile Plot)
2. Shapiro-Wilk Test
3. Kolmogorov-Smirnov Test
4. Anderson-Darling Test
5. Lilliefors Test
6. Jarque-Bera Test
7. D'Agostino's K-squared Test

1. Graphical Methods

Histograms and Q-Q Plots are visual tools to assess normality.

• Histogram: Displays the frequency distribution of the data. If the data are normally distributed, the histogram will have a bell-shaped curve.
• Q-Q Plot: Plots the quantiles of the data against the quantiles of a normal distribution. If the points fall roughly along a straight line, the data are likely normally distributed.

2. Shapiro-Wilk Test

The Shapiro-Wilk test evaluates the null hypothesis that a sample comes from a normally distributed population. It is considered one of the most powerful normality tests.

• Formula:
  $W = \frac{(\sum_{i=1}^{n} a_i x_{(i)})^2}{\sum_{i=1}^{n} (x_i - \bar{x})^2}$
  where 
  $x_{(i)}$$a_i$

• Interpretation:

  • Null Hypothesis (H0): The data are normally distributed.
  • Alternative Hypothesis (H1): The data are not normally distributed.
  • p-value < 0.05: Reject H0, suggesting the data are not normally distributed.

3. Kolmogorov-Smirnov Test

The Kolmogorov-Smirnov (K-S) test compares the sample distribution with a reference distribution (normal distribution).

• Interpretation:
  • Null Hypothesis (H0): The data follow a specified distribution.
  • Alternative Hypothesis (H1): The data do not follow the specified distribution.
  • p-value < 0.05: Reject H0, suggesting the data do not follow a normal distribution.

4. Anderson-Darling Test

The Anderson-Darling test is a modification of the K-S test, giving more weight to the tails of the distribution.

• Interpretation:
  • Null Hypothesis (H0): The data are normally distributed.
  • Alternative Hypothesis (H1): The data are not normally distributed.
  • Significance Levels: Provides critical values for different significance levels (e.g., 15%, 10%, 5%, 2.5%, and 1%).

5. Lilliefors Test

The Lilliefors test is an adaptation of the K-S test for situations where the parameters of the normal distribution (mean and standard deviation) are estimated from the data.

• Interpretation:
  • Null Hypothesis (H0): The data are normally distributed.
  • Alternative Hypothesis (H1): The data are not normally distributed.
  • p-value < 0.05: Reject H0, indicating non-normality.

6. Jarque-Bera Test

The Jarque-Bera test assesses whether sample data have the skewness and kurtosis matching a normal distribution.

• Formula:
  $JB = \frac{n}{6} \left( S^2 + \frac{(K-3)^2}{4} \right)$
  where 
  $S$$K$$n$

• Interpretation:

  • Null Hypothesis (H0): The data are normally distributed.
  • Alternative Hypothesis (H1): The data are not normally distributed.
  • p-value < 0.05: Reject H0, suggesting non-normality.

7. D'Agostino's K-squared Test

D'Agostino's K-squared test combines skewness and kurtosis to test for normality.

• Interpretation:
  • Null Hypothesis (H0): The data are normally distributed.
  • Alternative Hypothesis (H1): The data are not normally distributed.
  • p-value < 0.05: Reject H0, indicating non-normality.

Conclusion

Checking for normality is a crucial step in data analysis to ensure the validity of statistical tests that assume a normal distribution. Graphical methods like histograms and Q-Q plots provide a visual assessment, while statistical tests such as the Shapiro-Wilk, Kolmogorov-Smirnov, Anderson-Darling, Lilliefors, Jarque-Bera, and D'Agostino's K-squared tests offer more rigorous evaluations. Using these methods, you can confidently determine whether your data meet the normality assumption.