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What is the null hypothesis of the Shapiro test?

Published in Statistical Hypothesis Testing 3 mins read

The null hypothesis of the Shapiro-Wilk test is that a given data sample has been generated from a normally distributed population.

Understanding the Shapiro-Wilk Test

The Shapiro-Wilk test is a specific type of hypothesis test used to assess whether a sample of data comes from a normal distribution. It is widely applied in statistics because many statistical methods, such as t-tests and ANOVA, assume that the data they analyze are normally distributed.

In the context of the Shapiro-Wilk test, the hypotheses are formally stated as follows:

Hypothesis Statement
Null Hypothesis (H₀) The sample data is drawn from a normally distributed population.
Alternative Hypothesis (H₁) The sample data is not drawn from a normally distributed population.

Interpreting the Results

When you perform a Shapiro-Wilk test, the primary output you look at is the p-value. This p-value helps determine whether there is enough evidence to reject the null hypothesis.

  • If the p-value is low (typically less than a predetermined significance level, e.g., 0.05): We reject the null hypothesis. This indicates that there is sufficient evidence to conclude that the sample data does not come from a normal distribution. In simpler terms, the data is likely non-normal.
  • If the p-value is high (greater than the significance level): We fail to reject the null hypothesis. This suggests that there is not enough evidence to conclude that the data is non-normal. In other words, the data could reasonably be considered to come from a normal distribution.

Practical Implications

Testing for normality is a crucial preliminary step for many statistical analyses. If your data violates the assumption of normality for a particular test, the results of that test might be unreliable.

  • Example: Suppose you want to compare the means of two groups using an independent samples t-test. A common assumption for this test is that the data in both groups are normally distributed. You would run a Shapiro-Wilk test on each group's data.
    • If both p-values are high, you can proceed with the standard t-test.
    • If one or both p-values are low, suggesting non-normality, you might consider:
      • Transforming your data to achieve normality.
      • Using a non-parametric alternative test (e.g., the Mann-Whitney U test instead of the independent samples t-test), which does not assume normality.

Understanding the null hypothesis of the Shapiro-Wilk test is fundamental to correctly interpreting its output and making informed decisions about subsequent statistical analyses. For further reading on the Shapiro-Wilk test, you can consult resources like Wikipedia's page on the Shapiro-Wilk test.