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Shapiro-Wilk Test

Written by the Fiveable Content Team • Last updated August 2025
Written by the Fiveable Content Team • Last updated August 2025

Definition

The Shapiro-Wilk test is a statistical test used to determine if a dataset follows a normal distribution. It is commonly used in the context of one-way ANOVA analysis to assess the assumption of normality, which is a key requirement for the validity of the ANOVA test.

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5 Must Know Facts For Your Next Test

  1. The Shapiro-Wilk test is a more powerful normality test compared to other common tests, such as the Kolmogorov-Smirnov test, especially for small sample sizes.
  2. The Shapiro-Wilk test statistic, denoted as 'W', ranges from 0 to 1, with values closer to 1 indicating that the data is more likely to have come from a normal distribution.
  3. The Shapiro-Wilk test is a hypothesis test, where the null hypothesis is that the data follows a normal distribution, and the alternative hypothesis is that the data does not follow a normal distribution.
  4. If the p-value of the Shapiro-Wilk test is less than the chosen significance level (e.g., 0.05), the null hypothesis is rejected, and it is concluded that the data does not follow a normal distribution.
  5. Violation of the normality assumption in one-way ANOVA can lead to inaccurate p-values and potentially incorrect conclusions about the significance of the differences between group means.

Review Questions

  • Explain the purpose of the Shapiro-Wilk test in the context of one-way ANOVA.
    • The Shapiro-Wilk test is used to assess the assumption of normality, which is a key requirement for the validity of the one-way ANOVA analysis. If the data in each group does not follow a normal distribution, as indicated by a significant Shapiro-Wilk test (p-value < 0.05), then the ANOVA results may not be reliable, and alternative non-parametric tests should be considered. Ensuring the normality assumption is met is crucial for drawing accurate conclusions about the differences between group means.
  • Describe how the Shapiro-Wilk test statistic and p-value are interpreted in the context of the one-way ANOVA normality assumption.
    • The Shapiro-Wilk test statistic, 'W', ranges from 0 to 1, with values closer to 1 indicating that the data is more likely to have come from a normal distribution. The p-value of the Shapiro-Wilk test represents the probability of observing the given test statistic (or a more extreme value) if the null hypothesis of normality is true. If the p-value is less than the chosen significance level (e.g., 0.05), the null hypothesis is rejected, and it is concluded that the data does not follow a normal distribution. This violation of the normality assumption would then need to be addressed before proceeding with the one-way ANOVA analysis.
  • Analyze the implications of violating the normality assumption in one-way ANOVA and discuss potential alternative analyses that could be used in such cases.
    • Violating the normality assumption in one-way ANOVA can lead to inaccurate p-values and potentially incorrect conclusions about the significance of the differences between group means. This is because the ANOVA test is based on the assumption that the data in each group follows a normal distribution. If this assumption is violated, the test may not be robust, and the results may not be reliable. In such cases, alternative non-parametric tests, such as the Kruskal-Wallis test, can be used. These non-parametric tests do not rely on the normality assumption and can provide more accurate results when the data does not follow a normal distribution. Additionally, transforming the data (e.g., log transformation) or using robust ANOVA methods that are less sensitive to non-normality may also be considered to address the violation of the normality assumption.

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