6SQ ToolsWhat Is a Normality Test and Why Does It Matter in Quality Control?
If you work with statistical process control (SPC), capability analysis, or hypothesis testing, you've probably heard that your data should be "normally distributed." But how do you actually check that assumption? That's where a normality test comes in.
A normality test is a statistical procedure used to determine whether a set of sample data comes from a population that follows a normal (Gaussian) distribution. Many common quality tools—such as control charts for individual values, process capability indices (Cp, Cpk), and t-tests—rely on the assumption of normality. If your data are heavily skewed or have outliers, the results of those analyses can be misleading.
What It Is
In plain terms, a normality test answers one question: How likely is it that this sample was drawn from a normally distributed population? It does this by comparing the shape of your data's distribution (its histogram, skewness, and kurtosis) against the theoretical bell curve.
There are several well-known normality tests, each with slightly different strengths:
All of these tests share the same logic: they calculate a test statistic, then compare it to a known distribution to produce a p-value.
How It Works: The Steps
Regardless of which test you use, the procedure follows a standard framework:
- Null hypothesis (H₀): The data are normally distributed.
- Alternative hypothesis (H₁): The data are not normally distributed.
Each test has its own formula. For example, the Shapiro–Wilk statistic (W) measures how well the ordered sample values fit a straight line in a normal probability plot. The Anderson–Darling statistic (A²) gives extra weight to the tails of the distribution.
The p-value is the probability of observing a test statistic as extreme as (or more extreme than) the one you calculated, assuming the null hypothesis is true.
Choose a significance level (commonly α = 0.05).
- If p-value ≥ α, you fail to reject H₀. You have no strong evidence that the data are non-normal.
- If p-value < α, you reject H₀. The data are significantly non-normal.
A Worked Illustrative Example
Example data (illustrative only): Suppose you collect 30 measurements of a machined part's diameter (in mm). You run a Shapiro–Wilk test and obtain a test statistic of W = 0.972 with a p-value of 0.58.
Now imagine a second sample of 30 parts from a different process gives W = 0.901 with a p-value of 0.009.
Common Pitfalls
When to Use a Normality Test
Use a normality test whenever you plan to:
If your data fail the test, you still have options: transform the data (e.g., Box–Cox), use non-parametric tests, or model the distribution explicitly (e.g., Weibull or lognormal).
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Ready to check your own data? Use the free, interactive normality test tool at https://www.6sq.com/tools/normaltest/ — paste your data, choose your preferred test, and get an immediate p-value and probability plot to support your quality decisions.
A normality test is a statistical procedure used to determine whether a set of sample data comes from a population that follows a normal (Gaussian) distribution. Many common quality tools—such as control charts for individual values, process capability indices (Cp, Cpk), and t-tests—rely on the assumption of normality. If your data are heavily skewed or have outliers, the results of those analyses can be misleading.
What It Is
In plain terms, a normality test answers one question: How likely is it that this sample was drawn from a normally distributed population? It does this by comparing the shape of your data's distribution (its histogram, skewness, and kurtosis) against the theoretical bell curve.
There are several well-known normality tests, each with slightly different strengths:
- Shapiro–Wilk test: Generally considered the most powerful for small to medium sample sizes (n < 50).
- Anderson–Darling test: Sensitive to deviations in the tails of the distribution; commonly used in Minitab.
- Kolmogorov–Smirnov test: A general goodness-of-fit test, but less sensitive to tail differences.
- Jarque–Bera test: Based on skewness and kurtosis; often used in econometrics.
All of these tests share the same logic: they calculate a test statistic, then compare it to a known distribution to produce a p-value.
How It Works: The Steps
Regardless of which test you use, the procedure follows a standard framework:
- State the hypotheses.
- Null hypothesis (H₀): The data are normally distributed.
- Alternative hypothesis (H₁): The data are not normally distributed.
- Calculate the test statistic.
Each test has its own formula. For example, the Shapiro–Wilk statistic (W) measures how well the ordered sample values fit a straight line in a normal probability plot. The Anderson–Darling statistic (A²) gives extra weight to the tails of the distribution.
- Determine the p-value.
The p-value is the probability of observing a test statistic as extreme as (or more extreme than) the one you calculated, assuming the null hypothesis is true.
- Make a decision.
Choose a significance level (commonly α = 0.05).
- If p-value ≥ α, you fail to reject H₀. You have no strong evidence that the data are non-normal.
- If p-value < α, you reject H₀. The data are significantly non-normal.
Note on the normal probability plot: A quick visual check complements any formal test. If your points fall roughly along the diagonal reference line, normality is plausible. Formal tests are still recommended because visual judgment can be subjective, especially with small samples.
A Worked Illustrative Example
Example data (illustrative only): Suppose you collect 30 measurements of a machined part's diameter (in mm). You run a Shapiro–Wilk test and obtain a test statistic of W = 0.972 with a p-value of 0.58.
- Since 0.58 > 0.05, you fail to reject H₀.
- Conclusion: There is no significant evidence that the diameter data deviate from a normal distribution. You may proceed with capability analysis or parametric tests.
Now imagine a second sample of 30 parts from a different process gives W = 0.901 with a p-value of 0.009.
- Since 0.009 < 0.05, you reject H₀.
- Conclusion: The data are significantly non-normal. You should investigate the cause (e.g., tool wear, mixed batches) or consider non-parametric alternatives or data transformations before proceeding.
Common Pitfalls
- Small samples lack power. With n < 10, most normality tests rarely detect non-normality even when it exists. Don't over-interpret a "pass" on tiny samples.
- Large samples over-detect. With n > 200, even trivial deviations from normality can produce a significant p-value. In such cases, check the normal probability plot and the actual magnitude of skewness—practical significance matters more than statistical significance.
- Ignoring outliers. A single extreme value can break normality. Always plot your data first (histogram, boxplot) to spot anomalies before running the test.
- Testing the wrong variable. Normality applies to the residuals in regression or ANOVA, not necessarily to the raw response variable. Make sure you test the right data.
When to Use a Normality Test
Use a normality test whenever you plan to:
- Calculate Cp, Cpk, or Pp, Ppk for process capability.
- Use individual (I-MR) control charts.
- Run a t-test, ANOVA, or regression that assumes normal errors.
- Decide between parametric and non-parametric methods.
If your data fail the test, you still have options: transform the data (e.g., Box–Cox), use non-parametric tests, or model the distribution explicitly (e.g., Weibull or lognormal).
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Ready to check your own data? Use the free, interactive normality test tool at https://www.6sq.com/tools/normaltest/ — paste your data, choose your preferred test, and get an immediate p-value and probability plot to support your quality decisions.
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