Checking normality in R: Shapiro–Wilk and Q–Q plots

How to check the normality assumption in R the right way — combining the Shapiro–Wilk test with a Q–Q plot, and why the plot usually matters more than the p-value.

Many common tests assume roughly normal data (or normal residuals). Here’s how to check that assumption in R without over-relying on any single number. This is exactly the judgement call behind question 5 of our which-test tool.

Two tools, used together

• Shapiro–Wilk test (shapiro.test) — a formal test of the null “the data are normal.”
• Q–Q plot (qqnorm + qqline) — a visual check: normal data hug the diagonal line.

Use both, and trust the plot when they seem to disagree.

set.seed(5)
x <- rnorm(80, mean = 10, sd = 2)

shapiro.test(x)

    Shapiro-Wilk normality test

data:  x
W = 0.98151, p-value = 0.3009

par(mfrow = c(1, 2), mar = c(4, 4, 2, 1), family = "sans")
hist(x, col = "#2f6fed33", border = "#1b2a4a", main = "Histogram", xlab = "x")
qqnorm(x, col = "#17a2b8", pch = 19, main = "Normal Q-Q plot")
qqline(x, col = "#b02a37", lwd = 2)

Figure 1

Points close to the red line and a bell-shaped histogram: normality is a reasonable assumption here.

The big caveat: sample size

The Shapiro–Wilk test is sensitive to sample size. In small samples it can miss real non-normality; in very large samples it flags trivial, harmless departures as “significant.” So:

• Large n: a significant Shapiro–Wilk often doesn’t matter — the test you’re running is robust, and the Q–Q plot looks fine. Don’t panic over the p-value.
• Small n: the test has little power, so lean on the Q–Q plot and subject-matter knowledge.

If it’s not normal

You have good options: a nonparametric test (Mann–Whitney, Kruskal–Wallis), a transformation (log, square-root), or a method that doesn’t need normality of the raw data. For regression, remember it’s the residuals that should be normal, not the predictors.

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Assumption checks are where analyses quietly go wrong. We make them explicit.