How to Report a Normality Check in APA 7

Checking normality is the step everyone performs and almost nobody writes up. What reaches the reviewer is “normality was checked” and nothing else: no statistic, no degrees of freedom, no p, and crucially no account of what happened when the test came out significant. This is what that paragraph needs, in what order, and how the decision is worded. What a Shapiro-Wilk p actually means, and why it is not a traffic light, I cover separately in how Shapiro-Wilk gets misread; this article is about the reporting. The wider map of assumptions is in how to check statistical assumptions.

What the normality paragraph has to contain

Three required pieces and two that are required in practice. Required: the test statistic (W for Shapiro-Wilk, D for Kolmogorov-Smirnov), its degrees of freedom and an exact p value to three decimal places. The other two: skewness and kurtosis with their standard errors, and one sentence about the visual inspection. With those five things a reader can judge the departure themselves; with “normality was checked” they can judge nothing.

The degrees of freedom for Shapiro-Wilk are simply the n of the analysis, and that is exactly what SPSS prints in the df column of the Tests of Normality table. Strictly, W has no degrees of freedom the way a t does: what goes in the brackets is the number of cases, which SPSS labels df for the convenience of the table. R does not print them at all — shapiro.test returns only W and p-value — so you add the n yourself. If you check normality within each group, every group gets its own row and its own n, and all of them belong in the manuscript, not just the one that behaved.

Leading zeros are where most manuscripts slip. W ranges from 0 to 1 and so does D, so neither takes one: W = .98, D = .07. Skewness and kurtosis can exceed 1, so they keep it: skewness = 0.44, kurtosis = 0.24, and the same for their standard errors, SE = 0.17. It is a trivial distinction and it is one of the things copy editors fix by hand most often.

The paragraph belongs at the top of the Results, in a block usually headed Preliminary analyses or Data screening, and it has a twin in the data-analysis subsection of the Method: that is where you say what you were going to look at and by what criterion, before you had seen a single p value. Writing the criterion after seeing the result is precisely what an experienced reviewer can smell from a distance, and what preregistration was invented to stop.

Shapiro-Wilk or Kolmogorov-Smirnov

SPSS prints both tests in the same table, and that is where the trouble starts, because they often disagree and the temptation is to keep whichever one lets the analysis through. The convention in psychology and the health sciences is Shapiro-Wilk: it has more power than Kolmogorov-Smirnov across essentially the whole range of sample sizes we work with. Choose before you look at the output, say so in the Method, and report that one.

If you do report the SPSS Kolmogorov-Smirnov, you have to state that it carries the Lilliefors significance correction, because it is not the classical K-S test. The classical version assumes the mean and standard deviation of the reference distribution are known, whereas yours were estimated from the sample itself; without the correction the test is far too conservative. The note is one line: “Kolmogorov-Smirnov with Lilliefors significance correction, D(214) = .07, p = .012”.

What you cannot do is report both tests and then discuss only the convenient one, or switch tests between variables within the same study. If your normality table shows Shapiro-Wilk for four measures and Kolmogorov-Smirnov for two, someone will ask why, and the honest answer is rarely a good one.

There is a third route that is perfectly publishable: run no formal test at all and decide on skewness, kurtosis and the Q-Q plot. A growing number of reviewers prefer it, and with large samples it is the most defensible of the three. What is not an option is writing nothing, because a missing paragraph reads as a check you never carried out.

Skewness, kurtosis and the Q-Q plot

Skewness and kurtosis are reported with their standard errors, never bare. The standard error depends only on sample size — at N = 214 it comes out at SE = 0.17 for skewness and SE = 0.33 for kurtosis — and without it the reader cannot tell whether 0.44 is a lot or nothing. SPSS prints them next to the values in the Descriptives table of the Explore procedure; in R, describe() in psych gives you skewness and kurtosis and you compute the standard error from the formula, which needs nothing but the n; jamovi prints both with their standard errors in the Descriptives panel.

The ratio of the value to its standard error is read as a z, and that is how the sample-size problem sneaks back in: with 214 cases, a skewness of 0.44 gives a ratio of 2.6 and “comes out significant” even though the histogram is perfectly presentable. The most widely used working convention is the absolute one — skewness below 2 and kurtosis below 7, proposed by West and colleagues in 1995. It is a convention, and it should be presented as one: orientation, not a legal threshold.

For the Q-Q plot, write what you saw, not that it exists. “A Q-Q plot was inspected” carries no information; “points lay close to the diagonal except in the upper tail, where nine cases departed from it” does, and it tells the reader exactly where the departure lives. If that inspection is what your decision rests on, the plot goes in as a numbered figure or into the supplementary materials with its caption.

The histogram with a normal curve superimposed is genuinely useful to you and almost never to the paper: it eats half a page and says less than two well-chosen numbers. Keep it for the thesis, where an examiner does expect to see it, or for the supplement. In a manuscript the Q-Q plot earns far more per square centimeter.

A worked example

“Distributions were examined before the main analyses. Perceived stress scores (PSS-10) in the full sample (N = 214, M = 19.64, SD = 6.85) showed skewness of 0.44 (SE = 0.17) and kurtosis of 0.24 (SE = 0.33). The Shapiro-Wilk test, specified in the analysis plan, was significant, W(214) = .98, p = .004. The Q-Q plot showed points close to the diagonal except in the upper tail, where nine cases departed from it, and the 5% trimmed mean (19.42) differed from the mean by 0.22 points. Given the sample size, the magnitude of the skew and the robustness of the t test to departures of this order, the planned parametric analyses were retained.”

The order is not decorative. The variable is described first, then the size of the departure, then the formal test with its three numbers, then what the plot showed, and the decision with its reasons last. Written that way, a significant p stops being a problem and becomes one data point among four. Written the other way round — the p first and on its own — the paragraph reads like a confession.

Note the 5% trimmed mean as well. It is a cheap trick and a very persuasive one: SPSS prints it in the Explore table whether you ask for it or not, and if it barely moves away from the mean you have shown that the extreme cases are not dragging the center of the distribution anywhere. Two figures and a comma that save you a paragraph of argument.

If what you are looking at is the SPSS Tests of Normality table with its two blocks and you cannot tell which figure goes where, run it through the normality interpreter and it gives you the reading and the sentence in exactly this format.

Why a significant p in a large sample is not a decision

This is the heart of it, and the reason the paragraph is written the way you have just seen. Shapiro-Wilk tests whether your data are statistically distinguishable from a perfect normal distribution, not whether they are close enough for the analysis you intend to run. At N = 214, a W of .98 — an excellent fit — already returns p = .004. And the reverse holds: with n = 26 and a skewness of 1.18 that is plainly visible in the histogram, the same test returns W(26) = .94, p = .134 and waves you through.

The consequence for the write-up is concrete: the sentence cannot end on the p value. If it ends “p = .004, so non-parametric tests were used”, a reviewer is entitled to ask why you gave up statistical power over a departure nobody has been shown. And if it ends “p = .004, but parametric tests were retained”, it reads as though you are ignoring your own result without saying why.

The version that works names three things besides the test: the sample size, the magnitude of the departure (skewness, kurtosis, what the Q-Q plot showed) and what a departure of that size does to the analysis you are about to run. “Given the sample size and a skewness below 1, ANOVA was retained, as it is robust to departures of this magnitude with balanced groups” is two lines that close the question permanently.

And if you genuinely do decide to change analysis, justify that with data too rather than with the p. “The distribution showed a pronounced floor effect, with 41% of the sample at the minimum score, so rank-based tests were used” is a justification; “Shapiro-Wilk was significant” is not. How to write up whatever comes next is in how to report non-parametric tests.

What reviewers flag

“Normality was checked” and nothing more. No statistic, no degrees of freedom, no p, and no way of knowing which test was run. It is the statistical equivalent of saying nothing, and at journals with a methods checklist it gets returned before it reaches review.

Checking the raw outcome variable in a regression. The assumption concerns the model residuals, not the original Y. A two-humped Y can produce immaculate residuals if your predictors account for the two humps, and a beautifully symmetrical Y can hide residuals with an impossible tail.

Checking the whole sample when you are comparing groups. For a t test or an ANOVA the assumption applies within each group, or equivalently to the residuals. Two groups with different means widen when pooled, and look bimodal once the gap is large enough, precisely because the effect you are testing for is real, so the normality test ends up measuring the effect rather than the shape.

Formatting: W without degrees of freedom, and misplaced zeros. Without the n the W cannot be interpreted: the same .98 returns p = .004 with 214 cases and p = .83 with 30, because the p value comes from combining the statistic with the sample size. And W = 0.98 alongside skewness = .44 has both leading zeros exactly the wrong way round.

Showing normality only for the variables where it went well. If the study measures six things and three appear in the manuscript, it shows. Either the normality table covers every variable, or there is no table and one honest sentence saying what you looked at.

Treating the assumption as an entry condition. Writing “the data met the assumption of normality (p = .21)” with n = 26 claims something the test cannot deliver: failing to reject is not proof, and at that sample size the test has very little power. “No appreciable departures from normality were detected” is what you can defensibly write.

Frequently asked questions

How do you report a Shapiro-Wilk test in APA 7?

With the W statistic, its degrees of freedom and an exact p value: W(214) = .98, p = .004. The degrees of freedom are the n of the analysis, and W takes no leading zero because it cannot exceed 1. Skewness and kurtosis with their standard errors go alongside it, plus a sentence on what the Q-Q plot showed, and the paragraph ends on your decision rather than on the p value.

What are the degrees of freedom for Shapiro-Wilk?

They are simply the n of the analysis. SPSS prints them in the df column of the Tests of Normality table, matching the number of valid cases for that variable; R's shapiro.test does not print them at all, so you supply the n yourself. If you check normality within groups, each row carries the n of its own group and all of them belong in the write-up.

Should I report both Shapiro-Wilk and Kolmogorov-Smirnov?

No. Pick one before you look at the output, state it in the analysis plan and report that one. Shapiro-Wilk is the convention in psychology because it has more power at the sample sizes we typically work with. If you report the SPSS Kolmogorov-Smirnov, you must say it carries the Lilliefors significance correction, because it is not the classical K-S test.

What should I do if Shapiro-Wilk is significant in a large sample?

Write the full paragraph instead of switching analysis by reflex. At N = 214 a W of .98, which is an excellent fit, already gives p = .004: the test is detecting departures with no practical consequence. The wording that convinces a reviewer names the sample size, the magnitude of the departure (skewness, kurtosis, the Q-Q plot) and what a departure of that size does to the test you intend to run.

Do you test normality of the data or of the residuals?

In regression, the residuals of the model, never the raw outcome variable. For a t test or an ANOVA, within each group or, equivalently, on the residuals: the pooled sample mixes groups with different means and departs from normality precisely because the effect exists. It is one of the most commonly flagged errors in review.

What skewness and kurtosis values are acceptable?

The most cited working convention, from West and colleagues (1995), puts the practical limit at 2 for skewness and 7 for kurtosis in absolute value. Below those, parametric tests with reasonable sample sizes hold up well. It is a convention rather than a rule, so present it as one and always report the standard error alongside each statistic.

With the statistic, its degrees of freedom, an exact p, skewness and kurtosis with their standard errors and one sentence explaining the decision, your normality check is reported the way APA 7 expects. If the distribution gets genuinely awkward — floor effects, transformations that change the interpretation, models that do not assume normality at all — we can work through it together in my statistical consulting.