Kolmogorov-Smirnov or Shapiro-Wilk: which normality test to use?

Direct answer: The Kolmogorov-Smirnov test checks whether a sample follows a stated distribution. In a dissertation it is most often used to ask whether your data are normal. It compares the sample against the reference distribution and flags a gap that is too large to put down to chance.

NIST/SEMATECH (n.d.) states the hypotheses plainly: the null hypothesis is that the data follow a specified distribution, and the alternative is that they do not. The reference distribution must be continuous. The same page lists what it regards as the most serious limitation, that the distribution has to be fully specified. If you estimate the mean and standard deviation from your own sample, as every thesis student does when testing for normality, the standard critical values are no longer valid and would have to be found by simulation.

The estimated-parameter problem is the reason a corrected version exists. Lilliefors (1967) published the adjustment for the case where the mean and variance are unknown, and SPSS applies it in its normality output. IBM (n.d.-a) documents that Explore reports the Kolmogorov-Smirnov statistic with a Lilliefors significance level alongside the Shapiro-Wilk statistic. So when you quote "the K-S test" from SPSS Explore, you are quoting the Lilliefors-corrected version, which matters when a supervisor asks how you ran it.

Direct answer: For most thesis datasets, the sources we checked favour Shapiro-Wilk. Razali and Wah (2011) found it the most powerful of the four tests they compared and Kolmogorov-Smirnov the least. Mishra et al. (2019) pair Shapiro-Wilk with samples under 50 and Kolmogorov-Smirnov with 50 or more.

Razali and Wah (2011) compared Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and Anderson-Darling by simulation. Their conclusion reads: "Shapiro-Wilk test is the most powerful test for all types of distribution and sample sizes whereas Kolmogorov-Smirnov test is the least powerful test" (Razali & Wah, 2011, p. 32). Power here means the chance of catching a real departure from normality. They also found that the Lilliefors version always outperformed plain Kolmogorov-Smirnov, which supports using SPSS's corrected output and not a hand-built comparison.

Ghasemi and Zahediasl (2012) go further. Their guide for non-statisticians describes Shapiro-Wilk as better powered than Kolmogorov-Smirnov even after the Lilliefors correction, and they argue against Kolmogorov-Smirnov when parameters are estimated from the data. Their stance is stronger than many statisticians would take, so cite it as the authors' recommendation and not as a rule every field accepts. Ghasemi and Zahediasl (2012) recommend Shapiro-Wilk as the test to use.

Neither test rescues a tiny sample. Razali and Wah (2011) warn that all four tests perform poorly at sample sizes of 30 and below, so with such a small n, read a normality plot as well and report your reasoning.

Direct answer: Shapiro-Wilk is the usual first choice for small and medium samples, while Kolmogorov-Smirnov is printed beside it in the same SPSS table. Both tests are sensitive to sample size, and both should be read together with a Q-Q plot or histogram.

The two rows on sample size and sensitivity are where students most often misread the table. Small samples most often pass because the test lacks the power to object (Ghasemi & Zahediasl, 2012), and a very large sample can fail because the test objects to differences no one would see on a chart. Ghasemi and Zahediasl (2012) make the second point: with a large sample, even a small deviation from normality can give a significant result that need not matter for parametric tests. That is why Kent State University Libraries (2026) tells readers not to rely on the test alone and to look at graphs too.

Direct answer: Open Analyze, then Descriptive Statistics, then Explore, move your variable into the Dependent List, and tick Normality plots with tests under Plots. SPSS prints a Tests of Normality table with both Kolmogorov-Smirnov and Shapiro-Wilk, plus Q-Q plots you can check by eye.

Both Ghasemi and Zahediasl (2012) and Mishra et al. (2019) describe this path. Ghasemi and Zahediasl add that the Kolmogorov-Smirnov row in this table carries the Lilliefors significance correction. Kent State University Libraries (2026) show the same table with the columns Statistic, df and Sig., and a footnote naming the Lilliefors correction.

One trap is worth knowing about. IBM's support notes say the older One-Sample Kolmogorov-Smirnov option under Nonparametric Tests did not apply the Lilliefors correction before version 21.0.0.1, so the p-values from the two menus can differ for the same data (IBM, n.d.-b). If your university guide or a classmate used the legacy menu, the numbers in your table may not match theirs even though the data are identical. State in your methods chapter which menu you used, and keep the syntax or output file so the result can be reproduced.

If your data are grouped, move the grouping variable into the Factor List so SPSS tests each group separately. Normality is usually judged within groups, not across the whole column.

Direct answer: There is no good or bad statistic value on its own. You read the Sig. column. A p-value above your alpha, usually .05, means you do not reject normality; a p-value at or below it is evidence that the data depart from normality. Always add a plot before you decide.

NIST/SEMATECH (n.d.) describes the Kolmogorov-Smirnov statistic as a measure of the largest gap between the sample's cumulative distribution and the reference distribution. A smaller gap means a closer fit, but the number only has meaning next to its df and p-value, so quote all three.

Kent State University Libraries (2026) give the wording for a p-value above alpha: you do not reject the null hypothesis, and there is not enough evidence to conclude that the data are non-normal. Mishra et al. (2019) say the null hypothesis is "accepted" when P is above .05, but strictly the test cannot prove normality, so write "did not reject" or "found no evidence against normality". Kent State University Libraries (2026) also note that the two tests can disagree, and their worked SPSS table shows it. For Height, Kolmogorov-Smirnov gives .045 (df 408, Sig. .049) while Shapiro-Wilk gives .993 (df 408, Sig. .070), so one test rejects normality at .05 and the other does not. For Weight, both reject: Kolmogorov-Smirnov .084 and Shapiro-Wilk .944, both with df 376 and Sig. .000. With 376 cases that is the large-sample situation where a plot decides whether the departure matters.

For reporting, MAAS suggests an APA-style line such as, using the Kent State Weight example above, "A Shapiro-Wilk test indicated a departure from normality, W(376) = .944, p < .001". This format is our suggestion for APA-style reporting, and none of the sources above prescribes it, so check your department's guide.

Direct answer: Do not abandon your planned analysis automatically. Check the plot, check the group sizes, then decide between a parametric test you can defend, a transformation, or a non-parametric alternative. Say in your methods chapter which route you took and why it fits.

Ghasemi and Zahediasl (2012) note that a significant result from a large sample need not stop you using a parametric test, so a Q-Q plot, skewness and kurtosis, and the size of your groups all belong in the decision. Razali and Wah (2011) recommend combining graphs with a formal test and with skewness and kurtosis. For a deeper treatment of which test follows a failed assumption, see our guide to help with thesis statistics, which separates advice you may ask for from analysis that must stay yours.