SPSS Kolmogorov-Smirnov Test for Normality

An alternative normality test is the Shapiro-Wilk test.

SPSS Npar Tests One Sample Kolmogorov Smirnov Output

What is a Kolmogorov-Smirnov normality test?

The Kolmogorov-Smirnov test examines if scores
are likely to follow some distribution in some population.
For avoiding confusion, there's 2 Kolmogorov-Smirnov tests:

In theory, “Kolmogorov-Smirnov test” could refer to either test (but usually refers to the one-sample Kolmogorov-Smirnov test) and had better be avoided. By the way, both Kolmogorov-Smirnov tests are present in SPSS.

Kolmogorov-Smirnov Test - Simple Example

So say I've a population of 1,000,000 people. I think their reaction times on some task are perfectly normally distributed. I sample 233 of these people and measure their reaction times.
Now the observed frequency distribution of these will probably differ a bit -but not too much- from a normal distribution. So I run a histogram over observed reaction times and superimpose a normal distribution with the same mean and standard deviation. The result is shown below.

Kolmogorov Smirnov One Sample Test What Is It Histogram

The frequency distribution of my scores doesn't entirely overlap with my normal curve. Now, I could calculate the percentage of cases that deviate from the normal curve -the percentage of red areas in the chart. This percentage is a test statistic: it expresses in a single number how much my data differ from my null hypothesis. So it indicates to what extent the observed scores deviate from a normal distribution.
Now, if my null hypothesis is true, then this deviation percentage should probably be quite small. That is, a small deviation has a high probability value or p-value.
Reversely, a huge deviation percentage is very unlikely and suggests that my reaction times don't follow a normal distribution in the entire population. So a large deviation has a low p-value. As a rule of thumb, we reject the null hypothesis if p < 0.05. So if p < 0.05, we don't believe that our variable follows a normal distribution in our population.

Kolmogorov-Smirnov Test - Test Statistic

So that's the easiest way to understand how the Kolmogorov-Smirnov normality test works. Computationally, however, it works differently: it compares the observed versus the expected cumulative relative frequencies as shown below.

Kolmogorov-Smirnov Test - Line Chart of Observed and Expected Relative Frequencies

The Kolmogorov-Smirnov test uses the maximal absolute difference between these curves as its test statistic denoted by D. In this chart, the maximal absolute difference D is (0.48 - 0.41 =) 0.07 and it occurs at a reaction time of 960 milliseconds. Keep in mind that D = 0.07 as we'll encounter it in our SPSS output in a minute.

The Kolmogorov-Smirnov test in SPSS

There's 2 ways to run the test in SPSS:

We'll demonstrate both methods using speedtasks.sav throughout, part of which is shown below.

SPSS Kolmogorov Smirnov Test One Sample Variable View

Our main research question is which of the reaction time variables is likely
to be normally distributed in our population?
These data are a textbook example of why you should thoroughly inspect your data before you start editing or analyzing them. Let's do just that and run some histograms from the syntax below.

*Run basic histograms for inspecting if distributions look plausible.

frequencies r01 to r05
/format notable
/histogram normal.

*Note that some distributions do not look plausible at all!


SPSS One Sample Kolmogorov Smirnov Test Histogram

Note that some distributions do not look plausible at all. But which ones are likely to be normally distributed?

SPSS Kolmogorov-Smirnov test from NPAR TESTS

Our preferred option for running the Kolmogorov-Smirnov test is under Analyze SPSS Menu Arrow Nonparametric Tests SPSS Menu Arrow Legacy Dialogs SPSS Menu Arrow 1-Sample K-S... as shown below.

SPSS Kolmogorov Smirnov Test Nonparametric Legacy Dialogs Menu

Next, we just fill out the dialog as shown below.

SPSS Kolmogorov Smirnov Test Nonparametric Dialog

Clicking Paste results in the syntax below. Let's run it.

Kolmogorov-Smirnov Test Syntax from Nonparametric Tests

*One-sample Kolmogorov-Smirnov test from analyze - nonparametric tests - legacy dialogs - 1 sample ks-test.

/K-S(NORMAL)=r01 r02 r03 r04 r05

*Only reaction time 4 has p > 0.05 and thus seems normally distributed in population.


SPSS Npar Tests One Sample Kolmogorov Smirnov Output

First off, note that the test statistic for our first variable is 0.073 -just like we saw in our cumulative relative frequencies chart a bit earlier on. The chart holds the exact same data we just ran our test on so these results nicely converge.
Regarding our research question: only the reaction times for trial 4 seem to be normally distributed.

SPSS Kolmogorov-Smirnov test from EXAMINE VARIABLES

An alternative way to run the Kolmogorov-Smirnov test starts from Analyze SPSS Menu Arrow Descriptive Statistics SPSS Menu Arrow Explore as shown below.

SPSS Kolmogorov Smirnov Test Analyze Explore Menu SPSS Kolmogorov Smirnov Test Explore Dialog

Kolmogorov-Smirnov Test Syntax from Nonparametric Tests

*One-sample Kolmogorov-Smirnov test from analyze - descriptive statistics - explore.

EXAMINE VARIABLES=r01 r02 r03 r04 r05

*Shorter version.

EXAMINE VARIABLES r01 r02 r03 r04 r05
/missing pairwise /*IMPORTANT!*/.


SPSS Kolmogorov Smirnov Test Analyze Explore Output

As a rule of thumb, we conclude that a variable is not normally distributed if “Sig.” < 0.05. So both the Kolmogorov-Smirnov test as well as the Shapiro-Wilk test results suggest that only Reaction time trial 4 follows a normal distribution in the entire population.
Further, note that the Kolmogorov-Smirnov test results are identical to those obtained from NPAR TESTS.

Reporting a Kolmogorov-Smirnov Test

For reporting our test results following APA guidelines, we'll write something like “a Kolmogorov-Smirnov test indicates that the reaction times on trial 1 do not follow a normal distribution, D(233) = 0.07, p = 0.005.” For additional variables, try and shorten this but make sure you include

Kolmogorov-Smirnov Test Reporting in APA Style

Wrong Results in SPSS?

If you're a student who just wants to pass a test, you can stop reading now. Just follow the steps we discussed so far and you'll be good.

Right, now let's run the exact same tests again in SPSS version 18 and take a look at the output.

SPSS One Sample Kolmogorov Smirnov Test Output V18 Exact Tests

In this output, the exact p-values are included and -fortunately- they are very close to the asymptotic p-values. Less fortunately, though, the SPSS version 18 results are wildly different
from the SPSS version 24 results
we reported thus far.
The reason seems to be the Lilliefors significance correction which is applied in newer SPSS versions. The result seems to be that the asymptotic significance levels differ much more from the exact significance than they did when the correction is not implied. This raises serious doubts regarding the correctness of the “Lilliefors results” -the default in newer SPSS versions.
Converging evidence for this suggestion was gathered by my colleague Alwin Stegeman who reran all tests in Matlab. The Matlab results agree with the SPSS 18 results and -hence- not with the newer results.

One-Sample Kolmogorov-Smirnov Test - Matlab Output

Kolmogorov-Smirnov normality test - Limited Usefulness

The Kolmogorov-Smirnov test is often to test the normality assumption required by many statistical tests such as ANOVA, the t-test and many others. However, it is almost routinely overlooked that such tests are robust against a violation of this assumption if sample sizes are reasonable, say N ≥ 25.The underlying reason for this is the central limit theorem. Therefore, normality tests are only needed for small sample sizes if the aim is to satisfy the normality assumption.
Unfortunately, small sample sizes result in low statistical power for normality tests. This means that substantial deviations from normality will not result in statistical significance. The test says there's no deviation from normality while it's actually huge. In short, the situation in which normality tests are needed -small sample sizes- is also the situation in which they perform poorly.

Thanks for reading.

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