How do you test for normality?

Use a combination of the Shapiro-Wilk test, a visual Q-Q plot or histogram, and the skewness and kurtosis values. Relying on any one of these alone can be misleading, so read them together.

What does the Shapiro-Wilk test tell you?

It tests whether a sample departs from a normal distribution. A p-value above .05 means there is no evidence against normality, so the assumption holds, while a value below .05 suggests a departure.

What should I do if my data is not normally distributed?

You can switch to a nonparametric test such as the Mann-Whitney U, transform the variable with a log or square-root function, or rely on the robustness of t-tests and ANOVA when the sample is reasonably large. Document whichever route you choose.

Does my raw data have to be normal?

Usually not. For t-tests and ANOVA the assumption concerns the distribution within groups or the residuals, and for regression it concerns the model residuals, rather than the raw data overall.

How to test for normality (and what to do when data is not normal)

You test for normality to decide whether your data meets the assumption behind parametric tests such as the t-test, ANOVA, and regression. You check it with a combination of a statistical test (Shapiro-Wilk), a visual plot (a Q-Q plot or histogram), and the skewness and kurtosis values, never on one of these alone.

What is actually assumed to be normal

A frequent misunderstanding sends researchers down the wrong path: most tests do not require your raw data to be normal. For a t-test or ANOVA the assumption is about the residuals or the distribution within each group, and for regression it is about the residuals of the model. Checking the right quantity matters, because perfectly valid data can look non-normal until you examine residuals. If the underlying test choice is still unsettled, start with choosing a statistical test before worrying about this assumption.

The Shapiro-Wilk test in SPSS and R

The Shapiro-Wilk test is the standard formal check. A p-value above .05 means you do not have evidence against normality, so the assumption holds; below .05 suggests a departure. In SPSS it appears under Analyze, Descriptive Statistics, Explore, in the Tests of Normality table. In R it is a single line, shapiro.test(x). Use Shapiro-Wilk rather than Kolmogorov-Smirnov for the sample sizes typical of a dissertation.

Why you should not rely on the test alone

The significance test has a known weakness: with a large sample it flags trivial, harmless deviations as significant, and with a small sample it can miss real ones. That is why a Q-Q plot matters. If the points fall close to the diagonal line, the data is near-normal regardless of what the p-value says. Read the plot and the test together, and bring in skewness (values roughly within -1 to 1 are generally fine) as a tiebreaker.

Reading a Q-Q plot, point by point

A Q-Q plot compares your data's quantiles against the quantiles a perfectly normal distribution would produce, drawing one point per observation against a reference diagonal. When the points sit on the diagonal, the observed values match the expected normal values closely, which is the visual signature of a near-normal variable. Small wobbles around the line are ordinary sampling noise and not a cause for concern.

The shape of the departures tells you what kind of problem you have. An S-shaped curve, where the points bend off the line at one end, points to skew, with the data piling up on one side. Points that curl away at both tails, sitting above the line at the top and below it at the bottom, signal heavy tails, meaning more extreme values than a normal distribution expects. A plot reads better than a single p-value precisely because it shows you the direction and severity of the deviation rather than collapsing everything into one pass-or-fail number.

On a Q-Q plot, points that track the diagonal reference line signal a near-normal variable; systematic bends away from it reveal skew or heavy tails.

How sample size changes the decision

Sample size quietly drives how you should weigh the evidence. With a large sample, the Shapiro-Wilk test gains the power to flag deviations so small they have no practical effect on your results, so a significant p-value there often means very little. With a small sample, the same test loses power and can wave through genuinely non-normal data, giving false reassurance.

The practical rule is to shift your reliance with the sample. In small samples, lean on the Q-Q plot and informed judgement, because the formal test is weak. In large samples, lean on the plot and the central limit theorem, which makes the sampling distribution of the mean approximately normal regardless of the raw distribution, rather than over-reacting to a significant but trivial test result. Either way, the plot stays in the picture; the test never decides alone.

What to do when your data is not normal

A failed normality check is not a dead end. You have several legitimate routes:

Use a nonparametric test: a Mann-Whitney U for a t-test, a Kruskal-Wallis for ANOVA, a Spearman correlation for Pearson.
Transform the variable: a log or square-root transform can normalise skewed data, though it changes interpretation.
Rely on robustness: with a reasonably large sample, t-tests and ANOVA tolerate moderate non-normality thanks to the central limit theorem.

Choose the route deliberately and document why, because an examiner will ask. Whichever you pick, report the assumption check itself, following how to report statistics in APA style.

Build the check into your workflow

Assumption testing belongs early in your analysis, right after data cleaning and before you run the main test, not as an afterthought once results are in. Doing it in that order keeps your results chapter honest and defensible. For doctoral models with more demanding assumptions, see PhD statistics help, and to have the full sequence handled for a collected dataset, see dissertation data analysis help.