How to Test Normality in Medical Research: Shapiro-Wilk vs Kolmogorov-Smirnov Explained

Q: When should I use Shapiro-Wilk vs Kolmogorov-Smirnov for normality testing?

Use the Shapiro-Wilk test for most medical research applications. It is consistently more powerful (better at detecting non-normality) than the Kolmogorov-Smirnov test, particularly for small to medium samples (n < 50). The Kolmogorov-Smirnov test with Lilliefors correction (as implemented in SPSS) is acceptable for larger samples, but even then Shapiro-Wilk is generally preferred. For sample sizes below 50, Shapiro-Wilk is the clear recommendation in most statistical guidelines for health sciences. The standard Kolmogorov-Smirnov test (without Lilliefors correction) should not be used when population parameters are estimated from the sample.

Q: What p value in a normality test means data is normally distributed?

In a normality test (Shapiro-Wilk or Kolmogorov-Smirnov), a p value GREATER than 0.05 means you fail to reject the null hypothesis of normality meaning your data are consistent with a normal distribution. A p value of 0.03, for example, means p 0.05) is what you want to see to confirm normality. However, never rely on the p value alone always examine histograms and Q-Q plots alongside the statistical test result.

Q: Does normality testing apply to the raw data or to the residuals?

For regression and ANOVA, the normality assumption technically applies to the model residuals (errors), not the raw data. A common misconception is that ANOVA requires the raw data to be normally distributed within each group but what the assumption requires is that the residuals from the fitted model are approximately normal. Testing raw data normality is a reasonable proxy check, especially in simple two-group comparisons. However, for linear regression analysis, always test normality on the standardized residuals rather than on the raw outcome variable. SPSS can generate standardized residuals for this purpose via the regression menu's Plots Normal P-P Plot of Regression Standardized Residuals.

Q: Can I use parametric tests if my sample is large even if data are not normal?

Yes. The Central Limit Theorem states that the sampling distribution of the mean approaches normality as sample size increases, regardless of the shape of the underlying distribution. This approximation becomes reliable at n 30 per group for most biological data. For groups of n 50, parametric tests (independent t-test, paired t-test, one-way ANOVA, Pearson correlation) are generally robust to non-normality. However, if the data show extreme skewness (skewness statistic > 2 or 7), or multiple peaks (bimodal distribution), even large samples may warrant non-parametric alternatives or at minimum, a sensitivity analysis comparing parametric and non-parametric results.

Q: How do I interpret a Q-Q plot for normality in SPSS?

In a Q-Q (quantile-quantile) plot, your data points are plotted against the theoretical quantiles of a perfect normal distribution. If your data are normally distributed, all points will fall approximately along the straight diagonal reference line. Systematic deviation from this line indicates non-normality: a curved S-shape (points above the line at the bottom and below the line at the top, or vice versa) indicates skewness. Points curving away from both ends of the line indicate heavy tails (leptokurtosis) or light tails (platykurtosis). Isolated extreme points that diverge sharply from the line at either end indicate outliers. In practice, minor random scatter around the line is expected and acceptable judge the overall pattern, not individual points.

Before selecting any statistical test in medical research t-test, ANOVA, Pearson correlation you must answer one foundational question: is your data normally distributed? Getting this wrong does not simply produce a lower-quality result; it can produce a fundamentally invalid one. A researcher who applies an independent t-test to severely skewed cortisol data, or who uses Pearson correlation on non-normally distributed hospital length-of-stay records, is drawing inferences from a model whose assumptions have been violated. This guide builds a complete, practical understanding of normality testing in medical research what the normal distribution is, how to assess it visually and statistically, when to use Shapiro-Wilk versus Kolmogorov-Smirnov, how to interpret the results, and exactly what to do when your data fails the test.

What Is a Normal Distribution?

The normal distribution sometimes called the Gaussian distribution after mathematician Carl Friedrich Gauss is a continuous probability distribution that describes many naturally occurring measurements in biology and medicine. When data follow a normal distribution, they form a characteristic bell-shaped curve that is perfectly symmetric around the mean.

Three properties define a normal distribution:

Symmetry: The distribution is perfectly symmetric around its center. The left and right halves are mirror images.
Mean = Median = Mode: All three measures of central tendency coincide at the peak of the bell curve.
The 68-95-99.7 Rule (Empirical Rule): A fixed proportion of observations fall within defined distances from the mean, measured in standard deviations.

The 689599.7 Empirical Rule Normal Distribution

68%

of values fall within
+/-1 SD of the mean

95%

of values fall within
+/-2 SD of the mean

99.7%

of values fall within
+/-3 SD of the mean

Medical Variables That Are Typically Normally Distributed

Many physiological measurements in healthy adult populations follow an approximately normal distribution when measured in a representative sample:

Adult height and weight in a homogeneous population
Hemoglobin levels in a healthy adult sample
Systolic blood pressure in normotensive adults
Serum sodium and potassium in a general population
Cognitive test scores across a large educational cohort

Medical Variables That Are Often NOT Normally Distributed

Many clinical measurements are inherently skewed particularly those that cannot go below zero but have a long upper tail:

C-reactive protein (CRP), interleukins, and other inflammatory markers typically right-skewed
Cortisol, adrenaline, and other hormone levels right-skewed
Hospital length of stay right-skewed (most patients stay 15 days; some stay months)
Platelet count in disease states often non-normally distributed
Serum creatinine in chronic kidney disease cohorts skewed by disease severity
Questionnaire scores from Likert scales often ordinal and not continuous

Why Normality Testing Matters Before Choosing a Statistical Test

Parametric statistical tests the independent t-test, paired t-test, one-way ANOVA, Pearson correlation, and linear regression all rest on a common mathematical foundation: they assume that the data (or the model residuals) follow a normal distribution. This assumption is not cosmetic. It is baked into the formulas that generate p-values and confidence intervals. When the assumption is violated, those p-values and confidence intervals may be incorrect sometimes severely so.

What Happens When You Ignore Normality? Applying a parametric test to severely non-normal data can: (1) produce inflated or deflated p-values that misclassify a result as significant or non-significant; (2) generate confidence intervals that do not accurately reflect the true uncertainty in your estimate; (3) make the test susceptible to outlier influence, since parametric tests use mean and variance which are highly sensitive to extreme values. The consequence in medical research is potential false conclusions about treatment efficacy, diagnostic accuracy, or risk factor association.

The normality assumption is the gateway to all subsequent analytical decisions. Test it first before running any inferential analysis on continuous data. The specific consequences vary by test type, but the principle is universal: know your distribution before applying your test.

There is one important nuance to this: the Central Limit Theorem. For sufficiently large samples (n 30 per group is a widely cited heuristic), the sampling distribution of the mean becomes approximately normal regardless of the underlying data distribution. This means that parametric tests retain their validity with large samples even when raw data are not perfectly bell-shaped. However, this protection weakens with extreme skewness, heavy outliers, or bimodal distributions and it does not apply to all parametric tests equally. Normality testing remains important even in moderately large samples.

Visual Methods for Assessing Normality: Histogram and Q-Q Plot

Statistical tests for normality are valuable but must always be interpreted alongside visual inspection of the data. Over-reliance on p-values from normality tests particularly in large samples where even trivial departures from normality become statistically significant is one of the most common methodological errors in clinical research. Always start with visual assessment.

Method 1: The Histogram

A histogram bins your continuous data into intervals and displays the frequency (or relative frequency) of observations in each bin. A normally distributed variable will produce a histogram that approximates the bell curve: roughly symmetric, with the tallest bars in the center and progressively shorter bars in both directions.

Signs of Normality in a Histogram

What to Look For

Symmetric, single-peaked (unimodal) shape
Bars taper gradually and evenly on both sides
No extreme outlier bars isolated far from the main distribution
Mean, median, and mode appear to coincide near the center
Approximately bell-shaped overall silhouette

Signs of Non-Normality in a Histogram

Red Flags to Identify

Long tail extending to the right (positive skew) or left (negative skew)
Two distinct peaks (bimodal) often indicates two subpopulations mixed
Flat, uniform distribution (rectangular) no clear central peak
Extremely tall, narrow peak with almost no spread (leptokurtic)
Isolated bars far from the main mass of data (outliers)

In SPSS, generate a histogram via: Analyze Descriptive Statistics Frequencies Charts Histograms With normal curve. The "with normal curve" option overlays the theoretical normal distribution on your histogram, making the comparison immediate and visual.

Method 2: The Q-Q Plot (Quantile-Quantile Plot)

The Q-Q plot is a more sensitive visual tool than the histogram. It plots the observed quantiles of your data against the theoretical quantiles of a perfect normal distribution. If your data are normally distributed, the points will fall along a straight diagonal reference line. Deviation from this line reveals the nature and location of non-normality.

Normal Q-Q Plot Pattern

What Normal Looks Like

All points fall close to the straight diagonal line
Minor random scatter around the line is expected and acceptable
No systematic curves or deviations from the line
Points at the tails stay near the line (no sharp divergence)

Non-Normal Q-Q Plot Patterns

What to Look For

S-curve: Points bow above the line on left, below on right (or vice versa) skewness
Upward bow at both ends: Points curve away from line at upper and lower tails heavy tails
Downward bow at both ends: Points fall inside the line at both ends light tails
Single divergent point: One isolated extreme point far from the line outlier

Practical Rule The Q-Q plot is the single most informative visual for normality assessment. In your thesis or paper, SPSS generates a "Normal P-P Plot" by default (which plots cumulative probabilities) and a separate "Detrended Normal Q-Q Plot" (which should show points randomly scattered near the zero line if data are normal). Both are generated automatically by the Explore procedure request them for every continuous outcome variable you analyze.

Statistical Tests for Normality: An Overview

Visual inspection is subjective and experience-dependent. Statistical normality tests provide an objective, quantitative assessment. The null hypothesis (H) for all normality tests is the same: the data are normally distributed. A statistically significant result (p < 0.05) means you reject H evidence against normality has been found.

Several formal tests exist, each with different properties, power, and ideal sample size ranges. The two most commonly reported in medical research are:

Shapiro-Wilk Test (SW) most powerful for small to medium samples; the gold standard for medical research
Kolmogorov-Smirnov Test (KS) with Lilliefors correction more established historically, better suited for larger samples

Less commonly used tests include the Anderson-Darling test, D'Agostino-Pearson test, and Jarque-Bera test. These are available in R and Stata but are rarely reported in medical theses or clinical papers submitted to standard journals.

The Shapiro-Wilk Test Explained

The Shapiro-Wilk test was developed by Samuel Shapiro and Martin Wilk in 1965. It is consistently rated as the most powerful formal test for normality across a wide range of sample sizes, particularly for small to medium samples (n = 3 to approximately 50). It has been validated and recommended by most statistical guidelines for health and medical sciences, and it is the test you should report by default in your medical thesis or clinical paper.

How the Shapiro-Wilk Test Works

The test works by calculating the correlation between your ordered data and the corresponding expected order statistics from a normal distribution. Conceptually, it asks: how well do the actual sorted values match what they would look like if the data came from a perfect normal distribution?

The output of the test is the W statistic, which ranges from 0 to 1:

W = 1.000: Perfect normality data match the theoretical normal distribution exactly
W close to 1 (e.g., 0.97, 0.98): Data are approximately normal
W substantially below 1 (e.g., 0.82, 0.75): Significant departure from normality

The p-value associated with W tells you whether the departure from normality is statistically significant at your chosen alpha level (typically 0.05):

p > 0.05: Fail to reject normality data are consistent with a normal distribution. Proceed with parametric tests (if other assumptions are also met).
p < 0.05: Reject normality statistically significant evidence of non-normality. Consider non-parametric alternatives or data transformation.

Shapiro-Wilk Key Properties Recommended sample range: n = 3 to ~300 (most guidelines specify n < 50 as the sweet spot). Available in SPSS via Analyze Explore. Most powerful normality test for typical medical research sample sizes. The default choice for thesis and clinical paper methods sections. Sensitive to ties in data at small sample sizes.

The Kolmogorov-Smirnov Test Explained

The Kolmogorov-Smirnov (KS) test was developed in the 1930s by Soviet mathematicians Andrey Kolmogorov and Nikolai Smirnov. In its original form, it compares an observed sample distribution to a fully specified theoretical distribution meaning you must provide the exact mean and standard deviation of the reference normal distribution, rather than estimating them from the sample data.

This is an important limitation: in real research, you never know the population mean and standard deviation in advance you estimate them from the sample itself. The standard KS test does not account for this estimation, which makes it overly conservative (less likely to detect non-normality) and technically inappropriate when parameters are estimated from the data.

The Lilliefors Correction: What SPSS Actually Uses

To address this limitation, Hubert Lilliefors developed a correction in 1967 that adjusts the KS critical values to account for the fact that population parameters are estimated from the sample. The Kolmogorov-Smirnov test with Lilliefors correction (sometimes written as the KS-Lilliefors test or KSTEST) is what SPSS implements and labels as "Kolmogorov-Smirnov" in its Explore output.

The D statistic produced by the KS test represents the maximum absolute difference between the observed cumulative distribution function and the theoretical normal cumulative distribution function. A larger D value indicates greater departure from normality.

Important Distinction When your SPSS output shows the "Kolmogorov-Smirnov" test under the Tests of Normality table, it is automatically applying the Lilliefors correction. You should report it as "Kolmogorov-Smirnov test with Lilliefors correction" rather than simply "KS test" the distinction matters methodologically and some examiners will ask about it.

Shapiro-Wilk vs Kolmogorov-Smirnov: Side-by-Side Comparison

Feature	Shapiro-Wilk	Kolmogorov-Smirnov (Lilliefors)
Year developed	1965	1933 (KS); 1967 (Lilliefors correction)
Test statistic	W (0 to 1)	D (maximum CDF difference)
Ideal sample size	n = 3 to ~300 (optimal n < 50)	n = 50 to ~2000 (weaker for small samples)
Statistical power	High most powerful normality test for small/medium n	Lower less sensitive to departures from normality
Parameters estimated from data?	Yes built into the test	Yes via Lilliefors correction in SPSS
Sensitivity to tails	Good overall sensitivity, especially to tail departures	Better sensitivity to departures at the distribution center
Available in SPSS	Yes Analyze Explore	Yes Analyze Explore (with Lilliefors)
Recommended for medical thesis	Primary recommendation	Secondary report alongside SW if required
Limitation	Not designed for n > 5000 (though still functional)	Low power for small samples; may miss non-normality

Practical Recommendation For virtually all medical thesis and clinical research applications, use the Shapiro-Wilk test as your primary normality test. SPSS reports both tests simultaneously in the same output table report both results, but base your decision on Shapiro-Wilk. If the two tests give conflicting results (which occurs especially at sample sizes near 50), trust Shapiro-Wilk and corroborate with visual inspection of histograms and Q-Q plots.

How to Run Normality Tests in SPSS: Step-by-Step

SPSS generates both the Shapiro-Wilk and Kolmogorov-Smirnov tests, along with histograms and Q-Q plots, through a single procedure called "Explore." Follow these steps:

Open the Explore Procedure

Go to: Analyze Descriptive Statistics Explore. The Explore dialog box will open.

Analyze Descriptive Statistics Explore

Select Your Variables

Move your continuous outcome variable(s) hemoglobin, BMI, CRP, etc. into the Dependent List box. If you want to test normality by group (e.g., males vs females, intervention vs control), move the grouping variable into the Factor List box.

Request Normality Tests

Click the Plots button. Check Normality plots with tests. This activates both the Shapiro-Wilk and KS-Lilliefors tests AND generates the Normal Q-Q Plot. Also check Histogram to get the visual display. Click Continue.

Plots Normality plots with tests Histogram Continue

Set Display Options and Run

Back in the main Explore dialog, set Display to Both (statistics and plots). Click OK.

Read the Tests of Normality Table

SPSS generates a table titled "Tests of Normality." It shows: the Kolmogorov-Smirnov statistic (D), degrees of freedom, and p-value; and the Shapiro-Wilk statistic (W), degrees of freedom, and p-value. Look for the Shapiro-Wilk p-value first. If p > 0.05 normality not rejected. If p < 0.05 normality rejected.

Output: Tests of Normality table Shapiro-Wilk Sig. column

Inspect the Plots

Review the histogram for overall shape. Review the Normal Q-Q Plot do points follow the diagonal line? Review the Detrended Normal Q-Q Plot are points scattered randomly around zero? Use both tests AND both plots to form a judgment about normality. No single output element should be interpreted in isolation.

Interpreting Normality Results: Medical Research Examples

Example 1 Hemoglobin (g/dL) in a General Population Study

Dataset: 85 adult males aged 1845 from a general hospital outpatient clinic. The researcher plans to compare hemoglobin levels between smokers and non-smokers using an independent t-test.

SPSS Normality Output:
Shapiro-Wilk: W = 0.974, df = 85, p = 0.082
Kolmogorov-Smirnov: D = 0.071, df = 85, p = 0.200
Histogram: Symmetric, single peak, approximately bell-shaped.
Q-Q Plot: Points fall close to the diagonal reference line; minor random scatter only.

Decision: Both tests yield p > 0.05. Normality is not rejected. The histogram and Q-Q plot confirm approximate normality. Proceed with the independent samples t-test as planned. Report: "Hemoglobin values were approximately normally distributed in both groups (Shapiro-Wilk, W = 0.974, p = 0.082), supporting the use of the independent samples t-test."

Example 2 Serum CRP (mg/L) in a Rheumatoid Arthritis Cohort

Dataset: 60 patients with rheumatoid arthritis. Serum C-reactive protein is measured at baseline. The researcher intends to compare CRP between responders and non-responders to biologic therapy.

SPSS Normality Output:
Shapiro-Wilk: W = 0.741, df = 60, p < 0.001
Kolmogorov-Smirnov: D = 0.196, df = 60, p < 0.001
Histogram: Strongly right-skewed; majority of patients cluster near 515 mg/L but a long tail extends to 120+ mg/L.
Q-Q Plot: Points curve above the diagonal line at the high end classic right-skew S-curve pattern.

Decision: Both tests reject normality (p < 0.001). CRP is biologically right-skewed by nature. Proceed with the Mann-Whitney U test instead of the independent t-test. Alternatively, apply log transformation to CRP values, re-test normality on the log-transformed data, and use parametric tests on the log-CRP if normality is achieved. Report: "CRP values were not normally distributed (Shapiro-Wilk, W = 0.741, p < 0.001) and were therefore analyzed using the Mann-Whitney U test."

Example 3 BMI in a Hypertension Study Large Sample

Dataset: 350 adult patients enrolled in a hypertension registry. The researcher plans to use one-way ANOVA to compare mean BMI across three treatment groups (n 117 per group).

SPSS Normality Output:
Shapiro-Wilk (Group 1): W = 0.961, df = 117, p = 0.003
Histogram: Slightly right-skewed but reasonably symmetric for a BMI dataset.
Skewness statistic: +0.68 (within acceptable range of 2 to +2)

Decision: Shapiro-Wilk is significant (p = 0.003) due to the large sample size even small, clinically unimportant deviations produce significant results at n = 117. The histogram and skewness statistic confirm only mild departure. With n = 117 per group, the Central Limit Theorem provides sufficient protection. Proceed with one-way ANOVA and note in your methods: "Although Shapiro-Wilk indicated significant departure from normality (p = 0.003), visual inspection confirmed only mild positive skewness (skewness = 0.68). Given the large sample size (n 100 per group), parametric ANOVA was considered appropriate by virtue of the Central Limit Theorem."

Example 4 Platelet Count in a Dengue Fever Study

Dataset: 40 dengue fever patients admitted to a tertiary care center. The researcher records daily platelet count to compare between patients who develop dengue hemorrhagic fever (n = 16) and those who do not (n = 24).

SPSS Normality Output:
Shapiro-Wilk (DHF group): W = 0.832, df = 16, p = 0.007
Shapiro-Wilk (non-DHF group): W = 0.901, df = 24, p = 0.023
Histograms: Both groups show bimodal patterns and non-symmetric shapes.

Decision: Normality is rejected in both groups (p < 0.05). With small group sizes (n = 16 and 24), the Central Limit Theorem does not provide sufficient protection. The Mann-Whitney U test is the appropriate alternative. Report: "Platelet counts were not normally distributed in either group (Shapiro-Wilk, W = 0.832, p = 0.007 for DHF group; W = 0.901, p = 0.023 for non-DHF group). Group comparison was performed using the Mann-Whitney U test."

What to Do If Your Data Is Not Normally Distributed

A failed normality test is not the end of your statistical analysis it is the beginning of a different analytical path. You have three main options, and the best choice depends on your sample size, the degree of non-normality, and your research question.

Option 1: Use Non-Parametric Tests (Recommended Default)

Non-parametric tests make no assumption about the shape of the underlying population distribution. They work with the ranks of the data rather than the raw values, making them robust to skewness, outliers, and non-normality. Every common parametric test has a non-parametric equivalent:

Research Situation	Parametric Test	Non-Parametric Alternative
Compare 2 independent groups	Independent t-test	Mann-Whitney U test
Compare 2 paired/repeated measurements	Paired t-test	Wilcoxon Signed-Rank test
Compare 3+ independent groups	One-way ANOVA	Kruskal-Wallis test
Compare 3+ repeated measurements	Repeated-Measures ANOVA	Friedman test
Correlation between 2 continuous variables	Pearson correlation	Spearman rank correlation

Option 2: Apply a Data Transformation

For right-skewed biological variables, a logarithmic transformation (log or natural log) often converts non-normal data to approximate normality. This is standard practice for variables such as cortisol, CRP, platelet count, hospital length of stay, and viral loads.

To apply a log transformation in SPSS: Transform Compute Variable create a new variable (e.g., "log_CRP") using the function LG10(CRP). Then run the normality test on the transformed variable. If the log-transformed variable is normally distributed, proceed with parametric tests on the transformed values and note this in your methods section.

Common Transformations for Medical Data Log transformation (log or ln): for right-skewed continuous biological variables (CRP, cortisol, viral loads, platelet counts). Square root transformation (x): for count data with moderate skewness. Reciprocal (1/x): for extreme right skew, less commonly used. Box-Cox transformation: a general family of power transformations, available in R. Always re-test normality on the transformed data before proceeding.

Option 3: Invoke the Central Limit Theorem (Large Samples)

When group sizes are large (generally n 30 per group as a minimum; n 50 for added confidence), the sampling distribution of the mean approximates normality regardless of the raw data distribution. This permits the use of parametric tests even when the raw data fail normality tests. The caveat: this protection weakens with extreme skewness (skewness statistic > 2), heavy kurtosis (> 7), or clearly bimodal data.

The Parametric vs Non-Parametric Decision Framework

Use the following decision process when selecting between parametric and non-parametric tests for continuous outcome variables. Apply this framework before choosing any specific test:

Inspect Histograms and Q-Q Plots

Before running any formal test, look at your data visually. Severe skewness or bimodality visible in a histogram already signals the need for non-parametric methods. Visual inspection should never be skipped it contextualizes the formal test results.

Run the Shapiro-Wilk Test

Use SPSS Explore to run the Shapiro-Wilk test for each group separately. Record the W statistic and p-value. A p > 0.05 in all groups supports parametric tests. A p < 0.05 in any group signals non-normality.

Assess Sample Size Per Group

If n < 30 per group and Shapiro-Wilk rejects normality: use non-parametric tests. If n 50 per group and the departure is mild (skewness statistic between 2 and +2 with no major outliers): parametric tests are generally acceptable due to the Central Limit Theorem. Document your reasoning in the methods section.

Check for Outliers

SPSS boxplots (generated by the Explore procedure) flag outliers (-) and extreme values (*). Outliers can cause a non-significant result in the Shapiro-Wilk test to be unreliable, or can cause a false normality failure in a large dataset. Investigate every flagged outlier is it a real clinical value, a data entry error, or a measurement artifact?

Make and Document Your Decision

Write a clear methods statement: state which normality test you used, the result (W or D statistic, p-value), the visual findings, and the test you chose as a result. Reviewers and examiners expect to see this documented explicitly normality assessment that is not reported is assumed to have been skipped.

Common Mistakes Researchers Make When Testing Normality

Mistake 1: Skipping Normality Testing Entirely

The most prevalent error: applying t-tests, ANOVA, or Pearson correlation to continuous data without ever checking the normality assumption. This is especially common when researchers use default test selections without understanding the underlying assumptions. Examiners and peer reviewers will ask about this, and "I assumed it was normal" is not an acceptable answer.

Fix: Make normality testing a non-negotiable first step in your analysis plan. Run SPSS Explore on every continuous outcome variable before selecting any inferential test. Document the results in your methods section under "Statistical Analysis."

Mistake 2: Relying on a Significant Normality Test to Reject Parametric Tests in Large Samples

In large samples (n > 200), the Shapiro-Wilk test is so sensitive that even trivial, clinically irrelevant departures from normality produce significant p-values. A researcher with n = 400 might obtain SW p = 0.003 for data that a histogram shows to be almost perfectly bell-shaped. Mechanically concluding "not normal, use non-parametric" in such a case discards statistical power unnecessarily.

Fix: In large samples, weight visual inspection more heavily than the formal test p-value. Calculate the skewness and kurtosis statistics and their standard errors. A skewness statistic within +/-2 and kurtosis within +/-7, combined with a reasonably bell-shaped histogram, supports parametric analysis even with a significant Shapiro-Wilk p-value.

Mistake 3: Testing Normality on the Entire Sample Instead of by Group

When comparing two groups (e.g., diabetics vs non-diabetics), the normality assumption applies to the data within each group separately, not to the pooled sample. A researcher who combines all 100 patients and runs a single normality test may obtain a misleading result one group may be normally distributed while the other is severely skewed, or the combination of two separate groups may itself appear non-normal even when each group individually is normal.

Fix: Always run normality tests separately for each group using the Factor List in SPSS Explore. Report the result for each group. The parametric test is appropriate only when normality is satisfied within both (or all) groups.

Mistake 4: Reporting Only the KS Test and Ignoring Shapiro-Wilk

Some older statistical textbooks recommend the KS test, and some researchers report it as their primary normality test by habit. For sample sizes typical in medical research (n = 20200), the Kolmogorov-Smirnov test has substantially lower statistical power than Shapiro-Wilk it is more likely to miss real departures from normality, leading to incorrect application of parametric tests.

Fix: Always report both tests (SPSS generates both automatically), but clearly state that Shapiro-Wilk was used as the primary criterion. If the two tests disagree, note the discrepancy and base your decision on Shapiro-Wilk, corroborated by visual inspection.

Mistake 5: Neglecting Visual Inspection and Over-Relying on the p-Value

A researcher who reports only "Shapiro-Wilk p = 0.12, therefore data are normal" without examining the histogram or Q-Q plot may miss clinically important features: a bimodal distribution (two patient subpopulations mixed together), extreme outliers that inflate variance without affecting the test significantly, or a flat distribution that technically passes the normality test at low power.

Fix: Always examine the histogram AND the Q-Q plot in addition to formal test results. These plots are automatically generated by SPSS Explore alongside the normality tests. If the visual patterns and the formal test disagree, investigate further before making a final decision.

Mistake 6: Failing to Report Normality Testing in the Methods Section

Many researchers perform normality tests during their analysis but never mention them in the manuscript or thesis. Readers and reviewers are then unable to evaluate whether the chosen statistical tests were appropriate. This omission is particularly problematic in thesis viva examinations, where it can raise serious concerns about methodological rigor.

Fix: Include a dedicated sentence in the Statistical Analysis subsection of your Methods chapter. Example: "Normality of continuous variables was assessed using the Shapiro-Wilk test and visual inspection of histograms and Q-Q plots. Variables not meeting the assumption of normality were analyzed using the corresponding non-parametric tests."

Frequently Asked Questions

When should I use Shapiro-Wilk vs Kolmogorov-Smirnov for normality testing? +

Use the Shapiro-Wilk test for most medical research applications. It is consistently more powerful better at detecting real departures from normality than the Kolmogorov-Smirnov test, particularly for small to medium samples (n < 50). The KS test with Lilliefors correction is acceptable for larger samples, but even then Shapiro-Wilk is generally preferred. For samples below 50, Shapiro-Wilk is the clear recommendation in most statistical guidelines for health sciences. SPSS generates both tests automatically in the Explore output report both, but base your decision on Shapiro-Wilk.

What p value in a normality test means data is normally distributed? +

In a normality test, a p value greater than 0.05 means you fail to reject the null hypothesis of normality meaning your data are consistent with a normal distribution. A p value of 0.03 means p < 0.05 and you reject normality. The direction is the opposite of how most researchers use p values: for normality tests, a non-significant p value (p > 0.05) is what you want to see to proceed with parametric tests. However, never rely on the p value alone always examine histograms and Q-Q plots alongside the statistical test result, especially when sample sizes are very small or very large.

My data failed the normality test what should I do next? +

First, do not panic many real-world biological datasets are not normally distributed. Your options are: (1) Use the non-parametric equivalent of your intended test (Mann-Whitney instead of independent t-test, Kruskal-Wallis instead of one-way ANOVA, Wilcoxon signed-rank instead of paired t-test, Spearman instead of Pearson correlation). (2) Apply a log transformation (works well for right-skewed biological variables like cortisol, CRP, or platelet counts) and re-test normality on the transformed data. (3) If your sample is large (n > 30 per group), invoke the Central Limit Theorem the sampling distribution of the mean becomes approximately normal even when raw data are not, justifying parametric tests. Document your decision clearly in your methods section.

Does normality testing apply to the raw data or to the residuals? +

For regression and ANOVA, the normality assumption technically applies to the model residuals (errors), not the raw data. A common misconception is that ANOVA requires raw data within each group to be normally distributed what is actually required is that the residuals from the fitted model are approximately normal. Testing raw data normality is a reasonable proxy check, especially in simple two-group comparisons. However, for linear regression analysis, always test normality on the standardized residuals rather than on the raw outcome variable. SPSS can generate standardized residuals for this purpose via the Regression menu's Plots Normal P-P Plot of Regression Standardized Residuals.

Can I use parametric tests if my sample is large even if data are not normal? +

Yes, with caveats. The Central Limit Theorem states that the sampling distribution of the mean approaches normality as sample size increases, regardless of the underlying distribution. This approximation becomes reliable at n 30 per group for most biological data, and is very robust at n 50. For groups of n 50, parametric tests (independent t-test, paired t-test, one-way ANOVA, Pearson correlation) are generally robust to non-normality. However, if the data show extreme skewness (skewness statistic > 2 or < 2), heavy tails (kurtosis > 7), or multiple peaks (bimodal distribution), even large samples may warrant non-parametric alternatives or at minimum, a sensitivity analysis comparing parametric and non-parametric results.

How do I interpret a Q-Q plot for normality in SPSS? +

In a Q-Q plot, your data points are plotted against the theoretical quantiles of a perfect normal distribution. If data are normally distributed, all points will fall approximately along the straight diagonal reference line. Systematic deviation indicates non-normality: an S-shaped curve (points above the line at the bottom and below at the top, or vice versa) indicates skewness. Points curving away at both ends of the line indicate heavy tails (leptokurtosis). Isolated extreme points that diverge sharply at either end indicate outliers. The Detrended Normal Q-Q Plot (also generated by SPSS Explore) should show points randomly scattered around the zero horizontal line if data are normal any systematic pattern in the detrended plot is a sign of non-normality. In practice, minor random scatter around the line is expected and acceptable judge the overall pattern, not individual points.

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