Example Data Table
| Case |
Values |
Alpha |
Transform |
Purpose |
| Lab batch A |
12.1, 11.8, 12.4, 12.0, 11.9, 12.3 |
0.05 |
None |
Check assay normality |
| Reaction times |
210, 225, 231, 219, 240, 260 |
0.05 |
Natural log |
Reduce right skew |
| Count measure |
4, 6, 5, 7, 8, 6, 5 |
0.10 |
Square root |
Inspect transformed counts |
Formula Used
First, sort the observations from smallest to largest.
Let x(i) be each ordered value. Let x̄ be the sample mean.
W = [sum ai x(i)]² / sum [xi − x̄]²
The ai weights come from expected normal order scores.
This page estimates those weights with standard normal quantiles.
The p value uses a Royston style logarithmic approximation.
Skewness and excess kurtosis use adjusted sample formulas.
How to Use This Calculator
Enter one sample in the data box.
Choose an alpha level before reading the decision.
Select a transform only when it fits the measurement process.
Press Calculate to show the result above the form.
Use the CSV option for spreadsheet records.
Use the PDF option for a quick report.
Review plots separately when the decision is important.
Understanding the Shapiro Wilk Test
The Shapiro Wilk test checks whether a sample looks normal. It compares ordered observations with expected normal scores. A high W value means the pattern is close. A low W value signals curvature, skew, or heavy tails. The p value helps decide whether that evidence is strong.
Why Normality Matters
Many statistical methods assume normal errors or normal data. Examples include t tests, analysis of variance, and regression checks. The test is useful before choosing those methods. It is also useful after removing obvious entry mistakes. It should not replace a histogram or a probability plot. Visual checks often explain why a result fails.
Reading the Output
This calculator reports sample size, mean, standard deviation, W, p value, skewness, and kurtosis. The decision uses your selected alpha level. When p is below alpha, the calculator rejects normality. When p is above alpha, it does not reject normality. That does not prove perfect normality. It means the sample did not show enough evidence against it.
Good Data Practice
Use independent measurements from one clear process. Avoid mixing groups with different centers. Keep original units when possible. Use a log or square root transform only when it makes scientific sense. Very small samples have weak power. Very large samples may flag tiny departures that do not matter in practice.
Advanced Use
The calculator can test raw or transformed values. It also lists sorted observations for review. Outlier flags help you inspect unusual points. Export tools make reporting easier. Use the CSV file for spreadsheets. Use the PDF report for records or quick sharing. Always write the alpha level in your report. Also state whether the p value is approximate.
Practical Decision Making
A normality test is one part of analysis. Consider sample design, plots, domain knowledge, and model purpose. If the test fails, try robust methods or nonparametric tests. You may also model transformed data. If the test passes, continue with caution. Assumptions still need context, clean data, and sensible measurement rules.
Report W and p together. Include sample size and alpha. Describe transformations clearly. Mention that rounding can change displayed values. Keep raw data available for audit. Recheck results when new observations arrive during final review.
FAQs
What does the Shapiro Wilk test measure?
It measures how closely ordered sample values match values expected from a normal distribution. The W statistic gets closer to one when the sample pattern is more normal.
What is a good p value?
A p value above your alpha level means you do not reject normality. It does not prove the data are perfectly normal.
Can I use fewer than three values?
No. The test needs at least three observations. More values usually give a clearer view of distribution shape.
Should I transform my data?
Use a transform only when it matches the data process. Log transforms often help positive right-skewed values. Square roots can help count-like values.
Why does a large sample fail normality?
Large samples can detect very small departures. Check plots and practical impact before changing your whole analysis plan.
What does W mean?
W is the normality statistic. Values near one suggest a closer normal pattern. Smaller values suggest skew, tails, curvature, or outliers.
Is the p value exact?
This calculator uses an approximation. It is suitable for quick checks, teaching, and screening. Critical studies should confirm results with statistical software.
Can I export my result?
Yes. Use CSV for spreadsheets. Use PDF for a simple report that includes the main statistics and the normality decision.