How to check your distribution
- Paste one sample and choose its number format. Each number is one equally weighted observation. Do not pool different groups simply to get a larger sample.
- Choose Check distribution. Start with the histogram, then inspect whether the Q–Q points follow a roughly straight pattern. Read the plot values below each chart if you prefer a table.
- Open Shapiro–Wilk test details for W, the p-value, and its calculation method. Read the warnings and consider why the measurements may depart from a normal model.
- Copy normality results includes the method and interpretation cautions. Editing the sample or number format hides the old result. Reset example restores the reproducible sample below.
A reproducible example
The default observations are 12, 15, 14, 18, 16, 19, 21, 17, 15, 20, 13, and 16. These invented numbers illustrate the output; they do not establish a sampling design.
There are 12 observations and 10 distinct values. The Shapiro–Wilk statistic is W = 0.9757210794 and the approximate p-value is 0.960625836. The histogram has five equal-width bins with counts 2, 3, 3, 2, and 2. This sample provides little evidence against the normal model, but the large p-value does not establish normality. The sample is small and has repeated values.
Try replacing the last observation with 80. The histogram and Q–Q plot reveal the isolated high observation. Investigate how it arose; do not remove a valid measurement just to increase the p-value.
Methods behind the plots and test
Shapiro–Wilk compares the ordered observations with weights derived from normal order statistics. W is a weighted squared contrast divided by the sample’s centered sum of squares. Smaller W generally indicates stronger departure from the normal model.
The implementation uses the uncensored Royston AS R94 algorithm, adapted from SciPy with its BSD license retained in the source. The p-value is exact for n = 3 and approximate for larger samples, with separate approximations for n = 4–11 and n = 12–5,000. P-values are diagnostic approximations, not guarantees of measurement or model validity.
The histogram uses equal-width bins, with the bin count equal to the ceiling of 1 + log₂(n), known as Sturges’ rule. Intervals include the lower bound; only the final interval also includes the upper bound. Constant data have one bin.
The normal Q–Q plot compares sorted observations with standard-normal quantiles at Blom plotting positions (i − 3/8) / (n + 1/4), for ranks i = 1 through n. The dashed reference line passes through sample quartiles calculated with linear interpolation, R type 7. It is omitted when the interquartile range is zero. Every observation is plotted; no observations are sampled or dropped.
What this check cannot tell you
- Independence and representative sampling come from the study design. A normality test cannot establish them. Time trends, repeated measurements, mixtures of groups, or dependence need separate attention.
- A normal population is continuous. Rounded values, discrete counts, many ties, truncation, or censoring can alter the test’s calibration. This tool does not implement a censored-data test.
- For a paired t-test, the relevant distribution is the within-pair differences. For regression, inspect the appropriate residuals alongside model-specific diagnostics; fitted residuals are not generally independent observations, so this p-value should not be treated as a calibrated residual test.
- A cutoff such as 0.05 is not a universal decision rule. Choose an analysis using the scientific question, design, effect size, and robustness. Do not use this page as an automatic switch between a t-test and a rank test.
The supported size is 3–5,000 observations. Larger samples require a suitable analysis elsewhere; do not discard observations merely to fit this limit. Constant observations receive an undefined test result instead of a misleading p-value.
Calculations normalize the range internally to avoid scale-related overflow or underflow. Inputs are still browser numbers with roughly 15–17 significant digits. Values that differ beyond that precision may become identical; rescale or center the original measurements before pasting. The accepted nonzero magnitude range is 1e−100 to 1e150. A numerical or approximation floor is labeled as a bound rather than an exact zero.
Sources and further reading
Keep exploring
All statistics tools · Welch’s t-test calculator · Worked statistics guide