Information
Target: To check if the normal distribution model fits the observations.
The tool combines the following methods:
1. A formal normality test: Shapiro-Wilk test. This is one of the most powerful normality tests.
2. Graphical methods: QQ-Plot chart and Histogram.
The Shapiro-Wilk test uses only the right-tailed test. When performing the test, the W statistic is only positive and represents the difference between the estimated model and the observations. The bigger the statistic, the more likely the model is not correct. The left-tailed may represent a value that is too small, the W statistic can't be too small.
How the p-value is calculated
The calculator uses Royston's AS R94 algorithm (swilk) for every sample size, which is the same algorithm R's shapiro.test() uses. It applies the exact result for n = 3, a gamma transform for n ≤ 11 and the normal approximation above that, so the p-value is accurate across the whole range.
Small sample size (n ≤ 50)
For reference, the tool also reports the p-value read from the original Shapiro-Wilk exact tables (p-values 0.01, 0.02, 0.05, 0.1, 0.5, 0.9, 0.95, 0.98, 0.99). The W value usually falls between two cells, and that reference figure is a harmonic interpolation between the two p-values, so it is most precise around the common significance levels. The two figures should agree closely; where they differ, the AS R94 value is the reported one.
Compared to other tests the Shapiro-Wilk has a good power to reject the normality, but as with any other test it needs to have a sufficient sample size, around 20 depending on the distribution, see examples.
In this case, the normal distribution chart is only for illustration.
Large sample size (n > 50)
Since the sample size is large the normal approximation inside AS R94 is good for any p-value.
* The maximum sample size is 5000, but since no distribution is exactly normal distribution, a very large sample size has the power to reject the normality assumption for almost any distribution even if the difference from the normal distribution is minimal.
If you need to test the average, and the sample size is large and reasonably symmetrical, even if the population distribution is not normal the average's distribution will be approximately normal (Central Limit Theorem).
Normality effect size
There is no agreed way to calculate the effect size, hence we use the Kolmogorov-Smirnov effect size to measure the deviation from the normality:
We know only the sample effect size!.
We defined the following effect levels of the effect size:
- Small - 0.0063, the effect size of Chi-squared(df=40)
- Medium - 0.00224, the effect size of Chi-squared(df=20)
- Large - 0.0427, the effect size of Chi-squared(df=5)
For example, Small is any value between 0.026 and 0.03.
We calculated the effect size of Chi-squared sample data using a simulation with 10,000 repeats, each run over a sample size of 1000. The effect level is only a wild rule of thumb, we still recommend looking at the Q-Q plot.
How to use the Shapiro-Wilk calculator
- Enter the data — one column of values in the grid.
- Type values directly; press Enter for a new row. New rows are added automatically once you reach the last one.
- Paste data copied from Excel (tab-delimited).
- Paste data with another delimiter (comma, space, or custom) by enabling it under More options.
- Set the significance level (α).
- Open More options to exclude outliers or change the rounding.
- Press Calculate.