Normality Calculator
Paste a column of numbers and run six normality tests in one go. Each result shows a p-value and its test statistic, together with the mean, range and standard deviation of your data.
Created by the Reeliy Editorial Team. Last updated: Oct 19, 2026
Using the normality test calculator
Paste or type your numbers into the Data to test box, separated by spaces, commas, tabs or new lines, with a dot as the decimal mark. A column copied from a spreadsheet works as it is. You need at least 10 values. The Alpha box holds the threshold you plan to judge the p-values against (0.05 by default); it must be a number from 0 up to, but not including, 1, and it does not change any of the numbers shown.
The results list one line per test, in the form "p-value (statistic)", followed by the number of data points, the mean, the range and the standard deviation. The standard deviation reported here divides by n, matching the way the distance tests below standardise your data. Characters that are not part of a number are ignored, and a token with no digits counts as 0, so check the Data Points line to confirm that every value was read.
The Shapiro-Francia line reads the values in the order you typed them. Enter the data sorted from smallest to largest to get a meaningful figure there; all the other tests ignore the order.
What is a normality test?
A normality test asks whether a sample is consistent with having been drawn from a normal (bell-shaped) distribution. The null hypothesis is that the data are normal, so a small p-value is evidence against normality and a large one is merely the absence of such evidence. Many procedures, such as t-tests, ANOVA and regression diagnostics, assume normally distributed data or residuals, which is why checking is common.
The tests fall into three families. Regression and correlation tests (Shapiro-Wilk, Shapiro-Francia) compare the ordered data with the values expected from a normal sample. Distance tests (Anderson-Darling, Cramer-von Mises) measure how far the empirical cumulative distribution lies from the fitted normal curve. Moment tests (D'Agostino-Pearson, Jarque-Bera) look at skewness and kurtosis.
Run the tests on all of the data. Remove an outlier only when you know it is a recording error, not simply because it spoils the fit. Also note that a low p-value can come from non-independent observations as well as from a non-normal shape.
Interpreting the outcome
- p-value below alpha (conventionally 0.05): the data are hard to reconcile with a normal distribution. Consider a transformation, another distribution or a non-parametric method.
- p-value above alpha: no evidence of non-normality. It does not prove normality; with few observations every test has little power.
- Tests disagree: this is common for small or nearly normal samples because each test is sensitive to different departures. One clearly low p-value among several is enough reason to look at a histogram or Q-Q plot.
- Very large samples: tiny, harmless departures can produce small p-values, so judge the size of the departure as well as the p-value.
Supported tests
| Test | Statistic | Idea | Sample size here |
|---|---|---|---|
| Shapiro-Wilk | W | squared correlation with normal order statistics | 10 to 50 |
| Shapiro-Francia | W’ | Royston's version for larger samples | 10 to 5,000 |
| Cramer-von Mises | W | distance between ecdf and fitted normal cdf | 10 and up |
| Anderson-Darling | W (A²) | like Cramer-von Mises, weights the tails more | 10 and up |
| D'Agostino-Pearson | K² | skewness and kurtosis combined | 10 and up |
| Jarque-Bera | JB | skewness and kurtosis, large-sample | 10 and up |
The Shapiro-Wilk and Shapiro-Francia tests
Shapiro and Wilk (1965) compare a weighted sum of the ordered observations with the sample variance:
W = ( Σ aᵢ × (x₍ₙ₊₁₋ᵢ₎ − x₍ᵢ₎) )² ÷ Σ (xᵢ − x̄)²
The weights aᵢ come from the expected values and covariances of normal order statistics. They were tabulated for samples of up to 50, so the Shapiro-Wilk line appears for 10 to 50 values. W is at most 1, and values close to 1 indicate normality. For the p-value, ln(1 − W) is treated as approximately normal using Royston's polynomial approximation. Shapiro and Francia (1972) extended the idea to larger samples, and Royston's algorithm handles up to 5,000 values; the Shapiro-Francia line is shown for every sample size.
The Cramer-von Mises test
W² = 1 ÷ (12n) + Σ ( pᵢ − (2i − 1) ÷ (2n) )²pᵢ is the fitted normal probability of the i-th smallest value
It sums the squared gaps between the fitted cumulative probability and the evenly spaced plotting positions, so every part of the distribution counts equally. The p-value comes from the usual small-sample adjustment W²(1 + 0.5/n).
The Anderson-Darling test
A² = −n − (1 ÷ n) × Σ (2i − 1) × [ ln pᵢ + ln(1 − p₍ₙ₊₁₋ᵢ₎) ]
The logarithms weight the two tails heavily, which makes the test good at spotting heavy tails or outliers. Each logarithm is taken from the tail area itself, so a value dozens of standard deviations from the mean still gives a finite statistic. The p-value uses the adjusted statistic A²(1 + 0.75/n + 2.25/n²).
The D'Agostino-Pearson test
Skewness measures asymmetry and kurtosis measures the weight of the tails. Each is converted into a roughly standard normal score, Z(√b₁) and Z(b₂), and K² = Z(√b₁)² + Z(b₂)² is compared with a chi-square distribution with 2 degrees of freedom, so p = e^(−K²/2). The test is strong against skewed and uniform-like data but can miss non-normal shapes whose skewness and kurtosis are close to normal.
The Jarque-Bera test
JB = n ÷ 6 × ( S² + (K − 3)² ÷ 4 )S is the sample skewness and K the sample kurtosis
JB is also compared with a chi-square distribution with 2 degrees of freedom. Its approximation is only reliable for samples of a few hundred values, so treat small-sample JB results as rough.
Worked example
Take ten values: 1.5, 2.2, 2.9, 3.1, 3.8, 4.4, 5.0, 5.9, 7.3, 9.9. The mean is 4.60 and the standard deviation (dividing by n) is 2.4170. For the Shapiro-Wilk weights at n = 10 (0.5739, 0.3291, 0.2141, 0.1224, 0.0399) the paired gaps give 0.5739 × 8.4 + 0.3291 × 5.1 + 0.2141 × 3.0 + 0.1224 × 1.9 + 0.0399 × 0.6 = 7.3980. The sum of squared deviations is 58.42, so W = 7.3980² ÷ 58.42 = 0.9368.
All six p-values come out between 0.26 and 0.61 (Shapiro-Wilk 0.549, Shapiro-Francia 0.547, D'Agostino-Pearson 0.260, Jarque-Bera 0.555, Cramer-von Mises 0.608, Anderson-Darling 0.523). None is below 0.05, so there is no evidence against normality, although ten observations can only reveal large departures.
Now replace the largest value with something far out, giving 2.3, 3.1, 3.3, 4.4, 4.7, 5.2, 5.5, 5.9, 6.1, 6.4, 7.8, 8.1, 9.6 and 18. Every test now falls below 0.05: Shapiro-Wilk W = 0.7931 (p = 0.0041), Anderson-Darling 0.9282 (p = 0.0131), Jarque-Bera 16.58 (p = 0.00025). The heavy right tail is the cause.
Limits to keep in mind
- Approximate p-values. The p-values come from published formulas: Royston's normal approximation for the two Shapiro tests, and the fitted curves of D'Agostino and Stephens for Anderson-Darling and Cramer-von Mises. They were derived for the average case and are accurate to a couple of digits, so treat the many decimals as a guide and the tiniest p-values as indicative only. The normal curve inside them (fitted probabilities, tail areas and quantiles) is evaluated to about 15 significant digits.
- Fitted parameters. The distance tests estimate the mean and standard deviation from the same data; the adjustments mentioned above account for that.
- Ties and rounding. Heavily rounded data, with many repeated values, can fail normality tests even when the underlying quantity is continuous and normal.
- Multiple tests. Running six tests raises the chance that at least one is below 0.05 by luck; look at the pattern rather than a single result.
- Next steps after checking normality: the normal distribution calculator and the standard deviation calculator cover the distribution itself and its spread.
Frequently asked questions
Which normality test should I use?
The Shapiro-Wilk and Shapiro-Francia tests are generally the most powerful for small and moderate samples. Anderson-Darling is good for tail departures, D'Agostino-Pearson and Jarque-Bera look at skewness and kurtosis. Reading several together is better than relying on one.
How many data points do I need?
At least 10 for this calculator. With so few values the tests rarely reject unless the departure from normality is severe, so a high p-value should be read cautiously. Larger samples give the tests more power.
What does a p-value below 0.05 mean?
If the data really were normal, results this extreme or more would turn up less than 5% of the time. It is evidence against normality, not proof, and it says nothing about which other distribution fits.
Why is my Shapiro-Francia W’ very low when the other tests look fine?
That line reads the numbers in the order you entered them. If the values are not sorted from smallest to largest the statistic is meaningless. Sort the data and calculate again.
References
- Shapiro SS, Wilk MB. An analysis of variance test for normality (complete samples). Biometrika 1965;52:591-611.
- Shapiro SS, Francia RS. An approximate analysis of variance test for normality. Journal of the American Statistical Association 1972;67:215-216.
- Royston P. A pocket-calculator algorithm for the Shapiro-Francia test for non-normality: an application to medicine. Statistics in Medicine 1993;12:181-184.
- Anderson TW, Darling DA. A test of goodness of fit. Journal of the American Statistical Association 1954;49:765-769.
- D'Agostino RB, Pearson ES. Tests for departure from normality. Biometrika 1973;60:613-622.
- Jarque CM, Bera AK. A test for normality of observations and regression residuals. International Statistical Review 1987;55:163-172.
- Cody WJ (1969). Rational Chebyshev approximations for the error function. Mathematics of Computation 23(107), 631-637.
Cite this calculator and page
If you use this page in your work, you can cite it as:
Reeliy Editorial Team. (2026). Normality Calculator. Reeliy. https://reeliy.com/calculators/normality-test-calculator“Normality Calculator.” Reeliy, 2026, https://reeliy.com/calculators/normality-test-calculator.