Skewness and Kurtosis Calculator

StatisticsLast updated: August 22, 2026

Skewness measures how far a distribution departs from symmetry, and kurtosis measures how heavy the tails and how peaked the center are. Both are zero for a normal distribution, which is why the two coefficients together are the most common way to judge whether the normality assumption is reasonable.

This calculator returns the sample-corrected g₁ and g₂ coefficients (the same formulas as SPSS Descriptives and the Excel SKEW and KURT functions), their standard errors (SES and SEK), and the corresponding z-ratios. It also runs the D'Agostino-Pearson K² omnibus normality test, reports the decision at the significance level you choose, and draws a histogram alongside a normal Q-Q plot so you can confirm the numbers visually.

At least 8 values are required. Separate the numbers with commas, spaces, or line breaks.
Sets the false-positive risk. In the social sciences the usual choice is 5%.

Skewness and Kurtosis Formulas

Central moment:   mₖ = Σ(xᵢ − x̄)ᵏ / n
Skewness:         g₁ = [n / ((n−1)(n−2))] · Σ((xᵢ − x̄)/s)³
Kurtosis:         g₂ = {n(n+1) / [(n−1)(n−2)(n−3)]} · Σ((xᵢ − x̄)/s)⁴ − 3(n−1)² / [(n−2)(n−3)]
SE of skewness:   SES = √[ 6n(n−1) / ((n−2)(n+1)(n+3)) ]
SE of kurtosis:   SEK = 2·SES·√[ (n²−1) / ((n−3)(n+5)) ]
z-ratios:         z₁ = g₁ / SES ,  z₂ = g₂ / SEK
Omnibus test:     K² = Z(√b₁)² + Z(b₂)² ~ χ²(2)
Pearson 2:        Sk = 3·(x̄ − median) / s
Under normality:  g₁ = 0 ,  g₂ = 0  (b₂ = 3)

There is more than one definition of skewness and kurtosis. This tool reports the sample-corrected G₁/G₂ coefficients used by SPSS and Excel, while the D'Agostino-Pearson test runs on the uncorrected moment coefficients √b₁ and b₂.

How to Calculate

  1. Paste your data values into the box; at least 20 observations are recommended for a dependable result (8 is the minimum).
  2. Choose the significance level; it sets the critical z value and the decision of the K² test.
  3. Look at the sign of the skewness: positive means right-skewed (long tail on the right), negative means left-skewed.
  4. Compare the z-ratios with the ±critical z bound; if the bound is exceeded, that departure is statistically significant.
  5. Read the p-value of the D'Agostino-Pearson K² test: below α, the normality assumption is rejected.
  6. Check the histogram and the Q-Q plot to confirm the numbers visually; in large samples the plots are more informative than the test.

Worked Examples

Right-skewed, income-like data

For 20 observations, skewness is g₁ = 2.0406 with a standard error of SES = 0.5121, so z(g₁) = 3.985; this exceeds the ±1.96 bound, making the departure from symmetry significant. Excess kurtosis is g₂ = 4.7649 with SEK = 0.9924, giving z(g₂) = 4.801. The D'Agostino-Pearson test yields K² = 19.198 with p < 0.001, so normality is rejected; the mean is 23.10 while the median stays at 19.50.

Number of values (n): 20 · Skewness (g₁): 2.0406 · Standard error of skewness (SES): 0.5121

Near-normal symmetric data

For 16 measurements that are perfectly symmetric around 50, skewness is g₁ = 0.0000 and excess kurtosis is g₂ = -0.0022; the z values are 0.000 and -0.002, both inside the ±1.96 bound. With K² = 0.051 and p = 0.975, normality cannot be rejected; the mean and the median coincide at 50, and the Q-Q points track the line closely.

Number of values (n): 16 · Skewness (g₁): 0.0000 · Standard error of skewness (SES): 0.5643

Left-skewed exam scores

For 20 scores showing a ceiling effect, skewness is g₁ = -1.9565 (z = -3.820) and excess kurtosis is g₂ = 4.4718 (z = 4.506); the long tail is on the left, meaning a few low scores such as 45 and 62 drag the mean down. With K² = 18.072 and p < 0.001 normality is rejected, and because the mean is 81.75 while the median is 84.50, the median is the better measure of center.

Number of values (n): 20 · Skewness (g₁): -1.9565 · Standard error of skewness (SES): 0.5121

Frequently Asked Questions

What range of skewness and kurtosis counts as 'normal'?
There is no exact rule, but the common convention is that |g₁| < 1 and |g₂| < 1 are acceptable and the ±2 range is the limit. Some sources, especially in structural equation modeling, use |g₁| < 3 and |g₂| < 10. Never read the number on its own: judge it together with the histogram and the Q-Q plot.
What is a z-ratio and why is it used?
The z-ratio divides a coefficient by its own standard error, so it measures whether the departure from zero is meaningful given the sample size. A |z| above 1.96 is treated as significant at α = 0.05. However, the standard error shrinks as n grows, so even tiny departures become significant in large samples; there it is more sensible to read the size of the coefficient itself.
Should kurtosis be 3 or 0?
Both are correct, depending on the definition. The raw moment kurtosis b₂ equals 3 for a normal distribution; excess kurtosis subtracts 3 from b₂ and is therefore 0 under normality. SPSS, Excel, and this tool report excess kurtosis, so the reference value is 0 — but both coefficients are shown separately in the results.
Is the D'Agostino-Pearson test different from Shapiro-Wilk?
Yes. D'Agostino-Pearson combines only the departures in skewness and kurtosis, which makes it diagnostic: it also tells you in which direction the data depart. Shapiro-Wilk is a more general normality test and is usually more powerful in small samples. The two agree on most data sets; when they disagree, the Q-Q plot should settle it.
What should I do if normality is rejected?
First look for the cause: outliers, a mixture of two subgroups, or ceiling and floor effects are the usual suspects. For right-skewed data a log or square-root transformation often fixes the distribution. If transformation does not help, switch to nonparametric tests such as Mann-Whitney U or Kruskal-Wallis; in large samples, mean-based tests are already robust thanks to the central limit theorem.
How many observations do I need for skewness and kurtosis?
The kurtosis formula mathematically requires at least 4 observations, but this tool asks for a minimum of 8 for reliability. In practice, both coefficients fluctuate wildly from sample to sample below n = 20 and can be misleading. For a solid assessment, 50 or more observations are recommended.
Do Excel, SPSS, and R give the same result?
The Excel SKEW and KURT functions and the SPSS Descriptives output use the same sample-corrected formulas and match this tool exactly. Base R has no built-in function; skewness(x, type = 2) and kurtosis(x, type = 2) from the e1071 package give the same values, while type = 1 returns the uncorrected moment coefficients.