Skewness Calculator
Paste a comma-separated list of numbers to get the skewness and excess kurtosis of your data set. The calculator reports the bias-corrected Fisher-Pearson coefficient G1 (the value Excel and SPSS use), the moment skewness g1, Pearson median skewness, standard errors for both measures, and supporting statistics such as the mean, median, sample standard deviation, variance, range, and count. A gauge shows where your skewness sits on the symmetric-to-highly-skewed scale, and a step-by-step panel shows all the arithmetic with your actual numbers.
Formula
Worked example
For 2, 4, 4, 5, 9, 12, 20 (n = 7): the mean is 8 and the median is 5. The second central moment m2 = 34 and the third m3 = 203.14, giving moment skewness g1 = 203.14 / 34^1.5 = 1.025. Applying the Fisher-Pearson adjustment sqrt(7*6)/5 = 1.296 gives G1 = 1.328, highly right-skewed. Pearson median skewness = 3*(8-5)/s = 1.28. Excess kurtosis = 0.17, near-normal tails.
What skewness tells you about a distribution
Skewness is a single number that captures the asymmetry of a data set around its mean. A perfectly symmetric distribution, like the classic bell curve, has a skewness of zero, with the left and right halves mirroring each other. A positive (right) skew means the distribution has a long tail trailing toward larger values: most observations cluster on the low side while a few large outliers stretch the upper end, which typically drags the mean above the median. A negative (left) skew is the reverse, with a long tail toward smaller values that pulls the mean below the median. Recognising skew matters because many statistical methods assume symmetry, and a strongly skewed variable can distort averages, confidence intervals, and forecasts if it is treated as if it were normal.
The three skewness formulas this calculator reports
Three complementary measures of asymmetry are reported here. The moment coefficient g1 divides the average cubed deviation from the mean by the cube of the population standard deviation; it is simple but slightly biased for small samples. The adjusted Fisher-Pearson coefficient G1 multiplies g1 by sqrt(n(n-1)) / (n-2), correcting that bias so the estimate better reflects the wider population, and it is the value Excel's SKEW function and SPSS report by default. The Pearson median skewness SK2 = 3(mean - median) / s takes a different approach: it measures how far the mean sits from the median in standard-deviation units, making it more robust against extreme outliers because it does not cube the deviations. For large samples all three converge, but for small or outlier-rich data they can differ noticeably.
Excess kurtosis and the shape of the tails
Kurtosis measures how heavy or light the tails of a distribution are relative to a normal distribution, and this calculator reports the excess (Fisher) version so that a normal distribution scores zero. A positive excess kurtosis (leptokurtic) means the data has heavier-than-normal tails and a sharper central peak, often a sign of outliers. A negative excess kurtosis (platykurtic) means lighter tails and a flatter peak. A rule of thumb treats values between -1 and +1 as approximately normal-tailed, and values outside that range as notable. Excess kurtosis requires at least four data points because its formula divides by (n-2)(n-3). The standard errors of skewness and kurtosis let you judge whether the values are large relative to their sampling variation: if the absolute value exceeds roughly twice the standard error, the departure from normality is likely real rather than sampling noise.
Using skewness to choose the right statistical test
Many common tests, including t-tests, ANOVA, and Pearson correlation, assume that the data (or the residuals from a model) are approximately normally distributed. High skewness is often the first visible sign that this assumption is violated. As a rough guide, if |G1| is below 0.5 the data is unlikely to cause problems with these tests for moderate sample sizes. Between 0.5 and 1 you should examine your data more closely, consider transformations such as a square root or logarithm for right-skewed data, or switch to a non-parametric alternative. Above 1, a transformation or non-parametric approach is usually advisable. That said, sample size matters: large samples tolerate more skewness because the central limit theorem brings the sampling distribution of the mean closer to normal, while very small samples produce unreliable skewness estimates regardless of the value.
Interpreting the standard error of skewness
The standard error of skewness (SES) is computed as sqrt(6n(n-1) / ((n-2)(n+1)(n+3))). It decreases as the sample size grows, reflecting that larger samples yield more stable estimates. A common rule of thumb is that if the absolute skewness G1 exceeds 2 * SES, the skewness is statistically notable, meaning it is unlikely to arise from random sampling alone in a symmetric population. For example, with n = 30 the SES is about 0.43, so |G1| > 0.86 would be notable at roughly the 95% level. Similarly the standard error of kurtosis is approximately twice the SES multiplied by sqrt((n^2-1) / ((n-3)(n+5))). These are rough guidelines; a formal test such as the Jarque-Bera test or D'Agostino's test gives a more rigorous answer.
Quick-reference skewness and kurtosis thresholds
| Measure | Range | Interpretation |
|---|---|---|
| |G1| skewness | 0.0 to 0.5 | Fairly symmetric |
| |G1| skewness | 0.5 to 1.0 | Moderately skewed |
| |G1| skewness | Above 1.0 | Highly skewed |
| Skewness sign | Positive | Long tail to the right (mean > median) |
| Skewness sign | Negative | Long tail to the left (mean < median) |
| Excess kurtosis | Above +1 | Leptokurtic: heavy tails, sharp peak |
| Excess kurtosis | -1 to +1 | Approximately normal tails (mesokurtic) |
| Excess kurtosis | Below -1 | Platykurtic: light tails, flat peak |
| |G1| vs 2*SES | G1 > 2*SES | Skewness is statistically notable |
Common rule-of-thumb thresholds used in statistics textbooks and software documentation. These are conventions, not hard rules.
Frequently asked questions
What is the difference between positive and negative skew?
Positive (right) skew means the distribution has a longer tail stretching toward larger values, so a few high outliers pull the mean above the median. Negative (left) skew is the mirror image: the longer tail points toward smaller values, dragging the mean below the median. A skewness near zero indicates a roughly symmetric shape where the two tails are about the same length.
What is the difference between G1 and g1?
g1 is the raw moment skewness: the average cubed deviation from the mean divided by the cube of the population standard deviation. It is slightly biased downward for small samples. G1 is the adjusted Fisher-Pearson coefficient, which multiplies g1 by sqrt(n(n-1)) / (n-2) to correct that bias. G1 is what Excel's SKEW function and SPSS report. For large samples the two values are nearly identical, but for a handful of points the adjustment can shift the figure noticeably.
What is Pearson median skewness and when should I use it?
Pearson median skewness SK2 = 3(mean - median) / s measures how far the mean sits from the median in standard-deviation units. Because it uses the median rather than cubing deviations, it is more robust to extreme outliers. It is a useful sanity check alongside G1: if they point in the same direction, the skew is consistent across methods. If they disagree, one or two extreme values may be dominating the cubic-moment calculation.
Why do I need at least three numbers?
The adjusted Fisher-Pearson formula divides by (n - 2), so it requires at least three values. Kurtosis divides by (n-2)(n-3) and needs at least four values. With one or two data points there is not enough information to describe a meaningful shape, so the calculator returns a blank result until you enter at least three numbers that vary.
What does excess kurtosis measure?
Excess kurtosis (also called Fisher kurtosis) measures the tail-heaviness of a distribution relative to a normal distribution, which scores exactly zero. A positive value (leptokurtic) means the tails are heavier and the peak sharper than a normal curve, which often indicates the presence of outliers. A negative value (platykurtic) means lighter tails and a broader, flatter peak. The Excel KURT function reports this same excess-kurtosis measure.
How do I use the standard error of skewness?
The standard error of skewness (SES) quantifies sampling uncertainty: it shrinks as sample size grows. A common rule of thumb is that if |G1| exceeds 2 times SES, the skewness is statistically notable at roughly the 95% confidence level, meaning the departure from symmetry is unlikely to be purely due to sampling noise. For more formal testing, use the Jarque-Bera test or D'Agostino and Pearson's omnibus test.
Which skewness formula does Excel SKEW use?
Excel's SKEW function computes the adjusted Fisher-Pearson coefficient G1. The formula is G1 = [n / ((n-1)(n-2))] * sum((xi - mean)^3 / s^3), which is algebraically equivalent to G1 = sqrt(n(n-1)) / (n-2) * g1. This calculator's G1 output matches Excel exactly. The Excel KURT function computes excess kurtosis using the Fisher definition that subtracts 3, so a normal distribution gives zero.