Skewness
Compute sample and population skewness, excess kurtosis, and distribution shape from comma-separated data with interactive charts. Free online statistics calculator.
About This Calculator
About the Skewness and Kurtosis Calculator
This calculator computes skewness and kurtosis for any comma-separated dataset — two essential measures that describe the shape of a statistical distribution. Skewness tells you whether your data leans left or right, while kurtosis reveals whether the tails are heavier or lighter than a normal distribution. Students, researchers, data analysts, and quality control professionals use these metrics to assess normality before applying parametric tests.
The calculator computes both sample and population versions of skewness and kurtosis. Sample skewness uses the Excel-compatible formula (adjusted for bias with n/(n-1)(n-2) factor), while population skewness uses the raw moment formula. The same dual approach applies to kurtosis — we report excess kurtosis (Fisher definition) where zero matches a normal distribution. An automatic interpretation classifies your results as approximately symmetric, moderately skewed, or substantially skewed, and kurtosis as mesokurtic, leptokurtic, or platykurtic.
Regional Notes
India (IN): Skewness and kurtosis are widely used in Indian higher education (B.Sc. Statistics, MBA programs) and by market researchers analyzing consumer behavior. The formulas taught follow the same sample/population distinction used here.
United States (US): Used extensively in social sciences, psychology (SPSS/Excel output), Six Sigma quality control, and financial risk analysis. US textbooks commonly teach the sample moment coefficient of skewness (Fisher-Pearson standardized moment coefficient).
United Kingdom (UK): Applied in A-Level and university statistics curricula, medical research (epidemiology), and actuarial science. UK exam boards (Edexcel, AQA, OCR) require students to calculate and interpret both skewness and kurtosis.
Frequently Asked Questions
What is skewness?
Skewness measures the asymmetry of a probability distribution. A positive skew means the right tail is longer (more extreme values on the right), while a negative skew means the left tail is longer. A skewness near zero indicates a roughly symmetric distribution. Values between -0.5 and 0.5 are approximately symmetric, -1 to -0.5 or 0.5 to 1 are moderately skewed, and beyond -1 or 1 are substantially skewed.
What is kurtosis?
Kurtosis measures the tailedness or peakedness of a distribution relative to a normal distribution. Positive excess kurtosis (leptokurtic) means heavier tails and a sharper peak — more outliers. Negative excess kurtosis (platykurtic) means lighter tails and a flatter peak. Zero excess kurtosis (mesokurtic) matches the normal distribution.
How is sample skewness calculated?
Sample skewness uses the formula: skewness = n × Σ(xi - x̄)³ / [(n-1)(n-2) × s³], where x̄ is the sample mean, s is the sample standard deviation, and n is the number of observations. This matches the Excel SKEW function and requires at least 3 data points.
What is the difference between sample and population skewness?
Population skewness divides by N and uses population standard deviation (σ), while sample skewness adjusts for bias by using N(N-1)/(N-2)(N-3) factors and uses sample standard deviation (s). Sample skewness is recommended for most real-world data analysis since datasets are typically samples from a larger population.
What does a positive skewness mean?
A positive skewness (right-skewed) means the tail on the right side of the distribution is longer or fatter. The mean is typically greater than the median. Common examples include income distributions, house prices, and reaction times — where most values cluster on the left with a long tail of high values on the right.
What does a negative skewness mean?
A negative skewness (left-skewed) means the tail on the left side of the distribution is longer or fatter. The mean is typically less than the median. Examples include age at death (most people die at older ages with a tail of younger deaths) and test scores when the test is easy (most scores are high with a tail of low scores).
How many data points do I need for skewness and kurtosis?
You need at least 3 data points to calculate skewness and at least 4 data points to calculate kurtosis. For reliable estimates, statisticians generally recommend at least 20-30 data points. The calculator will show results for any sample meeting the minimum requirement but will note small-sample limitations.
What is excess kurtosis vs regular kurtosis?
Excess kurtosis subtracts 3 from the raw kurtosis value so that a perfect normal distribution has an excess kurtosis of 0. This calculator reports excess kurtosis (sometimes called Fisher kurtosis). Raw kurtosis of 3 means mesokurtic/normal. Positive excess kurtosis indicates heavier tails than normal, negative indicates lighter tails.