Skewness & Kurtosis Calculator

Introduction: Shape of a Distribution

Skewness and kurtosis describe how a distribution differs from the familiar bell-shaped normal curve. While the mean and variance tell you about the center and spread, skewness and kurtosis capture asymmetry and tail heaviness. They are especially useful when you want to check whether a normality assumption is reasonable for regression, hypothesis tests, or risk modeling.

This calculator takes a sample of numeric data, computes the sample skewness and sample kurtosis, and helps you interpret what those values say about the shape of your distribution. The explanations below use standard moment-based definitions and focus on practical interpretation rather than advanced theory.

Sample Skewness: Measuring Asymmetry

Skewness quantifies how symmetric (or asymmetric) your data are around the mean. A perfectly symmetric distribution has skewness equal to zero. When the right tail is longer or fatter than the left tail, the distribution is right-skewed (positive skewness). When the left tail is longer or fatter, it is left-skewed (negative skewness).

Intuitively:

• Skewness > 0: more extreme large values than small ones (right tail).
• Skewness < 0: more extreme small values than large ones (left tail).
• Skewness ≈ 0: roughly symmetric around the mean.

Sample Skewness Formula

Let x₁, …, x_n be your sample of size n, with sample mean \bar{x} and sample standard deviation s. A commonly used definition of the (bias-adjusted) sample skewness is based on the third standardized moment:

The calculator uses a standard sample formula (or an equivalent algebraic variant) so that skewness is comparable across different sample sizes.

Sample Kurtosis: Measuring Tail Heaviness

Kurtosis summarizes how heavy or light the tails of your distribution are compared with a normal distribution. It is based on the fourth standardized moment and is most often used in one of two forms:

• Raw kurtosis: kurtosis of a normal distribution is 3.
• Excess kurtosis: kurtosis minus 3; a normal distribution has excess kurtosis 0.

This calculator reports sample kurtosis in a conventional moment-based form; you can mentally subtract 3 to get excess kurtosis if required.

Sample Kurtosis Formula

With the same notation as above, a commonly used bias-adjusted sample kurtosis is:

Depending on convention, some software reports the value above (an excess-style measure) while others add 3 back. The key interpretation is relative tail weight, not the exact constant used.

Interpreting Skewness and Kurtosis Results

There are no universal cutoffs for what counts as “high” or “low” skewness and kurtosis, but some rules of thumb are widely used in practice.

Typical Interpretation Ranges

• Skewness
  • |skewness| < 0.5: distribution is approximately symmetric.
  • 0.5 ≤ |skewness| < 1: moderate skewness.
  • |skewness| ≥ 1: substantial skewness; normality assumptions may be questionable.
• Excess kurtosis (kurtosis − 3)
  • ≈ 0: tails similar to a normal distribution.
  • > 0: leptokurtic — heavy tails and a sharper peak.
  • < 0: platykurtic — light tails and a flatter peak.

Always interpret these values in context: sample size, the subject-matter domain, and the presence of outliers can all change how concerning a given value should be.

Worked Example

Suppose you have the following small dataset, representing for example five measured values:

2, 3, 5, 7, 11

1. Enter the data into the calculator as comma-separated values: 2,3,5,7,11.
2. Compute: the tool calculates the sample mean, standard deviation, skewness, and kurtosis.

For this dataset (rounded to 4 decimal places), you would obtain values similar to:

• Skewness ≈ 0.26
• Kurtosis (about relative to normal) ≈ -1.36

The skewness is slightly positive, indicating a mild right tail: the largest value (11) lies farther above the mean than the smallest value (2) lies below it. The kurtosis is negative, indicating lighter tails and a flatter peak than a normal distribution. In practice, these values suggest that the distribution is not extremely non-normal, but also not perfectly bell-shaped.

Comparison Summary

The table below summarizes how skewness and kurtosis describe different aspects of distribution shape.

Practical Use Cases

Skewness and kurtosis appear in many applied settings:

• Quality control and manufacturing: detect whether process measurements deviate from a target symmetric distribution, which can indicate drifts or systematic shifts.
• Finance and risk management: characterize return distributions, where heavy-tailed (high-kurtosis) behavior often signals higher risk of extreme losses or gains.
• Experimental science: test whether measurement errors are approximately normal, a common assumption for many statistical models.
• Survey analysis: assess whether responses cluster strongly near one end of a scale (skewness) or have more extreme responses than expected (kurtosis).

Assumptions and Limitations

Although skewness and kurtosis are widely used, they have important limitations:

• Sample size dependence: for small samples, estimates of skewness and kurtosis can be highly variable. Apparent non-normality may simply be sampling noise.
• Sensitivity to outliers: both measures are based on high powers of deviations from the mean (third and fourth power). A single extreme observation can dominate the result and exaggerate skewness or kurtosis.
• Sample vs. population: the calculator reports sample skewness and kurtosis. These are estimates of unknown population values and are subject to sampling error.
• Different software conventions: statistical packages (R, Python, Excel, SPSS, etc.) sometimes use slightly different formulas or normalizations. As a result, you may see small numerical differences for the same data between tools.
• Not a full normality test: skewness and kurtosis describe shape but do not constitute a formal hypothesis test. For formal testing, consider normality tests such as Shapiro–Wilk or Jarque–Bera, which combine information from these moments with distributional assumptions.
• Context still matters: what counts as problematic skewness or kurtosis in one field (e.g., finance) may be perfectly acceptable in another. Always interpret the numbers in light of subject-matter knowledge.

Frequently Asked Questions

What is a “good” skewness value?

No single cutoff defines a “good” skewness. In many applied settings, |skewness| less than about 0.5 is considered close enough to symmetric for approximate normal-based methods, especially with moderate or large sample sizes. However, stricter or looser thresholds may be used depending on the analysis.

What is excess kurtosis?

Excess kurtosis is simply kurtosis minus 3. Because a normal distribution has kurtosis 3, its excess kurtosis is 0. Reporting excess kurtosis makes it easier to see whether your data have heavier (positive) or lighter (negative) tails than normal. If the calculator reports kurtosis close to 3, then your excess kurtosis is close to 0.

Can skewness and kurtosis detect outliers?

Large skewness or kurtosis values often signal the presence of outliers or heavy tails, but they do not identify which observations are outliers. They should be treated as global shape diagnostics. If you see extreme values, you can follow up with visual tools such as histograms, boxplots, or Q–Q plots, and apply formal outlier detection methods when appropriate.

How many data points do I need?

There is no strict minimum, but in very small samples (for example, fewer than 20 points), skewness and kurtosis estimates can fluctuate dramatically from sample to sample. With larger samples, the estimates become more stable and more informative about the underlying population.

Next Steps and Related Analysis

After examining skewness and kurtosis, you may want to compute other descriptive statistics such as the mean, median, variance, or standard deviation, or apply formal normality tests. These complementary measures give a fuller picture of your data and help you decide whether transformations, robust methods, or alternative models are needed.