Skewness measures how lopsided a distribution is. Kurtosis measures how much of its variance comes from rare, extreme values rather than from typical ones. Together they describe the parts of a dataset's shape that the mean and standard deviation cannot see. This skewness and kurtosis calculator computes both, and it computes them under every convention in common use, because the single biggest source of confusion in this corner of statistics is not the arithmetic — it is that two tools can give different answers and both be correct.
Arb Digital maintains a set of free statistics tools for exactly this kind of check. Our descriptive statistics calculator reports a skewness figure inside a broader summary of centre and spread; this page is the one that goes deeper on shape alone, prints the sample and population forms side by side, distinguishes excess from raw kurtosis, and explains which software uses which. If you already know your figures and just need the picture, the box plot calculator and the histogram calculator show the same shape visually.
What This Skewness and Kurtosis Calculator Does
It parses your data, computes the mean and the second, third and fourth central moments, and derives four numbers: the population skewness g1, the sample-adjusted skewness G1, the population excess kurtosis g2, and the sample-adjusted excess kurtosis G2. You choose which pair drives the headline, and whether kurtosis is shown as excess or raw, but all four are printed in the panel underneath regardless, with the raw kurtosis values alongside.
Below the numbers the tool draws a simple binned sketch of your data so you can see whether the sign of the skewness figure matches what the distribution actually looks like, and it writes a plain-language interpretation of both statistics. Sample size is shown prominently, because these two statistics are far more unstable in small samples than most people expect.
How to Use It
- Paste your data. Any separator works, so a copied spreadsheet column needs no cleaning.
- Pick the definition you need. Choose sample if you are reconciling against a spreadsheet's SKEW and KURT functions, population if you are reconciling against a scientific computing library's defaults.
- Choose excess or raw kurtosis. Excess is the more common reporting convention and puts the normal distribution at zero.
- Read the interpretation, not just the sign. A skewness of 0.1 and a skewness of 2.0 are both "positive" and mean very different things.
- Check the sample size. Under about 30 observations, treat both statistics as rough indications rather than measurements.
The Formulas and Which Definition Is Which
Start with the central moments. For a dataset of n values with mean x̄, the k-th central moment is mk = Σ(xᵢ − x̄)ᵏ ÷ n. The population, or moment, skewness is then g1 = m₃ ÷ m₂3/2, and the population kurtosis is b₂ = m₄ ÷ m₂², with excess kurtosis g2 = b₂ − 3. The 3 is subtracted because a normal distribution has raw kurtosis of exactly 3, so subtracting it puts the normal case at zero and makes the sign directly interpretable.
Those moment forms are biased estimators for a sample. The adjusted Fisher-Pearson versions correct them: G1 = g1 × √(n(n−1)) ÷ (n−2), and G2 = [(n+1)g2 + 6] × (n−1) ÷ [(n−2)(n−3)]. G1 is what a spreadsheet's SKEW function returns and G2 is what its KURT function returns. The NIST/SEMATECH e-Handbook section on measures of skewness and kurtosis sets out both forms and explicitly flags the excess-kurtosis convention as a source of disagreement between sources.
The Default Dataset Worked Through
Take the eight values 2, 4, 4, 5, 7, 9, 12, 15. The mean is 7.25. The central moments are m₂ = 17.4375, m₃ = 44.15625 and m₄ = 641.8008. Population skewness is 44.15625 ÷ 17.43751.5 = 0.6064. Applying the sample correction with n = 8 gives √(8 × 7) ÷ 6 = 1.2472, so G1 = 0.7563. Raw population kurtosis is 641.8008 ÷ 17.4375² = 2.1107, so population excess kurtosis is −0.8893, and the sample-adjusted excess kurtosis works out at −0.4675.
Notice how far apart those figures are. The two skewness values differ by 25% of the smaller one, and the two excess kurtosis values differ by nearly a factor of two — and they even sit on different sides of any threshold you might set at −0.5. Nothing is wrong with either. They are answers to slightly different questions on a sample of eight. That gap narrows as n grows and is negligible by a few hundred observations, which is why the convention matters most precisely when the sample is small enough that people are most tempted to over-read the result.
Reading the Skewness Number
A positive skewness means a longer tail on the right: most values sit low with a few large ones stretching the average upwards. Revenue per customer, page-load times, session durations and household income all look like this. Negative skewness means the tail runs left, which usually indicates a ceiling — exam scores capped at 100, or a satisfaction rating that most people max out.
As a rough guide, an absolute value below 0.5 is close to symmetric, 0.5 to 1 is moderately skewed, and above 1 is strongly skewed. Those bands are conventions, not tests. The consequence that matters is practical: in a strongly right-skewed dataset the mean sits well above the typical value, so reporting an average order value or an average time on page without also reporting the median actively misleads. The mean, median and mode calculator gives you both centres to compare, and a large gap between them is itself a skewness signal.
Reading the Kurtosis Number, and What It Is Not
Kurtosis is widely and wrongly described as "peakedness". It is better understood as tailedness: it measures how much of the variance comes from rare extreme deviations. A high excess kurtosis — a leptokurtic distribution — means occasional very large departures from the mean, with more of the data bunched near the centre than a normal curve would put there. A negative excess kurtosis, platykurtic, means the opposite: the data is spread more evenly, with tails that die off faster than a normal distribution's.
The uniform distribution has an excess kurtosis of −1.2, which is the practical floor for a bounded, evenly spread variable. The default dataset above sits near −0.5 by the sample definition, indicating a distribution that is more evenly spread than a bell curve — consistent with a small set of values spanning a wide range with no strong central cluster. High kurtosis matters most in risk work, because a variable with fat tails will produce extreme outcomes far more often than a normal model predicts, and any interval built on the normal assumption will be too narrow. Confirm suspected extremes with the outlier calculator before drawing conclusions.
Why Small Samples Make Both Statistics Unreliable
Skewness depends on cubed deviations and kurtosis on fourth powers. Raising deviations to a high power means the largest observation dominates the calculation: in a sample of twenty, a single extreme point can flip the sign of the skewness and double the kurtosis. Both statistics have large standard errors at small n — the standard error of skewness is roughly √(6/n), which for n = 20 is about 0.55, so a measured skewness of 0.5 is not distinguishable from zero at all.
The working rule is to treat these numbers as descriptive below about 100 observations and only as evidence above that. Below n = 4 the sample kurtosis formula is undefined outright, because its denominator contains (n−3), and this tool reports that rather than printing a meaningless value. Where a formal question about normality is at stake, a test built for the purpose is a better instrument than a threshold applied to a skewness number, and a plot is better than either. The NIST/SEMATECH catalogue of graphical techniques lists the standard plots for the job.
Why Two Tools Disagree, and How to Settle It
There are two independent choices, so four possible answers. First, sample-adjusted or population moment. Second, excess or raw kurtosis. A spreadsheet's SKEW and KURT return the sample-adjusted skewness and the sample-adjusted excess kurtosis. Many scientific libraries default to the population moment forms, and several of them default to reporting excess kurtosis while offering a switch for raw. Statistical packages vary again, and some let you choose between three skewness types.
When two numbers disagree, work through it in order: check whether one has subtracted 3 and the other has not, which shows up as a gap of almost exactly 3; then check the sample correction, which for moderate n makes the sample figure slightly larger in absolute value than the population one. This tool prints all four values at once specifically so that the reconciliation takes seconds. Penn State's open STAT 200 elementary statistics course is a good reference for the underlying definitions if you need to cite one.
Arb Digital builds analytics reporting that shows distribution and tail risk, not a single mean with nothing behind it.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Reporting a figure without naming the definition — "skewness = 0.76" is incomplete without saying sample or population.
- Comparing a raw kurtosis against an excess kurtosis, which produces an apparent disagreement of almost exactly 3.
- Calling kurtosis peakedness — it is driven by the tails, and two distributions with identical peaks can have very different kurtosis.
- Reading a sign from a sample of twenty, where the standard error of skewness is larger than most measured values.
- Treating a skewness threshold as a normality test instead of using a test built for that purpose alongside a plot.
Related Free Tools From Arb Digital
Get the full summary including centre and spread from the descriptive statistics calculator, measure dispersion alone with the standard deviation calculator, see the shape drawn with the histogram calculator, position a single value with the z-score calculator, or check how much data a bell curve would put inside each band with the empirical rule calculator. Everything else is on the free online tools hub.
Frequently Asked Questions
Population skewness is the plain moment coefficient. Sample skewness multiplies it by a correction factor built from the sample size to reduce bias, and it is what a spreadsheet's SKEW function returns.
Raw kurtosis minus 3. The subtraction puts a normal distribution at zero, so a positive value means heavier tails than normal and a negative value means lighter tails.
Almost always because one reports excess kurtosis and the other reports raw, or because one applies the sample correction and the other does not. A gap of about 3 points to the excess convention.
By common convention, an absolute skewness under 0.5 is roughly symmetric, 0.5 to 1 is moderate, and above 1 is strong. These are reporting conventions rather than statistical tests.
Not really. Kurtosis is driven by the tails of the distribution, so a high value means extreme values occur more often than a normal curve predicts, whatever the peak looks like.
The standard error of skewness is roughly the square root of six divided by the sample size, so small samples are very unstable. Treat both figures as descriptive below about a hundred observations.
Its correction factor divides by the sample size minus three, so with fewer than four observations the formula is undefined and the tool reports that instead of a number.
This tool is provided for educational and analytical use. Shape statistics describe a sample and are not a substitute for a formal test of distributional assumptions.
