How to use this calculator
Enter your values in the fields above and click Calculate to see your results instantly. All calculations run in your browser — no data is sent to a server and results appear immediately. Click Clear to reset all fields and start over.
Understanding your inputs
Each input field is labeled with the specific value it expects. Hover over the ? hint icons (where present) for additional guidance on what each field means and what units to use. For best results, double-check that all your input values use consistent units before calculating.
Interpreting the results
Results are shown immediately after clicking Calculate. The highlighted result card shows the primary output — the value most people need. Additional cards show supporting calculations that provide context and help you verify the primary result makes sense. If results seem unexpected, re-check your inputs for typos or unit mismatches.
About this statistical calculator
This calculator implements standard statistical formulas used by professionals and students alike. The underlying math has been verified against reference implementations and textbook examples. For critical applications, always cross-reference results with authoritative sources or a qualified professional.
What skewness tells you and when to use it
Skewness measures how lopsided a set of numbers is around its mean. A perfectly symmetric dataset, such as heights clustered evenly on both sides of the average, has a skewness near zero. When a few unusually large values stretch the right-hand tail, the skewness is positive; when a few unusually small values stretch the left-hand tail, it is negative. Because it is a single number, it gives a quick check on the shape of a distribution before you pick a summary statistic or a statistical test.
Use it when you need to decide whether the mean is a fair description of the center, whether a log or square-root transform might be needed, or whether a method that assumes roughly normal data is reasonable. Income, home prices, response times and waiting times are typically right-skewed; exam scores with a ceiling effect are often left-skewed. Paste your raw values into the field above, separated by commas or spaces, and at least three numbers are required.
The formula and its variables
The calculator reports two versions. Both start from the deviations of each value from the mean and cube them, which keeps the sign (so direction is preserved) and gives extreme values a very large influence.
- n is the number of values and x̄ is their mean.
- m2 = Σ(x − x̄)² divided by n is the population variance, and s is the sample standard deviation, using n − 1 in the denominator.
- Population skewness g1 = [Σ(x − x̄)³ / n] / (population standard deviation)³.
- Sample skewness G1 = n / ((n − 1)(n − 2)) × Σ(x − x̄)³ / s³. This bias-corrected form is the one spreadsheets return from their SKEW function.
The page labels the result as approximately symmetric below 0.5 in absolute value, moderately skewed from 0.5 to 1, and highly skewed at 1 or above. These bands are common rules of thumb rather than hard statistical tests.
Worked example
Take the five values 2, 4, 4, 5, 9. The mean is 24 / 5 = 4.8. The deviations are −2.8, −0.8, −0.8, 0.2 and 4.2. Squared, they sum to 7.84 + 0.64 + 0.64 + 0.04 + 17.64 = 26.8. Cubed, they sum to −21.952 − 0.512 − 0.512 + 0.008 + 74.088 = 51.12.
The population variance is 26.8 / 5 = 5.36, so the population standard deviation is about 2.3152 and its cube is about 12.410. Then g1 = (51.12 / 5) / 12.410 = 10.224 / 12.410 ≈ 0.8239. The sample standard deviation is √(26.8 / 4) ≈ 2.5884, whose cube is about 17.343, so G1 = 5 / (4 × 3) × 51.12 / 17.343 ≈ 1.2282. The calculator shows Mean = 4.8000, Std Dev (sample) = 2.5884, sample skewness 1.2282 and population skewness 0.8239, and labels the data positively skewed and highly skewed. The single value 9 sitting far above the cluster near 4 is what drives that result.
Common mistakes and how to interpret the result
- Confusing the two versions. Software differs on whether it reports g1 or G1. With small samples they can differ a lot, as in the example above, so state which one you used.
- Trusting skewness from tiny samples. With only 3 to 10 values a single outlier dominates the cubed term. Treat the number as a rough hint, not proof of a skewed population.
- Reading zero as proof of normality. A skewness of zero only shows balance; a symmetric distribution can still have heavy tails or two humps. Check kurtosis and a histogram as well.
- Mixing up the tail direction. Positive skew means the long tail points right, and the bulk of the data sits to the left of the mean, not the right.
Related tools: Kurtosis Calculator for tail weight, Standard Deviation Calculator for the spread used in the denominator, Mean, Median and Mode Calculator to see how skew pulls the mean away from the median, and Normal Distribution Calculator for the symmetric benchmark.