What this calculator does
Skewness measures which way a distribution leans. A positive value means a long tail to the right, negative means a long tail to the left, and zero means symmetric. A perfectly even data set like 1, 2, 3, 4, 5 has a skewness of exactly zero.
The measure is dominated by extreme values, because each deviation is cubed before being summed. Adding a single large outlier to an otherwise tame data set can take the skewness from near zero to well above 2, which is both the strength and the weakness of the statistic: it is sensitive to exactly the thing it is designed to detect, and to nothing else.
The formula
Each value is expressed as a z-score, the deviation from the mean divided by the sample standard deviation, and those z-scores are cubed and summed. The sum is scaled by n divided by (n−1)(n−2), which is the adjustment that makes the estimate unbiased for a sample. Cubing preserves sign, so values below the mean contribute negatively and values above contribute positively. At least three data points are required.
| Term | Meaning |
|---|---|
| g₁ | The sample skewness statistic, adjusted for sample size. |
| Right-skewed | Positive skew: a long tail to the right, with the mean pulled above the median. |
| Left-skewed | Negative skew: a long tail to the left, with the mean pulled below the median. |
| Symmetric | Skewness near zero, with the tails balanced either side of the centre. |
The inputs explained
| Field | What to enter |
|---|---|
| Data (comma or space separated) | Your data, comma or space separated. At least three values are needed, and the statistic is unreliable for fewer than about twenty. |
When to use it
Checking a normality assumption
Many tests assume roughly symmetric data, and skewness is the quickest numerical check on that.
Understanding income or price data
Such data is almost always right-skewed, which is why the median is usually reported rather than the mean.
Deciding whether to transform data
Strong positive skew is often reduced by a log transform, and the skewness before and after shows whether it worked.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What does skewness look like for different data?
Symmetric, left-skewed and right-skewed data compared.
| Data set | Skewness (g₁) | Direction | Mean |
|---|---|---|---|
| 1, 2, 3, 4, 5 | 0 | Roughly symmetric | 3.000 |
| 5, 7, 9, 9, 9, 10, 12 | -0.3998 | Left-skewed: long tail to the left | 8.714 |
| 2, 4, 4, 4, 5, 5, 7, 9 | 0.8185 | Right-skewed: long tail to the right | 5.000 |
| 1, 1, 2, 2, 3, 3, 4, 20 | 2.682 | Right-skewed: long tail to the right | 4.500 |
Questions
What is a normal range for skewness?
Between −0.5 and 0.5 is usually called approximately symmetric, −1 to −0.5 or 0.5 to 1 moderately skewed, and beyond ±1 highly skewed. These are conventions rather than tests, and they are less meaningful for small samples where skewness is estimated poorly.
Does skewness tell me whether data is normal?
Only partly. A normal distribution has zero skewness, but zero skewness does not make a distribution normal, since a symmetric distribution can have quite the wrong tail weight. Check kurtosis as well, and prefer a proper normality test if the answer matters.
Why is income data right-skewed?
Because income has a floor at zero but no ceiling. Most people cluster in a band while a small number earn very much more, producing a long right tail. This is why median income is the standard reported figure: the mean is pulled upward by that tail.
How many data points do I need?
Formally three, but the estimate is very unstable below about twenty and still noisy below fifty. For small samples, a single value can dominate the result, so treat a skewness from ten points as a rough indication rather than a measurement.
For tail weight rather than lean, see the kurtosis calculator. For the full summary, see the descriptive statistics calculator.