What Is Skewness?
Skewness describes the direction and degree of asymmetry in a distribution. A positive value means a longer right tail, a negative value means a longer left tail, and a value near zero means the data are roughly symmetric.
Software does not all use the same formula. This calculator reports the moment coefficient (type 1), the adjusted Fisher-Pearson coefficient used by Excel's SKEW (type 2), the type 3 coefficient and Pearson's second coefficient, so you can match the one your course or package uses.
Below you will find the formulas, a variables table, a worked example and answers to common questions such as what counts as a large skew and how skewness relates to mean and median.
How to Use the Skewness Calculator
- Paste at least three values.
- Pick which definition to feature. All three are listed anyway.
- Read the interpretation, the z-test and the histogram. A skewness that is large in size but not significant simply reflects a small sample.
Skewness Formula
Moment coefficient (type 1)
Adjusted Fisher-Pearson (type 2)
Type 3
Standard error of skewness
Pearson’s second coefficient
Skewness Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| g₁ | Moment coefficient of skewness | Type 1 |
| G₁ | Adjusted Fisher-Pearson coefficient | Type 2, as in Excel SKEW |
| n | Sample size | Needs at least 3 values |
| s | Standard deviation | |
| SE | Standard error of skewness | Used to judge significance |
Skewness Worked Example
With the example values loaded in the calculator above, the steps are:
- Mean = 4.66667; second central moment m₂ = Σ(x − x̄)² ÷ n = 12.0556; third central moment m₃ = Σ(x − x̄)³ ÷ n = 66.2593.
- Type 1: g₁ = m₃ ÷ m₂^1.5 = 1.5829.
- Type 2: G₁ = g₁ × √(n(n − 1)) ÷ (n − 2) = 1.8187.
- Type 3: b₁ = g₁ × ((n − 1)/n)^1.5 = 1.3893.
- Standard error of skewness = 0.6373, so z = 1.8187 ÷ 0.6373 = 2.854.
Result: Skewness (G₁ adjusted, type 2) = 1.8187; z = G₁ / SE = 2.854.
How to Read the Skewness Result
The sample data are 1, 2, 2, 3, 3, 3, 4, 4, 5, 6, 9, 14. The mean, 4.67, is well above the median, 3.5, and the skewness is 1.82. A positive value means a longer tail to the right, here created by 9 and 14. The standard error is 0.637, so z = 2.85, which is more than twice its standard error and suggests the skew is not just chance.
With strongly right-skewed data, the median usually describes a typical value better than the mean. A log transformation is a common way to make such data more symmetric before using methods that assume a bell shape.
How to interpret skewness values
The thresholds are conventions, not laws. In small samples even a normal population can produce skewness of ±0.5 by chance, which is why the z-test matters: dividing by the standard error tells you how unusual your value would be if the true distribution were symmetric. A common cut-off is |z| above 1.96 for a 5% significance level.
| Skewness | Description | Typical example |
|---|---|---|
| Below −1 | Highly left-skewed | Age at retirement, exam scores with ceiling |
| −1 to −0.5 | Moderately left-skewed | Test scores on an easy exam |
| −0.5 to 0.5 | Approximately symmetric | Heights, measurement error |
| 0.5 to 1 | Moderately right-skewed | Reaction times |
| Above 1 | Highly right-skewed | Income, house prices, wait times |
Why skewness matters
- Mean vs median: with right skew the mean overstates the typical value; the median is usually the better summary.
- Parametric tests (t-tests, ANOVA, regression) assume roughly normal residuals; strong skew can distort p-values in small samples.
- Transformations such as log or square root can reduce right skew.
- Skewed outcomes in risk analysis mean the extremes on one side are much more likely than a normal model predicts.
Which skewness formula should I report?
Use the one your software or journal uses. For a quick check of a single sample, G₁ is the most common default (Excel, SPSS, SAS). For large samples all three are nearly identical. If you are comparing with R’s e1071::skewness, check its type argument: type 1 is g₁, type 2 is G₁ and type 3 is b₁.
| Tool | Function | Matches |
|---|---|---|
| Excel / Google Sheets | SKEW | G₁ |
| SPSS, SAS | Skewness | G₁ |
| Minitab | Skewness | b₁ |
| R e1071 | skewness(type=1/2/3) | g₁ / G₁ / b₁ |
| SciPy | skew(bias=True) | g₁ |
Where Skewness Is Used
Symmetry check
Measure whether data lean left or right.
Software comparison
See the type 1, 2 and 3 definitions side by side.
Normality
Use the z-score of skewness to judge whether it is large.
Choosing a summary
Decide between the mean and the median.
Common Skewness Mistakes to Avoid
- Always look at the histogram; skewness is a single number and can miss multi-modal shapes.
- Outliers can dominate skewness because it cubes the deviations.
- Do not expect exact agreement between packages unless you match the formula.
- Aim for at least 30 observations before leaning on the significance test.
Skewness Terms and Definitions
- Third central moment
- The average of cubed deviations from the mean; the numerator of skewness.
- Positive skew
- A longer right tail; the mean is greater than the median.
Skewness FAQs
What does skewness tell you?
The direction and degree of asymmetry in a distribution. Positive means a long right tail, negative means a long left tail, near zero means roughly symmetric.
What is a good skewness value?
Between about −0.5 and 0.5 is considered approximately symmetric. Between −1 and 1 is often acceptable for methods that assume normality.
Why are there different skewness formulas?
Different packages apply different small-sample corrections. g₁ is the raw moment coefficient; G₁ and b₁ adjust for sample size. They converge as n grows.
What does it mean if the mean is greater than the median?
It usually indicates right skew, because the long upper tail pulls the mean upward more than the median.
Is my data normal if skewness is zero?
Not necessarily. Zero skewness means symmetry, but the tails could still be much heavier or lighter than normal. Check kurtosis and a histogram too.
How is the standard error of skewness used?
Dividing the skewness by its standard error gives a z-score. A value beyond about ±1.96 suggests the distribution is genuinely skewed rather than the sample being a chance fluctuation.
How do I reduce skewness?
Common transformations are log, square root or reciprocal for right skew, and squaring or exponentiation for left skew. Check the histogram after transforming.
How do I interpret a skewness value?
The sign gives the direction: positive means a longer right tail, negative a longer left tail. Rules of thumb for "large" skew vary, so use the standard error and significance test shown in the results rather than a fixed cut-off.
Which skewness formula does Excel use?
Excel's SKEW function uses the adjusted Fisher-Pearson coefficient, shown here as type 2 (G₁). Excel's SKEW.P gives the population version, which matches the moment coefficient.
Which skewness values does this calculator report?
It reports three: g1 (moment coefficient), G1 (adjusted Fisher-Pearson, the same as Excel SKEW) and b1 (Minitab style). They converge as n grows. Pick the one your software or course uses.
How do I test whether skewness is significant?
Divide G1 by its standard error. A ratio beyond about ±2 suggests real skew. For the sample data 1.8187 ÷ 0.6373 = 2.85.
Why do the mean and median differ in skewed data?
Extreme values in the tail pull the mean toward them while the median barely moves. In the sample data the mean is 4.67 and the median 3.5.