Skewness Calculator

How to Use This Calculator

1. Enter your data as a comma-separated list (at least 3 values).
2. Read the Fisher-Pearson sample skewness G1 and distribution shape instantly.
3. Use Pearson's Skewness tab for the simpler (mean−median)/SD and (mean−mode)/SD formulas.
4. Use Sample Skewness tab for G1 with its standard error and Z-score for significance testing.
5. Use Bowley's Skewness tab for the quartile-based skewness — more robust to outliers.
6. Professional tier adds mean vs median comparison and recommended summary statistic.

Formula

Example

Frequently Asked Questions

• What is skewness and what does it tell me about my data? ▼
  Skewness is a measure of the asymmetry of a probability distribution. A symmetric distribution (like the normal bell curve) has skewness = 0. A positively skewed distribution has a longer tail extending to the right — the bulk of values cluster at the lower end with a few very large outliers pulling the tail rightward. A negatively skewed distribution has a longer tail extending to the left — values cluster at the upper end with a few very small outliers pulling the tail leftward. Skewness tells you several things: first, whether your data has outliers on one side; second, whether the mean or median is a better measure of center; and third, which statistical tests and models are appropriate. Income distributions are the classic example of positive skewness — most people earn near the median, but a small number of very high earners pull the mean far above the median and create a long right tail. Exam scores in easy tests are often negatively skewed — most students score near the maximum, with a few very low scores pulling the tail left. Karl Pearson formalized skewness in 1895 as part of his system of frequency curves.
• What is the difference between positive and negative skewness? ▼
  Positive skewness (right skew) means the distribution has a longer tail on the right side. Most values cluster below the mean, and the mean is pulled above the median by extreme high values. The tail extends toward larger numbers. Common examples: income distributions, house prices, survival times for diseases, stock returns, city population sizes, and word frequencies in language. Negative skewness (left skew) means the distribution has a longer tail on the left side. Most values cluster above the mean, and the mean is pulled below the median by extreme low values. The tail extends toward smaller numbers. Common examples: age at death in developed countries (most people die in their 70s-80s, with the tail extending left toward early deaths), scores on easy tests (most students score high, few score very low), and reaction times with a left floor imposed by minimum possible speed. A useful memory trick: the direction of skew is the direction of the long tail, not the peak. A right-skewed distribution has a peak on the left with a tail going right — this confuses many beginners who expect the peak to indicate the direction.
• Why is skewness important when choosing between mean and median? ▼
  The mean is sensitive to extreme values (outliers) in a way the median is not. In a skewed distribution, the mean is pulled toward the long tail and away from the typical value. In a highly right-skewed income distribution with most people earning 40,000-60,000 and a few billionaires, the mean might be 80,000 while the median is 52,000 — the mean is not a useful description of a typical person's income. When skewness is high (|skewness| > 1), the median plus interquartile range (IQR) gives a more informative and robust description of the data than mean plus standard deviation. Standard deviation is also inflated by outliers and tail values. For approximately symmetric data (|skewness| < 0.5), the mean and median are close, and mean + SD is appropriate. For moderately skewed data (0.5 to 1), report both. Skewness also affects statistical tests: many parametric tests (t-test, ANOVA) assume normally distributed data, and severe skewness violates this assumption. In such cases, use nonparametric alternatives (Mann-Whitney U, Kruskal-Wallis) or transform the data (log, square root) to reduce skewness before applying parametric tests.
• How do I interpret skewness values (0.5, 1, 2)? ▼
  Standard interpretation thresholds for the Fisher-Pearson sample skewness (G1): |skewness| < 0.5 is approximately symmetric — the distribution is close enough to symmetric for most purposes, and mean and SD are appropriate summaries. |skewness| between 0.5 and 1 indicates moderate skewness — visible asymmetry, worth noting; consider reporting both mean and median; parametric tests still generally robust for large samples. |skewness| > 1 indicates high skewness — significant asymmetry; median and IQR are preferred summaries; parametric tests may be inappropriate without transformation; the distribution departs substantially from normality. |skewness| > 2 is extreme skewness — found in distributions like the lognormal, Pareto, or exponential distributions, common in financial returns, internet traffic, and natural phenomena following power laws. Note that these thresholds are guidelines, not firm rules. The sample size matters: in small samples (n < 30), the standard error of skewness is large (roughly sqrt(6/n)), making the estimate unreliable. A formal test of skewness uses the Z-score = G1 / SE_skewness; values beyond ±2 suggest statistically significant skewness at the 5% level.
• Can I transform skewed data to make it more symmetric? ▼
  Yes, data transformations are a standard tool for reducing skewness and making data more suitable for parametric statistical methods. For positive (right) skew, common transformations include: logarithm (log(x) or log(x+1) for zero values) — the most common transformation for exponential or multiplicative processes, effective when values span several orders of magnitude; square root (sqrt(x)) — milder transformation, useful for count data and moderate right skew; inverse (1/x) — strong transformation for highly skewed data. For negative (left) skew, reflect the data first (max(x) − x or c − x for some constant c), apply a right-skew transformation, then re-reflect. Box-Cox transformation is a systematic family that includes log and power transformations as special cases, with a parameter lambda chosen to optimize normality. The Johnson transformation system can handle virtually any skewness pattern. Important cautions: transformations change interpretation — results must be back-transformed and interpreted on the original scale; log(mean) does not equal mean(log), so the geometric mean corresponds to the median of the log-transformed data; always report which transformation was applied and justify the choice. Transformations should be chosen for scientific reasons, not just to pass normality tests.

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