Distributions with the same mean and standard deviation can still have different tail directions and tail thickness.
Income data often contain many middle values and a few very large ones, creating a long right tail. An easy exam may show the opposite pattern: scores pile up near the top while a tail stretches toward lower values.
The mean and standard deviation do not reveal which way a distribution is asymmetric or how much weight lies in the tails.
Key question
How can we summarize the direction of asymmetry and the thickness of a distribution's tails?
What does skewness describe?
| Skewness | Typical shape | Example |
|---|---|---|
| Near 0 | roughly symmetric | height or measurement error |
| Positive | long right tail | income or waiting time |
| Negative | long left tail | scores on an easy exam |
The sign of skewness refers to the direction of the tail, not the side where most observations are concentrated.
Is kurtosis just peak height?
Describing kurtosis only as “peakedness” can be misleading. Higher kurtosis more directly reflects heavier tails and a greater tendency for large deviations relative to a normal distribution.
Excess kurtosis
Excess kurtosis defines the normal distribution as 0. Some software reports ordinary kurtosis, where the normal distribution equals 3, so always check the convention.
Why do measures of center move?
| Distribution | Common pattern |
|---|---|
| Right-skewed | mode < median < mean |
| Symmetric and unimodal | mean ≈ median ≈ mode |
| Left-skewed | mean < median < mode |
Not an absolute law
These are common patterns for unimodal distributions. They may fail when a distribution has multiple peaks or a more complex shape.
How should the numbers be read safely?
Financial returns
Even when the average return is near zero, high kurtosis may indicate that large gains or losses occur more often than a normal model would suggest.
Change the distribution shape
Begin with a symmetric distribution and then change the skew direction. Next increase the tail-thickness setting and compare what changes in the center and in the extremes.
Skew setting
Symmetric
Tail emphasis
1.0
This is an illustrative shape control; the slider values are not computed sample skewness or kurtosis.
What to notice
Changing skewness shifts the tail direction and the relative position of the mean. Increasing kurtosis makes the tail behavior and extreme values more prominent relative to the center.