What it shows
Three distributions side by side: negatively skewed on the left, symmetric in the middle and positively skewed on the right. In each panel the mode, median and mean are vertical lines at their exact values, and an arrow marks the longer tail in the two skewed panels.
In the symmetric panel the three lines coincide. In the skewed panels they separate, with the mean furthest toward the long tail.
How to read it
The line styles distinguish the measures: the mean is dashed, the median solid and the mode dotted.
The arrow points along the long tail, which is the direction the mean has moved.
Under positive skew the order from left to right is mode, median, mean. Under negative skew it reverses.
This ordering is typical rather than guaranteed. It can fail in multimodal distributions, in distributions where one tail is long and the other heavy, and most often in discrete distributions (von Hippel 2005).
The numbers behind it
- Positive skew: a gamma distribution with shape k = 2 and scale θ = 1. Mode = 1, median ≈ 1.678, mean = 2.
- Negative skew: the mirror image, 7 − X. Mode = 6, median ≈ 5.322, mean = 5.
- Symmetric: a normal distribution with mean 3.5 and standard deviation 1, so mean = median = mode = 3.5.
- All three panels share the horizontal range 0 to 7 and the same vertical scale.
Ways to use it in teaching
Cover the line labels and have students identify each measure from its position.
Relate it to why incomes and house prices are usually reported as medians.
Close with von Hippel’s caveat: the pattern is a rule of thumb, so students should check it against their own data.
For the summary statistics behind this kind of chart, see our plain-language guide to descriptive statistics.
Sources
von Hippel, P. T. (2005), "Mean, median, and skew: correcting a textbook rule", Journal of Statistics Education 13(2), doi:10.1080/10691898.2005.11910556. Standard formulas for the gamma and normal distributions.
