Skewed distributions
- Concept
When data have a long tail on one side, the mean drifts toward the tail and stops describing the typical patient.
What it is
A distribution is skewed when its two sides are not mirror images. In positive (right) skew a few very high values stretch the right tail; in negative (left) skew a few very low values stretch the left tail. Spotting it matters because it changes the summary (median instead of mean) and may call for a transformation or a non-parametric test.
Worked example
- Data: days in intensive care for 9 patients: 1, 1, 1, 2, 2, 3, 4, 8 and 23.
- Mode = 1 day (the most frequent value).
- Median = the fifth of the 9 sorted values = 2 days.
- Mean = (1 + 1 + 1 + 2 + 2 + 3 + 4 + 8 + 23) / 9 = 45 / 9 = 5 days.
- Order: mode 1 < median 2 < mean 5, the signature of positive skew.
- Without the 23-day patient, the mean would fall to 22 / 8 = 2.75 days while the median would barely move (2 to 2).
- Reading: the typical patient stays about 2 days; the mean of 5 mostly reflects one patient with a very long stay.
How to recognise the shape of a distribution
| Shape | Tail | Order of the measures | Clinical example |
|---|---|---|---|
| Symmetric | None | Mean = median = mode | Adult height |
| Positive skew | To the right (high values) | Mode < median < mean | Length of hospital stay |
| Negative skew | To the left (low values) | Mean < median < mode | Age at death in a country with high life expectancy |
In practice
- If a paper’s mean is much larger than its median, the variable is right-skewed and the median is the better read.
- Variables such as viral load are often analysed on a log scale, which makes the distribution more symmetric.
Common pitfalls
- Naming the skew after the side where the data pile up: the name follows the direction of the tail.
- Running a t test on small, very skewed samples without transforming the data or using a non-parametric test.
Test yourself
Think of the answer, then tap the question to see it.
In a positively skewed distribution, what is the order of mean, median and mode?
Mode < median < mean: the mean shifts toward the right tail.
Which measure of centre should be reported for skewed data?
The median, together with the interquartile range.
If the mean is lower than the median, which way is the skew?
Negative, with the tail to the left: there are a few very low values.
Why is length of hospital stay usually right-skewed?
Because it cannot go below zero and most patients leave early, while a few complicated patients stay a long time.
Connections
Educational content on epidemiological and statistical methods; the examples use made-up numbers to teach the calculation. Not a clinical or legal guide: current Colombian rules are in the official sources.
Sources: Measures of Skewness and Kurtosis (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012); Principles of Epidemiology | Lesson 2 - Section 8 (Centers for Disease Control and Prevention, 2012).
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Representative model, for educational purposes only.