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# Skewed distributions

- Concept

When data have a long tail on one side, the mean drifts toward the tail and stops describing the typical patient.

Updated Nov 5, 2026

## Key points

- Positive skew (tail to the right): mode

## 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

| Shape | Tail | Order of the measures | Clinical example |
| --- | --- | --- | --- |
| Symmetric | None | Mean = median = mode | Adult height |
| Positive skew | To the right (high values) | Mode
- 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.

## Memory aid

‘The mean chases the tail’: wherever the tail stretches, the mean follows.

## 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

## 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.

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## Connections

## Related 4

- [Normal distribution](https://durgical.com/en/epidemiology/normal-distribution/)
- [Measures of centre and spread](https://durgical.com/en/epidemiology/measures-of-centre-and-spread/)
- [Non-parametric tests](https://durgical.com/en/epidemiology/non-parametric-tests/)
- [Types of variables](https://durgical.com/en/epidemiology/types-of-variables/)

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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](https://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm) (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012); [Principles of Epidemiology | Lesson 2 - Section 8](https://archive.cdc.gov/www_cdc_gov/csels/dsepd/ss1978/lesson2/section8.html) (Centers for Disease Control and Prevention, 2012).

## **Spotted a mistake?** We are always improving for the community. Tell us what is wrong and our team will review it. Report a mistake

Representative model, for educational purposes only.

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License: CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Cite as: DurgiPedia (durgical.com), https://durgical.com/en/epidemiology/skewed-distributions/