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# Measures of centre and spread

- Measure

Mean, median and mode tell where the centre of the data lies; range, interquartile range and standard deviation tell how spread out it is.

Updated Nov 5, 2026

## Key points

- Mean = sum of values / n. Median = middle value of the sorted data. Mode = most frequent value.
- With skewed data or outliers, report the median and interquartile range (IQR = Q3 − Q1).
- With symmetric data, report the mean and standard deviation (SD).
- Variance = sum of (x − mean)² / (n − 1); SD = square root of the variance, in the same units as the data.
- An extreme value moves the mean and the range far more than the median and the IQR.

## What it is

A set of numerical data is described by two ideas: its centre and its spread. The mean uses every value but shifts with extremes; the median is the middle value and resists outliers. The standard deviation goes with the mean in symmetric data, and the interquartile range goes with the median in skewed data.

## Formula and reading

- Mean = Σx / n.
- Median: with odd n, the value at position (n + 1)/2; with even n, the average of the two central values.
- Range = maximum − minimum. IQR = 75th percentile − 25th percentile (it holds the central 50%).
- Sample variance s² = Σ(x − mean)² / (n − 1); SD = √s².
- Reading: an SD small relative to the mean means tightly grouped data; a large one, widely spread data.

## Worked example

- Data: length of stay after surgery for 7 patients: 2, 3, 3, 4, 5, 6 and 19 days.
- Mean = (2 + 3 + 3 + 4 + 5 + 6 + 19) / 7 = 42 / 7 = 6 days.
- Median = the fourth of the 7 sorted values = 4 days. Mode = 3 days (it appears twice).
- Range = 19 − 2 = 17 days. By the halves method, Q1 = 3 and Q3 = 6, so the IQR = 3 days.
- Deviations from 6: −4, −3, −3, −2, −1, 0 and 13; their squares add up to 16 + 9 + 9 + 4 + 1 + 0 + 169 = 208.
- Variance = 208 / (7 − 1) = 34.7; SD = √34.7 ≈ 5.9 days (rounded).
- Reading: one 19-day patient lifts the mean to 6 and the SD to 5.9; the honest summary is a median of 4 days (IQR 3 to 6).

## Markers

- SD
- IQR
- Q1
- Q3

## How to read it

- In a paper’s tables, ‘mean ± SD’ suggests symmetric data; ‘median (IQR)’ suggests skewed data such as length of stay or costs.
- If the SD is more than half the mean for a variable that cannot be negative, the data are almost certainly skewed.

## Common pitfalls

- Confusing the standard deviation (spread of the patients) with the standard error (precision of the mean).
- Reporting the mean of very skewed data, such as income or days in hospital, without the median.
- Averaging an ordinal variable coded with numbers.

## Memory aid

‘The median doesn’t flinch’: one extreme value drags the mean, but the median stays in the middle.

## Test yourself

Think of the answer, then tap the question to see it.

## Which measure of centre best resists extreme values?

The median, because it depends only on the order of the data, not on their size.

## Which pair of measures is reported for skewed data?

The median with the interquartile range (Q1 to Q3).

## Why does the sample variance divide by n − 1 rather than n?

Because deviations are taken from the sample’s own mean; dividing by n − 1 corrects the tendency to underestimate the population variance.

## What share of the data lies inside the interquartile range?

The central 50%, between the 25th and the 75th percentile.

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## Connections

## Related 4

- [Types of variables](https://durgical.com/en/epidemiology/types-of-variables/)
- [Skewed distributions](https://durgical.com/en/epidemiology/skewed-distributions/)
- [Normal distribution](https://durgical.com/en/epidemiology/normal-distribution/)
- [Standard error](https://durgical.com/en/epidemiology/standard-error/)

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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: [Principles of Epidemiology | Lesson 2 - Section 6](https://archive.cdc.gov/www_cdc_gov/csels/dsepd/ss1978/lesson2/section6.html) (Centers for Disease Control and Prevention, 2012); [Principles of Epidemiology | Lesson 2 - Section 7](https://archive.cdc.gov/www_cdc_gov/csels/dsepd/ss1978/lesson2/section7.html) (Centers for Disease Control and Prevention, 2012); [Measures of Location](https://www.itl.nist.gov/div898/handbook/eda/section3/eda351.htm) (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012); [Measures of Scale](https://www.itl.nist.gov/div898/handbook/eda/section3/eda356.htm) (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012).

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Representative model, for educational purposes only.

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