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.
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.
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.
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: Principles of Epidemiology | Lesson 2 - Section 6 (Centers for Disease Control and Prevention, 2012); Principles of Epidemiology | Lesson 2 - Section 7 (Centers for Disease Control and Prevention, 2012); Measures of Location (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012); Measures of Scale (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012).
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Representative model, for educational purposes only.