Measures of spread
Also known as: measures of dispersion, measures of variability
Measures of spread describe how far the values in a data set are from one another. Range, interquartile range, variance, and standard deviation are the common ones, and they tell you how much variability sits behind an average.
A measure of center tells you where a data set sits; a measure of spread tells you how tightly the values cluster around it. Two groups can share an identical mean while one is uniform and the other is wildly variable, so a center reported without a spread is an incomplete summary.
The simplest measure is the range, the largest value minus the smallest. It is easy to compute but depends entirely on two observations, so a single outlier distorts it. The interquartile range (IQR) fixes that by taking the third quartile minus the first, capturing the middle 50% of the data and ignoring the tails — which is why it pairs with the median as a resistant summary and defines the whiskers and outlier rule on a boxplot.
Variance and standard deviation measure spread around the mean. Variance is the average of the squared deviations from the mean, and standard deviation is its square root, which returns the answer to the original units. Because the deviations are squared, both are sensitive to extreme values, so they pair naturally with the mean and suit roughly symmetric distributions. Note the sample-versus-population distinction: a sample variance divides by n − 1 rather than n. For comparing spread across data sets with different units or scales, the coefficient of variation expresses the standard deviation as a proportion of the mean.
Spread appears across quantitative exams. The CIMA Certificate BA business economics paper tests calculating and interpreting range, quartiles, and standard deviation for business data, while USMLE Step 1 biostatistics uses standard deviation to work with the normal distribution, the empirical rule, and confidence intervals around a study estimate.
Key takeaways
- Measures of spread quantify variability; a mean or median alone does not describe a data set.
- Range is simple but driven by the two most extreme values.
- The interquartile range covers the middle 50% of the data and resists outliers, pairing with the median.
- Variance is the mean squared deviation, and standard deviation is its square root in the original units.
- Sample variance divides by n − 1, while population variance divides by n.