Measuring spread with variance and standard deviation
An average alone does not describe data fully: the sets 48, 50, 52 and 10, 50, 90 both have a mean of 50, but the second is far more spread out. Measures of dispersion describe this spread. The range is the difference between the largest and smallest values, which is simple to find but depends on only two observations. The mean deviation is the average of the absolute distances of the observations from the mean or the median.
Variance is the mean of the squared deviations from the mean, σ² = Σ(x − x̄)² ÷ n, and the standard deviation σ is its square root, so it is measured in the same units as the data. For a frequency distribution, σ² = Σf(x − x̄)² ÷ Σf, and a useful shortcut is σ² = (Σx²/n) − x̄². To compare the variability of two data sets with different means, use the coefficient of variation, (σ ÷ x̄) × 100; the set with the smaller value is more consistent. Cumulative frequency curves called ogives help locate the median and quartiles of grouped data.
Why it matters
Standard deviation is used to judge the consistency of a batsman's scores, the reliability of a machine, the risk of an investment and the variation in repeated experimental measurements. In science, results are often reported as a mean plus or minus a standard deviation. These ideas lead on to the normal distribution and to statistics courses in economics, psychology and data science.
Key terms
- Dispersion
- The extent to which the values in a data set are spread out from their centre.
- Variance
- The average of the squared differences between each observation and the mean.
- Standard deviation
- The square root of the variance, measuring spread in the same units as the data.
- Coefficient of variation
- The standard deviation as a percentage of the mean, used to compare consistency.