Centre, Spread and Shape
Descriptive statistics answer three questions about a data set: where is the middle, how spread out is it, and what shape does it have? No single number captures a distribution, which is why the full set is reported together.
Sample or Population
The distinction changes one thing: the variance denominator. Dividing by n gives the population variance; dividing by n−1 gives the sample variance.
The reason is that a sample mean sits closer to its own data than the true population mean does, which makes the raw squared deviations systematically too small. Using n−1 — Bessel's correction — compensates. With large samples the difference is negligible; with ten values it is over 5%.
The Five-Number Summary
| Statistic | What it marks |
|---|---|
| Minimum | The smallest value |
| First quartile (Q1) | 25% of values fall below it |
| Median (Q2) | The middle value |
| Third quartile (Q3) | 75% of values fall below it |
| Maximum | The largest value |
The interquartile range, Q3 − Q1, contains the middle half of the data and is immune to extreme values. This makes it the preferred spread measure whenever outliers are present.
Identifying Outliers
The standard rule flags any value more than 1.5 × IQR beyond Q1 or Q3. It is a convention rather than a test — the 1.5 has no theoretical derivation and was chosen by John Tukey as a practical threshold.
An outlier is not automatically an error. It may be a data entry mistake, a measurement fault, or the most interesting observation in the set. Deleting outliers because they are inconvenient is one of the most common forms of quiet data manipulation.
Mean Against Median
When they are close, the distribution is roughly symmetric. When the mean exceeds the median, the data is skewed right — a long tail of high values. Income data is the standard example: mean household income substantially exceeds the median, because a small number of very high incomes pull the mean upward while leaving the middle untouched.
Coefficient of Variation
Standard deviation divided by the mean, expressed as a percentage. It makes spread comparable across data sets with different units or scales — a standard deviation of 5 means something very different around a mean of 10 than around a mean of 1,000.
Frequently Asked Questions
Which measure of centre should I use?
Median for skewed data or when outliers are present; mean for symmetric data, and whenever further calculation is needed since the mean has better mathematical properties. Mode for categorical data, where the others do not apply.
Why square the deviations in variance?
Because deviations from the mean sum to exactly zero otherwise. Squaring removes the signs and penalises large deviations more heavily; taking the square root at the end returns to the original units.
How many data points do I need?
Descriptive statistics work on any size, but their reliability grows with n. Below about 30 observations, the sample standard deviation is itself quite unstable.