Descriptive Statistics

Summarising Data

Descriptive statistics condense a dataset into a few informative numbers, separating the essential shape of the data from its raw bulk. Two questions matter most: where is the centre, and how much spread is there?

Measures of centre (central tendency):

• Mean — the arithmetic average, $\bar{x} = \frac{1}{n}\sum x_i$. Sensitive to every value, including outliers.
• Median — the middle value when the data are sorted. Robust: a few extreme values barely move it.
• Mode — the most frequent value. The only one usable for purely categorical data.

Measures of spread (dispersion):

• Range — largest minus smallest; crude and outlier-sensitive.
• Variance $\sigma^2 = \frac{1}{n}\sum (x_i - \bar{x})^2$ — the average squared deviation from the mean.
• Standard deviation $\sigma = \sqrt{\sigma^2}$ — the square root of the variance, in the same units as the data, so it is the interpretable measure of typical deviation.

The mean-versus-median distinction is the practical heart of this lesson. For a symmetric dataset they roughly coincide, but for a skewed dataset they diverge: the mean is pulled toward the long tail while the median stays central. A few billionaires make mean income far exceed median income; the median better represents a "typical" value when data are skewed or have outliers. Choosing the right summary depends on the data's shape.

Common pitfall: using the mean as "the typical value" when data are skewed or contain outliers. The mean is pulled toward extreme values, so for a long-tailed distribution it can misrepresent the centre — the median is more robust and usually more representative. (Likewise, report the standard deviation, not the raw variance, when you want spread in the data's own units.)