Guide 2 · Descriptive statistics

Describe What You Observe

Descriptive statistics summarize the data you actually have — nothing more. A good summary is honest about center, spread and shape, and it never replaces looking at the data itself.

Colored beads sorted into rows beside a handwritten tally chart — Illustrative photograph — not measured evidence.

Know what kind of variable you have

Before computing anything, decide what each column measures. Categorical variables place observations in groups: blood type, region, yes/no answers. Numerical variables are quantities: they may be discrete (number of children, defects per batch) or continuous (height, waiting time, temperature).

The type decides the tools. You can count and take proportions of categories, but an "average region" is meaningless. Codes such as 1 = north and 2 = south look numerical without being so. Ordinal scales, like survey answers from "strongly disagree" to "strongly agree", sit in between: the order is real, but the gaps between steps may not be equal, so medians and proportions are usually safer than means.

Worked example

One large value moves the mean, not the median

Seven people report how many minutes they waited for a bus: 2, 3, 3, 5, 7, 8, 40.

Mean = (2 + 3 + 3 + 5 + 7 + 8 + 40) ÷ 7 = 68 ÷ 7 ≈ 9.7 minutes. Median = the 4th sorted value = 5 minutes. Mode = 3 minutes.

Six of the seven people waited 8 minutes or less, yet the mean suggests almost 10. The 40-minute wait pulls the mean upward; the median barely notices it. Neither is "wrong": the mean tells a transport planner the average total waiting burden, while the median describes a typical passenger's experience. Report the one that answers your question — and when the data are skewed, report both and say why they differ.

Before deleting the 40, find out why it happened. An outlier can be a typing error, or it can be the most important observation in the data set — a cancelled service, for instance.

Spread: how far values sit from the center

Two classes can share an average score of 70 while one has every student between 65 and 75 and the other ranges from 30 to 100. A center without a spread hides that difference.

The range (maximum minus minimum) is simple but driven entirely by the two extremes. The interquartile range (IQR) measures the width of the middle half of the data, from the 25th to the 75th percentile, and resists outliers. The standard deviation is roughly the typical distance of values from the mean; it pairs naturally with the mean and, like the mean, is sensitive to extreme values.

A common rule of thumb flags values more than 1.5 × IQR beyond the quartiles as potential outliers — the rule John Tukey used for box plots in Exploratory Data Analysis (1977). It is a prompt to investigate, not an instruction to delete.

Shape

A histogram reveals whether data are symmetric, skewed, or have several peaks. Incomes and house prices are typically right-skewed: many moderate values and a long tail of large ones, so the mean exceeds the median. Two peaks often mean two groups mixed together — for example, adult and child heights in one data set.

Always draw the data

In 1973 Francis Anscombe published four small data sets with almost identical summaries: mean of x = 9, mean of y ≈ 7.50, correlation ≈ 0.816 and the same fitted line, y ≈ 3 + 0.5x. Plotted, one is a clean linear trend, one a curve, one a line with a single outlier, and one is driven by a single extreme point. Same numbers, four different stories.

Relationships, groups and percentages

The correlation coefficient r runs from −1 to 1 and measures how closely two numerical variables follow a straight line. It says nothing about curved relationships, and it says nothing on its own about cause.

Grouping can reverse a pattern. In a well-known study of 1973 graduate admissions at the University of California, Berkeley, the overall figures showed men admitted at a higher rate than women. Bickel, Hammel and O'Connell (1975) found that, department by department, the difference largely disappeared or favored women; women had applied more often to departments that admitted fewer applicants of any sex. This reversal is called Simpson's paradox, and it is a reason to ask "broken down by what?" before trusting a total.

Finally, watch the language of change. If a risk rises from 1 in 1,000 to 2 in 1,000, that is a 100% relative increase but only a 0.1 percentage point absolute increase. Both are true; reporting only the first makes a small change sound alarming.

Which summary to choose (scroll sideways on small screens)
Situation	Center	Spread	Best picture
Roughly symmetric, no outliers	Mean	Standard deviation	Histogram
Skewed or with outliers	Median	Interquartile range	Box plot plus histogram
Categorical data	Mode or proportions	Counts per category	Bar chart
Two numerical variables	—	Correlation, with care	Scatter plot

What a description can and cannot justify

A summary can justify statements about these observations: "in our seven responses, the median wait was 5 minutes". It cannot, by itself, justify claims about everyone who rides the bus. Reaching beyond the data needs a sampling design and the tools in Infer with Care.

Sources: Anscombe (1973), The American Statistician 27(1); Bickel, Hammel & O'Connell (1975), Science 187; Tukey (1977); NIST/SEMATECH e-Handbook, exploratory data analysis chapter.