Histograms

A rigorous guide to histogram bins, density, distribution shape, and defensible comparison.

Touching bins show frequencies across adjacent numeric intervals. A histogram uses adjacent equal-width bins on value-interval and frequency axes. One highlighted bin makes its interval and height easier to compare with neighbouring intervals.

Precise definition

A histogram displays a quantitative variable by partitioning its number line into adjacent intervals and using bar area to represent frequency or relative frequency. With equal bin widths, heights may be counts; with unequal widths, height must be frequency density so area remains proportional to count.

Notation and mathematical language

For class $i$, density is $f_i/w_i$, where $f_i$ is frequency and $w_i$ width. Boundaries such as $[a,b)$ must be consistent. Unlike a bar graph, the horizontal position and touching bars express numerical adjacency and continuity of intervals.

Conceptual picture

A histogram approximates distribution shape: modes, skew, centre, spread, gaps, and tails. The bins aggregate observations, so the picture depends on origin and width. A striking pattern that disappears under reasonable alternative bins may be a display artifact.

Fully worked example

Interpretation and application

Histograms summarize response times, measurements, incomes, and scores. Observed skew can guide robust summaries or transformations, but explanations for the skew require domain knowledge and appropriate study design.