histogram

A histogram is a graphical representation of the distribution of a numerical variable. It groups data into a series of consecutive intervals, called bins, and displays the count or proportion of observations in each bin as a rectangle. The height of each rectangle indicates the frequency or density of data within that bin, and the x-axis corresponds to the bin ranges while the y-axis shows the chosen measure (count, relative frequency, or density).

Construction involves selecting the data, choosing bin edges, tallying observations in each bin, and plotting rectangles whose bases lie on the bin intervals. In a standard histogram, adjacent bars touch to emphasize the continuous nature of the variable; gaps are used for categorical data or discrete steps.

Bin width and the number of bins influence the histogram’s appearance and the inferred features of the distribution. Common binning rules include Sturges’ rule, the Freedman–Diaconis rule, and Scott’s rule; each has trade-offs between sensitivity to data features and stability across samples. Histograms can be presented as frequency histograms (counts), relative frequency histograms (proportions), or density histograms (areas correspond to probabilities, integrating to one).

Interpretation focuses on the shape: symmetry or skewness, number of modes, spread, and tails. Histograms are useful for exploratory data analysis but are sensitive to binning choices and outliers, and they may be less reliable with small samples. For smoother estimates of the underlying distribution, kernel density estimates are a common alternative. Multi-dimensional data can be represented with 2D histograms or heatmaps.