Bin width

Every bin in a histogram spans the same width $w$, and the bar over a bin is the count of values falling in it, with boundary values assigned by a fixed convention so no value is counted twice. The consequence people underestimate is that the shape a histogram shows is a joint property of the data and the chosen $w$, not a property of the data alone.

Take 28 exercise times in minutes and bin them twice. At $w = 30$ the counts from 0 are 4, 17, 6 and 1: one peak in the 30 to 60 block and a tail to the right, which you would describe as unimodal and skewed right. At $w = 10$ the same 28 values give 0, 1, 3, 8, 1, 8, 3, 2, 1, 1, 0 and 0. The block of 17 has split into 8, 1 and 8, so the single peak has become two peaks separated by a near-empty stretch from 40 to 50 minutes, and the honest description is now bimodal.

The wrong sentence is "the histogram is unimodal, so the data are unimodal." Neither picture above is a mistake. The wide one is right that most times fall between 30 and 60 minutes; the narrow one is right that there are two clusters inside that block. What is wrong is describing shape from a single histogram without ever trying a second width, because the description you produce is partly a description of your own binning choice.

A common starting point is the range divided by the number of bins you want: these times run from 15 to 95, so eight bins suggests $(95 - 15) / 8 = 10$, rounded to something readable. Too few bins hides structure. Too many turns single observations into spikes, and in the limit, when no two values share a bin, every bar is 0 or 1 tall and the shape is gone entirely.

Topic 1.5 of the Fall 2026 course covers histograms for one quantitative variable, including how changing the bin width changes the shape you see.