Bin Width vs Histogram

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.

A histogram cuts the number line into bins of equal width and draws a bar over each bin whose height is the count of values inside it, or the relative frequency. The bars touch because the axis is a continuous scale rather than a list of labels. Every histogram needs a boundary rule, and the usual one puts a value in the bin whose lower bound it meets or exceeds and whose upper bound it falls below.

Thirty exam scores binned by tens give 9 scores in 70 to 80 and 12 in 80 to 90, and three of those students scored exactly 80. Switch to the other convention, where a value belongs to the bin whose upper bound it equals, and those three move down: the counts become 12 and 9, the tallest bar changes, and so does the shape you would report. Nothing in the drawn picture tells you which rule was used.

The misreading is "twelve students scored 85." The bar over 80 to 90 says twelve students scored somewhere in that interval and nothing more. The individual values have been thrown away, which is also why you cannot read a median off a histogram, only the bin the median lands in.

Equal width is doing real work in the definition. When bins differ in width, height stops encoding frequency fairly: a bin of width 20 holding 12 values and a bin of width 10 holding 9 draw bars of height 12 and 9, so the wide one looks bigger, while the densities are $12 / 20 = 0.6$ and $9 / 10 = 0.9$ values per unit and the narrow bin is the more crowded one. With unequal bins the vertical axis has to carry density and you read area, not height.

A histogram also destroys the order the values arrived in, which is what a time plot keeps, and its shape depends on the bin width you choose. Histograms are topic 1.5.