Probability Histogram vs Histogram

A probability histogram draws one bar for each value a discrete random variable can take, and the height of the bar is $P(X = x)$. The heights add to exactly 1, because together they are the whole distribution. Equal-width bars sit centered on their values, so when the width is 1 each bar's area equals its probability as well and the total area is 1.

Roll a fair die three times and let $X$ count the sixes. The four bars have heights $\frac{125}{216} \approx 0.579$ at $x = 0$, $\frac{75}{216} \approx 0.347$ at $x = 1$, $\frac{15}{216} \approx 0.069$ at $x = 2$, and $\frac{1}{216} \approx 0.005$ at $x = 3$. They sum to 1, and the picture leans hard to the right.

That picture sets up the standard misreading: "the tallest bar is at 0, so the average number of sixes is 0." The tallest bar marks the most likely single value, which is a different question from the average. The mean here is $\mu_X = 0.5$ sixes, and it falls in the empty space between the bars at 0 and 1. Height reads likelihood; the balance point of the whole picture reads the mean, and on a skewed distribution the two are never in the same place.

Because the heights are probabilities, none can exceed 1 and all of them together make 1. A frequency histogram of data is not built that way: its heights are counts and they add to the sample size. A probability histogram also only exists for a discrete variable. A continuous one gets a density curve, whose vertical axis is density rather than probability and can rise above 1, since a uniform curve over an interval of width 0.5 has height 2.

One last thing the bars are not: data. They describe what the chance process does over many repetitions. Roll three dice twenty times and the histogram you actually get will not match these heights. It drifts toward them as the repetitions pile up.

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.