# Can a histogram have just one value as its bin in a probability histogram?

On [this link], (http://www.henry.k12.ga.us/ugh/apstat/chapternotes/7supplement.html) there is a histogram where on the x-axis, all the sums of the two dice are there {2, 3, 4, 5, ... 12}, and on the y-axis, the probability that they will occur is there.

I thought histograms have to contain a range as their bin. Can someone help me understand this?

• Arguably a histogram is the wrong tool since only integer values are possible for the variable (and the display should make that explicit). If you do (for some unclear reason) actually use a histogram with each bin of unit width and only containing one integer, then the height should still give the correct count or proportion (as appropriate for whichever kind of histogram you do). One must be careful, though -- if you do it carelessly, some programs (I'm looking at you, R) will happily bin integer data like this: [0,1], (1,2], (2,3] ... (!) (Of course if you use the right tool, it's fine.) Oct 12 at 21:40

I thought histograms have to contain a range as their bin.

This is not correct. Histograms show how many units fall into one of a finite number of bins, or what probability there is for an observation to fall into one of these bins.

Sometimes you already have a "natural" finite number of bins, as in the dice throwing example here: 11 bins. Then everything is straightforward.

However, sometimes you have too many bins to plot usefully, or even an infinite number of possible distributions, e.g., when the underlying distribution is continuous. Then, to build a histogram, you discretize the observations into a small number of bins, where every bin corresponds to a range of possible observations (where ranges should of course not overlap) - and then you proceed as above.

• You seem to describe bar charts rather than histograms. Histograms use areas to represent frequencies, probabilities, or relative frequencies.
– whuber
Oct 13 at 17:18
• @whuber: True, but if you take counting measure on the integers as the underlying measure for the "area", those things are equivalent, no?
– Ben
Oct 13 at 17:21
• @Ben Only in an irrelevant mathematical sense. Histograms are used for visualization. The visual impact of a bar is directly proportional to the product of its height and its width. There's no getting around the visual effect of the bar's width. Histograms exploit this in an essential manner whereas bar charts rely on the constancy of the widths. They require different kinds of interpretation.
– whuber
Oct 13 at 17:23
• @whuber: Okay, but I've seen plenty of histograms with equal bar-width. Unequal bar widths is a feature that histograms can have, but equal widths are also common, so I would say that a barplot with bars for individual units is still technically a histogram. (I am being a bit of a Devil's advocate here; as I point out in my own answer, you wouldn't normally call a barplot of the PMF a histogram.)
– Ben
Oct 13 at 17:28
• @Ben You point to one of the main reasons people confuse barplots and histograms. One way to distinguish the two--although it's subtle--is to read the label on the height axis. For a barplot it's a count or proportion, whereas for a histogram it's a count, proportion, or probability per unit length. With that in mind it's possible still to confuse the two types of plots only when the histogram bin lengths are all equal to $1,$ in which case the only way they differ will be in the height axis label.
– whuber
Oct 13 at 21:19

As the other answer here points out, it is possible to have a histogram where each bin is a single point (e.g., if we have a "natural" finite number of bins). I will just supplement the other answer here by noting that in this case we usually would not call this a histogram at all (even though technically it is one). Usually we would just refer to this as a barplot of the probability mass function.