# Why do we use the histogram?

As somebody who never took a statistics course (but had to teach a few classes on it), I wondered why is the histogram introduced in a statistics course? Usually when something is introduced in a "watered-down" way, it is important in later more advanced treatment of the course. The undergraduate course never motivates why.

My reasoning is that, if we have a random variable, $X$, we can estimate $P(a\leq X\leq b)$ by counting how often outcomes appear in a certain interval. The smaller the interval the better. Therefore, the point of the histogram is to "piece-together" information for the distribution of $X$, which is ultimately the goal of statistics. Is this the main reason?

• I'm not sure I follow your question. Are you asking if there is a "more advanced treatment" / data visualization technique of which the histogram is the "watered-down" version, OR if "piecing-together information for the distribution of $X$" is a worthy goal, OR if plotting a histogram actually helps us do this, OR if there are other reasons beyond that for making a histogram? – gung - Reinstate Monica Apr 7 '15 at 2:32
• The histogram is a very simple density estimator, simple enough to teach to mathematically naive students. Under some simple conditions it is consistent. Many statistics packages use formulas that attempt to optimize the binwidth (in terms of integrated MSE or asymptotic IMSE from the true density), though as a diagnostic tool that tends to oversmooth. – Glen_b -Reinstate Monica Apr 7 '15 at 2:44
• Related to @NickCox's comments, I frequently prefer a spikey distribution summary, i.e., what some of my R functions call a "spike histogram" that shows up to 100 bins, and if there are < 100 unique values, all the points. If there are no ties this is essentially a rug plot. I like to see all the data in all their glory, which still allows me to see tendencies. – Frank Harrell Apr 7 '15 at 12:04
• Following @FrankHarrell in turn, in Stata this is done by spikeplot. The manual entry shows an example in which fine structure that would be hidden by most histograms (and density estimates) is evident otherwise. stata.com/manuals13/rspikeplot.pdf – Nick Cox Apr 7 '15 at 12:26
• When introducing probability density functions to students, I have been comforted by knowing they usually have experience with histograms. I can help them build on that experience--provided it was correct!--to understand PDFs. By "correct," I mean knowing that a histogram uses area, rather than bar height, to display relative frequencies. Thus we ought to distinguish between histogram-like bar charts of frequencies and true histograms. The distinction becomes clear when variable bin widths are used. – whuber Apr 7 '15 at 14:34

The histogram is an easily implemented and efficient tool to visualize the distribution of count data, say taken from a small sample, and check the adequacy of fitted models to that data.

If you would like a more advanced approach with regards to histograms, simulate a bunch of normal samples and plot it on a histogram. Now try changing the binwidths and see if you can find one that gives a good visual representation of the simulated samples. Then write a formula for an optimal binwidth based on the type and size of the data simulated.

Histograms, are perhaps the best-known statistical plots. If you Google images for “statistics,” you will see many histograms or histogram-like plots. Something like this:

So a down-to-earth answer to the question

why is the histogram introduced in a statistics course?

is because anyone who will be working with statistical reports of any kind is very likely to encounter histograms and he or she should know what those plots are.

Why histograms are popular? I think there are two factors here. First, exploring the data using histograms with different numbers of bins and different cut points between bins is useful in understanding the shape of the data. But this exploration should not be confused with manipulation of the data for presentation purposes. The latter also contributes to the popularity of histograms significantly.

• Your picture doesn't show histograms, but pseudo-3-dimensional bar charts. The implication of each graph is that some quantity is rising from left to right, possibly over time. That's a time series bar chart, not a histogram The arbitrary colours, irregular arrows and spurious third dimension are at best misguided attempts to add decoration and at worst irrelevant chartjunk. The substance of your answer is that exploring data by varying bin widths and origin is helpful, but the dependence of histograms on such choices is a downside for beginners who might be interested in this thread. – Nick Cox Jun 27 '16 at 7:49
• "The implication of each graph is that some quantity is rising from left to right" Even the middle one? "Your picture doesn't show histograms" The pictures may well be histograms: you can't say for sure without knowing how they were created (do you know? I don't). But they are definitely histogram-looking pictures. If you don't like them, google images for "statistics" and you will find "proper" histograms, I promise. Re beginners: everyone (especially beginners) should know about possibilities of data manipulation and should look at histogram critically. – Kostia Jun 27 '16 at 8:05
• Granted that the middle one is ambiguous, as one of the associated images shows rising curves, you're criticising your own illustrations if you admit that you can't be sure that they are histograms. I agree with your last sentence of your comment, but I can't see that your answer promotes critical appreciation beyond stating that it's important. I am familiar with histograms myself and have already commented on the question above. – Nick Cox Jun 27 '16 at 8:23