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Histograms give a good sense of the distribution of a variable. Box plots attempt to do the same thing however, don't give as good of a picture of the distribution of this variable.

I don't understand why people use box plots. Histograms are better in every way. Is there a reason I would use both of them?

The only thing I think that box plots provide is: outliers! It tells us which observations may be outliers.

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    $\begingroup$ Is Histogram worse in every way than a representation of the whole distribution ? $\endgroup$ – Anthony Martin Mar 20 '16 at 14:16
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    $\begingroup$ Depends on what you want, with a box plot you can have some precise values (eg median, P75), that you do not have with an histogram. It displays less information, but is more synthetic. My point is that even an histogram is a simplification and a waste of information compared to the whole distribution. But it can be easier to use $\endgroup$ – Anthony Martin Mar 20 '16 at 14:24
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    $\begingroup$ A contrary viewpoint about the utility of histograms has been cogently expressed, and well illustrated, in the highly upvoted post at stats.stackexchange.com/a/51753 (which can be found by searching our site for "histogram"). $\endgroup$ – whuber Mar 20 '16 at 14:35
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    $\begingroup$ Interesting thought--but increasing the bin size would reduce the histogram to a boxplot-like figure while retaining its unfortunate dependence on the choice of cutpoints. IMHO, the real merits of boxplots can best be appreciated by studying Tukey's use of the N-letter summary for exploratory analysis of multivariate data and remembering that he was calculating with pencil and paper at the time. For visualizations like a "wandering schematic trace" other univariate summaries of conditional responses, like histograms or violin plots, simply would not work. $\endgroup$ – whuber Mar 20 '16 at 15:24
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    $\begingroup$ The two failures (imo) of the histogram happen when there are few samples or when the boxes are the wrong sizes. The weakness of a good boxplot (and I'm thinking JMP variability when I say it) are multi-modality, and fine detail. One place where the boxplot shines is when there are few samples. I also like it when there are a number of interacting variables at different levels - thus the JMP variability plot. $\endgroup$ – EngrStudent Mar 21 '16 at 0:20
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The fact that box plots provide more of a summary of a distribution can also be seen as an advantage in certain cases. Sometimes when we're comparing distributions we don't care about overall shape, but rather where the distributions lie with regard to one another. Plotting the quantiles side by side can be a useful way of doing this without distracting us with other details that we may not care about.

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    $\begingroup$ This is the best answer. Boxplots are better for comparing distributions than histograms! $\endgroup$ – kjetil b halvorsen Mar 20 '16 at 19:48
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In the univariate case, box-plots do provide some information that the histogram does not (at least, not explicitly). That is, it typically provides the median, 25th and 75th percentile, min/max that is not an outlier and explicitly separates the points that are considered outliers. This can all be "eyeballed" from the histogram (and may be better to be eyeballed in the case of outliers).

However, the much bigger advantage is in comparing distributions across many different groups all at once. With 10+ groups, this is a tiring task with side-by-side histograms, but very easy with box plots.

As you mentioned, violin plots (or bean plots) are somewhat more informative alternatives. However, they require slightly more statistical knowledge than the box plots (i.e. if presenting to a non-statistical audience, it may be a little more intimidating) and box-plots have been around much longer than kernel density estimators, hence their greater popularity.

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    $\begingroup$ +1. Correction though, box-plots provide medians, not means. $\endgroup$ – Greenparker Mar 21 '16 at 0:09
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    $\begingroup$ Everyone can be right. Box plots as usually plotted show medians (I've seen this denied, but do not recall seeing an example). But some implementations allow you to show means as well. That's often a good idea. $\endgroup$ – Nick Cox Mar 21 '16 at 0:46
  • $\begingroup$ Thanks for pointing that out. I keep (incorrectly) thinking it's usually the mean, which could lead to some very weird plots in extreme cases. $\endgroup$ – Cliff AB Mar 21 '16 at 4:18
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    $\begingroup$ would be nice if there were images to go along with this to show the value of side-by-side comparisons with box plots vs histograms $\endgroup$ – Old account Jun 3 '18 at 23:37
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  1. If I show you a histogram and ask you where the median is, you might be quite some time figuring it out... and then you'll only get an approximation to it. If I do the same with a boxplot you have it immediately; if that's what you're interested in, boxplots obviously win.

  2. I agree that boxplots are not as effective as a description of the distribution of a single sample, since they reduce it to a few points and that doesn't tell you a lot.

    However, if you're comparing many dozens of distributions, having all the details of each may be more information than is easily compared -- you may want to reduce the information to a smaller number of things to compare.

  3. If more information is better, there are many better choices than the histogram; a stem and leaf plot, for example, or an ecdf / quantile plot.

    Or you could add information to a histogram:

histogram with marginal boxplot histogram rugplot with jitter histogram with stripchart

(plots from this answer)

The first of those -- adding a narrow boxplot to the margin -- gives you any benefits to be gained from either display.

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Bar plots provide only the range of frequency of observations while box plots are better in telling where several parameters of a distribution lie, example mean and variances that bar plots cannot. Box plots are thus used as an effective comparative tool if one has several distributions.

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  • $\begingroup$ It is rare for a boxplot to display a mean--almost always they use medians--and they never represent variances directly. Note, too, that these quantities are not usually considered "parameters of a distribution": they are descriptive statistics for a batch of data. $\endgroup$ – whuber Jun 19 '17 at 17:46
  • $\begingroup$ Exactly, they are a nice tool for describing a distribution without going too much calculations. And they display medians more, and since in lots of cases both measures coincide, box plots are a nice tool to approximate the mean too. $\endgroup$ – Shiv_90 Jun 21 '17 at 8:56
  • $\begingroup$ Your comment seems to continue confounding the data with the underlying distribution. It is very rare for the mean to equal the median in any batch of data. Moreover, one of the better and most common uses of the boxplot is to identify asymmetry, which usually implies an important difference between mean and median. One of the fundamental principles behind the original conception of the boxplot is that it be a robust exploratory tool--which implies it better not be based on sensitive statistics like the mean or variance. $\endgroup$ – whuber Jun 21 '17 at 13:51

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