Box plots and histograms
It is instructive to consider how the median and quartiles relate to a histogram of a data set.
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The data set is split into quarters by the median and quartiles, so each section of the box plot contains equal numbers of data values and therefore has relative frequency 1/4. Since histogram area is proportional to relative frequency, the median and quartiles therefore split the histogram into four equal areas.
Although this result does not hold exactly if the median and quartiles do not coincide with class boundaries, the median and quartiles always approximately split a histogram into equal areas.
The diagram below shows the box plot of a symmetric distribution under the corresponding histogram.
Use the pop-up menu to change the shape of the underlying distribution. Observe that the histogram is split into four equal areas, corresponding to the median and quartiles of the distribution, and therefore the sections of the box plot.
Change the extremes, median and quartiles by dragging them on the diagram. Observe how the histogram shape reflects their values — when any two are close together, the density must be high (since the corresponding histogram area is always a quarter of the total area.
What does a box plot tell you about the distribution?
The diagram below shows the jittered dot plot and box plot of a batch of 100 values.
From any box plot, you should now have a reasonable impression of the distribution of values, and should be able to sketch the corresponding histogram.