Dot Plots, Histograms and Box Plots

In this lesson you will: Build a dot plot, a histogram and a box plot by hand from a list of data; Read a count, a fraction of the group, and the five-number summary off a display; Choose the display that answers the question you were actually asked.

One list, three pictures

A list of twenty-four numbers tells you almost nothing while it is still a list. Drawn, it starts talking. The three displays Ohio expects you to handle are the dot plot, the histogram and the box plot, and each of them throws away something different in exchange for showing you something else.

A dot plot keeps everything. One dot per data value, stacked above a number line. You can still read every individual number back out of it, which is why it only works for short lists of whole-ish numbers.

Histograms: give up the individuals, gain the shape

When the list runs to fifty or five hundred values, dots stop working. A histogram sorts the data into intervals called bins, counts how many land in each, and draws a bar of that height. The bars touch, because the bins touch — that is the visual difference between a histogram and an ordinary bar graph.

The cost is real. Once the data is binned you can no longer recover a single value. A bar covering 40 to 60 minutes with a height of 15 tells you fifteen students fell somewhere in that twenty-minute window, and nothing more precise than that.

Bin width is a choice you make. Wide bins smooth the picture and can hide a gap; narrow bins show detail and can turn a smooth distribution into a row of spikes.

Box plots: five numbers, four equal groups

A box plot compresses a data set to five numbers: the minimum, the first quartile, the median, the third quartile and the maximum. Sort the list, find the median, then find the median of each half. In this book the median itself is not counted in either half when the list has an odd number of values.

The box runs from Q_1 to Q_3 and the line inside it marks the median. The four sections — left whisker, left half-box, right half-box, right whisker — each hold about a quarter of the data. That is the whole idea, and it is the part students misread most often.

Common mistake — Reading a wide part of a box plot as more data

A student looks at the food-truck box plot, sees that the right whisker is far longer than the left, and concludes that most hours were busy ones.

The reasoning is natural, because in almost every other chart in the world, bigger means more. On a bar graph a tall bar is more. On a histogram a wide bar covers a wider interval and usually holds more people.

A box plot does not work that way. Each of its four sections holds roughly 25% of the values no matter how long or short it is. A long whisker means those few values were spread out, not that there were many of them.

Here, the hours from 33 to 46 customers are about a quarter of the hours, and so are the hours from 14 to 21 — even though one stretch is nearly twice as wide as the other.

How to catch it: Say the sentence out loud with a number in it: \"about a quarter of the data lies between 33 and 46.\" If the sentence you were about to write says \"most\" or \"the majority,\" a box plot alone cannot support it.