Histograms

Measurements are grouped into classes

Nineteen students record how long they spent on a homework task, in minutes. A time is a measurement, so it can take any value, such as 12.5 minutes or 27.3 minutes, and very few students will have exactly the same time. A frequency table with one row for each separate time would show a frequency of 1 in almost every row, and no pattern.

So the times are grouped into classes of equal width, 10 minutes each. Write t for a time in minutes. The class $10\le t<20$10 ≤ t < 20 holds every time up to, but not including, 20 minutes. A time of exactly 20 minutes goes in the next class, $20\le t<30$20 ≤ t < 30. The classes meet end to end, so every time belongs to exactly one class.

Each class is named by its two ends, so 10–20 stands for $10\le t<20$10 ≤ t < 20. The frequencies 3, 8, 6 and 2 add up to the 19 students.

Drawing the histogram

A histogram draws this table. The horizontal axis is a continuous scale of time in minutes, from 0 to 40, and each class takes an equal length of it. The vertical axis is frequency.

Each class gets one bar, standing exactly over its interval: the first bar runs from 0 to 10 on the time axis, the second from 10 to 20, and so on. The height of each bar is the frequency of its class: 3, 8, 6 and 2.

The bars touch, with no gaps between them. That is because the intervals touch: the first class ends at 10 where the second begins, and no time lies between two classes. A gap in a histogram would claim that no student took a time inside it.

The bars stand side by side over 0–10, 10–20, 20–30 and 30–40 minutes, with heights 3, 8, 6 and 2. The tallest bar is over 10–20.

Reading a histogram

The tallest bar belongs to the class that holds the most values. It is called the modal class, and here it is $10\le t<20$10 ≤ t < 20. The modal class is an interval of times, not the number 8: 8 is its frequency.

Add the heights to find how many values the histogram shows: 3 + 8 + 6 + 2 = 19 students. Add the bars on one side of a class boundary to count the values on that side. The students who took 20 minutes or more are in the last two classes: 6 + 2 = 8, which is $\frac{8}{19}\approx 42$8/19 ≈ 42% of the students.

A histogram does not show where each value lies inside its class. It says that 8 students took from 10 to 20 minutes, but not whether their times were close to 10 or close to 20. Grouping gives up that detail in return for a clear picture of how the values are spread.

A histogram is not a bar chart

A bar chart shows categories, such as favorite sports. Its bars stand apart, because nothing lies between one category and the next, and their order can be changed without changing the chart’s meaning.

A histogram shows a measurement on a number scale. Its bars touch, because the classes run on from one another, and their order is fixed by the scale: the bar for 10–20 minutes must stand between the bars for 0–10 and 20–30.

Area and height

Every bar here is 10 minutes wide, so the area of each bar is 10 × its height, and a bar twice as tall has twice the area. With equal class widths, the heights and the areas of the bars are in the same ratio, so the height can be read as the frequency.

Looking ahead: when the classes have different widths, a wide class collects more values just by being wide, and the height alone misleads. Then the area of a bar shows its frequency, and the vertical axis becomes frequency density.