The frequency density formula

The frequency density formula is frequency density = frequency ÷ class width. You use it to draw a histogram when the groups, or classes, are different widths: the height of each bar is its frequency density, so the area of each bar shows the frequency. To go the other way, frequency = frequency density × class width.

What is frequency density?

In a bar chart, the height of each bar is the frequency. That works when every group is the same width. When the groups are different widths, it would mislead: a class from 0 to 20 would look busier than a class from 20 to 25 just because it covers more values.

Frequency density fixes this. It is the frequency per unit of class width: how crowded the data is inside each class. Using it as the bar height makes the area of each bar equal to its frequency, so wide classes no longer look bigger than they are.

How to work out frequency density

1. Find each class width: the upper boundary minus the lower boundary. For 10 < t ≤ 15, the width is 15 − 10 = 5.
2. Divide the frequency by the class width for each class.
3. Draw the histogram with the variable on the x-axis and frequency density on the y-axis. Bars touch, with no gaps, because the data is continuous.

Worked example: working out frequency density

A survey records how long 73 people waited at a doctor's surgery, t minutes.

Here is the histogram those numbers give.

Here x is the waiting time in minutes and y is the frequency density. The 30 to 50 class has 8 people, but spread over 20 minutes, so its bar is low. The 15 to 20 class has 20 people in just 5 minutes, so it is the tallest. The histogram shows correctly that waits of 10 to 20 minutes were the most common, which a plain bar chart of the frequencies would hide. The Histogram Maker draws this for any table and works out the frequency densities for you.

Try it

Histogram Maker

Draw histograms with frequency density and cumulative frequency

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Reading frequencies from a histogram

Rearrange the formula to get frequencies back from a histogram:

In other words, the frequency is the area of the bar. If a bar runs from 40 to 60 and has a height of 2.5, its frequency is 2.5 × 20 = 50.

Estimating the frequency in part of a class

A question might ask how many people waited between 12 and 25 minutes. That range cuts through three bars, so add up the area of each piece, assuming the values are spread evenly within each class:

• 12 to 15 minutes: 3 × 3.6 = 10.8
• 15 to 20 minutes: the whole bar, 20
• 20 to 25 minutes: 5 × 1.5 = 7.5

Total: 10.8 + 20 + 7.5 = 38.3, so about 38 people. It is an estimate, because you do not know exactly where in each class the values lie.

Estimating the median from a histogram

There are 73 values, so the median is the (73 + 1) ÷ 2 = 37th value. Add up frequencies from the left: 12 in the first class, 30 after the second, 50 after the third. So the median is in the 15 to 20 class, and you need 37 − 30 = 7 more values from it. The bar has a frequency density of 4 people per minute, so 7 people take 7 ÷ 4 = 1.75 minutes. The median is about 15 + 1.75 = 16.75 minutes.

Completing a table and histogram

A common exam question gives a histogram with some bars drawn and a table with some frequencies missing, and asks you to fill in both. Work each class whichever way it needs:

If you know the frequencies, divide by the width to get a height. If you know a height, multiply by the width to get a frequency. Then check: the frequencies should add up to any total the question gives.

Estimating the mean from grouped data

The same table gives an estimate of the mean. Multiply each class midpoint by its frequency, add them, and divide by the total frequency. For the waiting times: (5 × 12) + (12.5 × 18) + (17.5 × 20) + (25 × 15) + (40 × 8) = 1330, and 1330 ÷ 73 = 18.2 minutes. You do not know the exact values inside each class, so treating them all as sitting at the midpoint gives an approximate answer.

Frequency density with a scale factor

Some histograms, especially at A-level, do not label the y-axis with frequency density but say that the area is proportional to frequency. Then you need to find what one square, or one unit of area, represents.

For example, if a bar 2 cm wide and 5 cm tall represents a frequency of 20, its area of 10 cm² stands for 20 values, so 1 cm² stands for 2 values. A different bar of area 7 cm² then represents 7 × 2 = 14 values. Find the scale from a bar you know before reading any others.

Here is a quick check.

Common mistakes

• Dividing the wrong way round. It is frequency ÷ class width. Width ÷ frequency gives nonsense.
• Using the midpoint or upper boundary instead of the width. For 10 < t ≤ 15 the width is 5, not 12.5 or 15.
• Plotting frequency instead of frequency density when the classes are unequal.
• Reading the height as the frequency. On a histogram, the area is the frequency. Multiply by the width.
• Leaving gaps between bars. Histograms show continuous data, so the bars touch.
• Not labelling the axis. The y-axis should say "frequency density", not "frequency".

Frequency density in GCSE Maths

In AQA's GCSE Maths specification, constructing and interpreting histograms with equal and unequal class intervals is part of statistics, and is Higher tier only. The frequency density formula is not on the GCSE Higher formulae sheet, so learn it: frequency density = frequency ÷ class width. Histograms come back at A-level in statistics, often with the scale factor version.

Practise statistics in a short timed paper

GCSE Maths Higher and A-level Maths: Statistics each give you a short timed paper every day, with 8 questions worth 25 to 35 marks. Histograms come up regularly in both.

Read next

Maths · StatisticsFrequency polygons: how to draw and compare themHow to draw a frequency polygon from grouped data, compare two on one set of axes, and tell them apart from histograms and other frequency diagrams.