Histograms: frequency density and unequal class widths

Rearranged: $\text{frequency} = \text{frequency density} \times \text{class width}$.

Using frequency density stops wide classes looking more important just because they cover more values.

The heights of 45 plants (in cm):

Draw the horizontal axis as a continuous scale of height, and the vertical axis as frequency density. Each bar runs from the start to the end of its class, with height equal to the frequency density. The tallest bar (15 to 20 cm) shows where the data is most concentrated, even though the 0 to 10 class is wider.

To find the frequency of a class, multiply the bar's height by its width. A bar from 30 to 50 with a frequency density of 1.5 represents $1.5 \times 20 = 30$ items.

Some questions give no scale on the vertical axis but tell you one frequency. Use that bar to work out the scale first: if a bar of width 10 represents 20 items, its frequency density is 2, which fixes the scale for the rest.

Assume the data is spread evenly across each class. Using the plant table, estimate how many plants are between 12 cm and 18 cm:

• From 12 to 15 (3 cm of the 10 to 15 class): $2.4 \times 3 = 7.2$.
• From 15 to 18 (3 cm of the 15 to 20 class): $3 \times 3 = 9$.
• Total: $7.2 + 9 = 16.2$, so about 16 plants.

It's an estimate because we don't know exactly where the values lie inside each class.

From a histogram, first work out each class frequency (area). Then estimate the mean using the midpoint of each class, exactly as for a grouped frequency table; our frequency table mean calculator shows the working. For the median, find the class containing the middle value and estimate within it, as above. For the plants, the median is the 23rd value: 20 plants are up to 15 cm, so it's 3 of the 15 plants into the 15 to 20 class, about $15 + \tfrac{3}{15} \times 5 = 16$ cm.

• Plotting frequency instead of frequency density.
• Using the wrong class width (for $10 < h \le 15$ it's 5, not 15).
• Leaving gaps between bars.
• Reading the bar height as the frequency.

1. A class $20 < t \le 25$ has a frequency of 18. Find the frequency density.
2. A bar from 40 to 70 has a frequency density of 0.6. Find the frequency.
3. A class $0 < m \le 50$ has a frequency density of 0.4. Estimate how many values are between 30 and 50.

Answers: 1. $18 \div 5 = 3.6$. 2. $0.6 \times 30 = 18$. 3. $0.4 \times 20 = 8$.

More practice in our histogram questions, or compare with cumulative frequency.