Understanding Histograms

 

How Do I Interpret a Histogram?

Histograms show grouped continuous data. Unlike a bar chart, the area of each bar represents the frequency — not just the height.

$$\text{Frequency} = \text{Frequency Density} \times \text{Class Width}$$

Watch Out:

• The y-axis might be labelled frequency density or simply frequency — check carefully!
• If all class widths are the same, height might be proportional to frequency.

 

Steps for Interpreting a Histogram

1. Work Out Missing Frequencies

Use:

$$\text{Frequency} = \text{Density} \times \text{Class Width}$$

2. Estimate Frequencies From Parts of a Bar

Sometimes you’ll be asked for frequency between two points within a class interval.

Treat this like a proportion of the whole bar’s area.

3. Compare Distributions

You can compare shapes, centers, and spreads of two histograms, but only if:

• They use the same class intervals.
• They use the same frequency density scale.

 

Example

The histogram shows the time (in minutes) taken by runners to complete a trail race.

 

 

You’re given part of the frequency table:

 

Time (mins)	Frequency
$30 \leq t \lt 40$	6
$40 \leq t \lt 50$	12
$50 \leq t \lt 55$	10
$55 \leq t \lt 60$	15
$60 \leq t \lt 90$	?

 

(a) Use the Histogram to Find the Missing Frequency

The bar for $60 \leq t \lt 90$:

• Frequency density = 0.4
• Class width = 90 - 60 = 30

$$\text{Frequency} = 0.4 \times 30 = 12$$

So, the missing frequency is 12

 

(b) Use the Table to Complete the Histogram

The bar for $30 \leq t \lt 40$ is missing:

• Frequency = 6
• Class width = 10

$$\text{Frequency Density} = \frac{6}{10} = 0.6$$

Draw a bar from 30 to 40 with a height of 0.6

 

 

 
(c) Estimate the Number of Runners Who Took More Than 58 Minutes

You're looking at:

• Part of the $55 \leq t \lt 60$ bar: only 2 minutes (from 58 to 60)
• Plus all of $60 \leq t \lt 90$

Bar for 55 ≤ t < 60:

• Frequency density = 3
• Width from 58 to 60 = 2

$$\text{Frequency} = 3 \times 2 = 6$$

From part (a), we know there are 12 runners in the 60–90 interval.

$$6 + 12 = 18$$

Estimated number of runners who took more than 58 minutes is 18