Histograms provide a useful way to display data. Students often get confused between histograms and bar charts. However, histograms and bar charts are different in a number of ways. Firstly, histograms are used to present continuous grouped data whereas bar charts are used to display ordinal, nominal or discrete data. Click here to see some examples of bar charts. Secondly, bar charts are usually used to compare data whereas a histogram shows the distribution of the data.

Most importantly, histograms are not plotted against frequency, they are plotted against frequency density. This means that the frequency is represented in the area of the bar.

histograms

Since the area of a rectangle is width multiplied by height, we have:

FREQUENCY = FREQUENCY DENSITY \times INTERVAL WIDTH

Going through a detailed example is a good way to fully understand histograms – see the detailed ‘Histogram Example’ below. Then go through the two examples given.

Note that the area may be exactly the frequency but it is also possible to use proportional areas. In these cases:

FREQUENCY = k\times AREA

where k\ne 1 – see Example 1.


Histogram Example

The following table shows the times 100 people waited in a queue for KFC at a specific store in the first hour of reopening during the CoronaVirus lockdown in 2020.

Queue Time, t (minutes) Frequency
0\leq t\textless 10 11
10\leq t\textless 20 4
20\leq t\textless 35 18
35\leq t\textless 50 27
50\leq t\textless 90 40

In order to create a histogram to display this data, we must calculate the frequency densities. That is, we need to find the height of each bar. Once we know the largest frequency density, we can choose the vertical scale for our histogram. Using the above formula:

FREQUENCY DENSITY = FREQUENCY \div INTERVAL WIDTH

The interval widths and frequency densities are as follows:

Interval Width Frequency Density
10 1.1
10 0.4
15 1.2
15 1.8
40 1

These columns are often added to the original table. The largest frequency density is 1.8 and so we choose the vertical scale accordingly. The histogram is constructed as follows.


If the interval widths are uniform, a frequency polygon (see more on Frequency Polygons) can be added to a histogram by joining up the midpoints of the top of each bar of the histogram with straight lines. See Example 1d below.


More Histogram Examples

Example 1

The following histogram shows the masses of some domestic housecats in kilograms:

  1. Justify why a histogram might be used for this data.
  2. Given that the number of cats between 3kg and 3.5 kg is 18, construct the frequency table that accompanies this histogram. How many cats have been weighed in total?
  3. Estimate how many cats weight more than 3.8kg.
  4. Add the associated frequency polygon to this histogram.

Example 2

The incomplete histogram and frequency table for the distance travelled to get to work by 100 employees of a given company are as follows:

histograms

Distance, D (km) Frequency
0\leq D\textless 10
10\leq D\textless 15 15
15\leq D\textless 20
20\leq D\textless 25 10
25\leq D\textless 30 20
30\leq D\textless 40
40\leq D\textless 50
  1. Complete the histogram and the frequency table.
  2. Using midpoints, find an estimate for the mean time travelled (to one decimal place).
  3. What is the median distance travelled?
  4. Using linear interpolation, find estimates for the lower quartile Q1 and the upper quartile Q3.


Alternatively, click here to practice exam questions from Past Applied Maths Papers.


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