1.2.4 Histograms

A histogram is similar to a bar chart but with some key differences
- A histogram is for displaying grouped continuous data whereas a bar chart is for discrete or qualitative data
- There will never be any gaps between the bars of adjacent groups in a histogram
- Whilst in a bar chart the frequency is read from the height of the bar, in a histogram the height of the bar is the frequency density
On a histogram frequency density is plotted on the y – axis
- This allows a histogram to be plotted for unequal class intervals
- It is particularly useful if data is spread out at either or both ends
The area of each bar on a histogram will be proportional to the frequency in that class

You will not be asked to draw a histogram but you may have to add information to one so you should make sure you are familiar with the process for drawing one
Step 1. Always check that there are no gaps between the upper boundary of a class and the lower boundary of the next class
- If there are gaps you will need to close them by changing the boundaries before carrying out any calculations
  - Consider whether the values are rounded or truncated before closing the gaps
Step 2. Find the class width of each group by subtracting the lower boundary from the upper boundary
Step 3. Calculate the frequency density for each group using the formula:

$frequency density = k \times \frac{frequency}{class width}$

Step 4. The histogram will be drawn with the data values on the x – axis and frequency density on the y – axis
- Remember that the scale on both axes must be even, although the class widths may be uneven
  - Both axes should be clearly labelled and units included on the x – axis
- Most often, the bars will have different widths
Occasionally you will be asked to add a frequency polygon to the histogram
- This is done by joining up the midpoints at the top of each bar
- You should not join up the first or last midpoint to the x – axis (it is not really a polygon!)

It is important to remember that the y – axis does not tell us the frequency of each bar in the histogram
The area of the bar gives information about the frequency
- Most of the time, the frequency will be the area of the bar and is found by multiplying the class width by the frequency density
- Occasionally, the frequency will be proportional to the area of the bar
  - Frequency = $k \times area$
  - In these cases more information will be given to help you find the value of k
You may be asked to find the frequency of part of a bar within a histogram
- Find the area of that section of the bar using any information you have already found out
- You will need to have found the value of k first

The table below and its corresponding histogram show the mass, in kg, of some new born bottlenose dolphins.

2-2-3-histograms-we-diagram

(i)

Use the histogram to find the value of

k

in the formula

frequency density = k \times \frac{frequency}{class width}

(ii)

Estimate the number of dolphins whose weight is greater than 13 kg.

(i)

Use the histogram to find the value of in the formula

frequency density = k \times \frac{frequency}{class width}

2-2-3-histograms-we-solution-part-1

(ii)

Estimate the number of dolphins whose weight is greater than 13 kg.

2-2-3-histograms-we-solution-part-2

Remember that the area of a bar in a histogram is not always the frequency itself but could be proportional to the frequency. Look carefully at the scales on the axes, it will rarely be a simple 1 unit to 1 square.

Edexcel International A Level Maths: Statistics 1

1.2.4 Histograms