1. Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com
GCSE: Histograms
Dr J Frost
Objectives: To understand why a histogram is useful for displaying data, and how to both draw and interpret a histogram.
Last modified: 13th January 2016

2. Click to Start Bromanimation
Pablo is hosting a party. He counts how many people are between 15 and 20, and 20 and 50.
Age (years)
Frequency
15 ≤ a < 20 15
20 ≤ a < 50
Why is below graph somewhat unhelpful.
How could we fix it?
Click to Start Bromanimation 15
Frequency
Age

3. Click to Start Bromanimation
Let’s presume that within each age group, the ages are evenly spread.
Then there would people of each age in the group, and people of each age in the group.
Age (years)
Frequency
15 ≤ a < 20 15
20 ≤ a < 50 ? ?
Click to Start Bromanimation 3 2 1
The resulting diagram is known as a histogram.
The ‘frequency per age’ is known as the ‘frequency density’. In general, given the frequency and class width, we can calculate it using:
Frequency Density =
Frequency
Class Width
Frequency Density
Estimated Frequency ?
Age

4. Bar Charts vs Histograms
For discrete data.
Frequency given by height of bars.
Histograms
For continuous data.
Data divided into (potentially uneven) intervals.
Frequency given by area of bars. ? ? ? ?
Frequency Density
Frequency
1.0m m m m m
Height
Shoe Size

5. Copy and complete ? ? ? ? ? ? ? ? Weight (w kg) Frequency
Frequency Density
0 < w ≤ 10 40 4
10 < w ≤ 15 6
1.2
15 < w ≤ 35 52
2.6
35 < w ≤ 45 10 1 ? ?
Freq ?
F.D.
Width ?
Frequency = 40 ? 5 4 3 2 1
Frequency = 15 ?
Frequency = 25 ?
Frequency Density
Frequency = 30 ?
Height (m)

6. ? ? ? ? ? ? ? The Box of Helpfulness Freq F.D. Width
Start by adding a Frequency Density column
Frequency Density
30  30 = 1 ? 84 ?
4.2 (using graph) ? ? 60
6 (using graph) ?
40  20 = 2 ?
18  30 = 0.6 ? 1 2 3 4 5 6 7 8
The Box of Helpfulness
We don’t know the scale on the frequency density axis. Can we work it out using the first row of the table?
This triangle will help throughout.
Freq
F.D.
Width

7. Determining the frequency density scale
Copy the diagram and table, then work out the scale on the frequency density axis. 4 3 2 1 ? ? 16 12 8 4 ? 2 1
Frequency Density
Frequency Density
Frequency Density
Height (m)
0≤𝑥<15
Frequency 30
Height (m)
0≤𝑥<5
Frequency 60
Height (m)
20≤𝑥<32
Frequency 6

8. Questions
Provided collection of past GCSE questions.

9. Summary ? ? ? ? ? Tips you might give your classmates... Purpose:
Histograms allow us to display continuous data grouped into (potentially non-fixed) intervals.
Area:
The area of a bar is equal to the frequency*.
* Actually it’s only proportional to it, but you don’t need to worry about that till A Level. ? ?
Working out the F.D. scale:
If the frequency is known and the bar height is known, we can work out the scale using the formula on the left.
Frequency Density Formula:
Frequency Density is ‘frequency per unit value’, i.e: ? ?
Working out proportion of things (no FD scale given):
Use any arbitrary scale for FD axis. Use it to find area of region that matches description. Divide by total area.
Freq ?
F.D.
Width

10. Proportion Histogram Questions
Sometimes you have to find the proportion of people/things/animals within some range of values. Just find the total area, and the area you’re interested in. 8 7 6 5 4 3 2 1
Total area =50 ?
What proportion of people had a height:
Between 10 and 14m:
𝟏𝟔 𝟓𝟎 = 𝟖 𝟐𝟓
Between 14 and 18m:
𝟏𝟖 𝟓𝟎 = 𝟗 𝟐𝟓 ?
Frequency Density ?
Bro Tip: If the frequency density scale is missing, you can set it to what you like.
Height (m)

11. Solutions – Questino 2 ? Solution: Total apples:
(40 x 0.12) + (20 x 0.36) + (20 x 0.7) + (20 x 0.56) + (40 x 0.18) = 44.4
Apples in range g:
(20 x 0.36) + (20 x 0.7) + (20 x 0.56) = 32.4
Proportion = =𝟎.𝟕𝟑 ?
This is Q2 on your worksheet.

12. Questions
Provided collection of past GCSE questions.

13. Solutions – Question 1
Answer:
×285=86.2 ?

14. Solutions – Question 3 FD 16 4
2.4
4.8 16 12 8 4
Frequency Density 40 60 56 32 ?
B1 for Frequency density label or appropriate units B2 for 4 correct histogram bars sq (B1 for 2 bars correct)

15. Solutions – Question 4 ?
0.05
0.04
0.03
0.02
0.01 8 6 ?

16. Solutions – Question 5