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GCSE: Histograms
Dr J Frost ([email protected])www.drfrostmaths.com
Last modified: 30th August 2015
Objectives: To understand why a histogram is useful for displaying data, and how to both draw and interpret a histogram.
Age (years) Frequency
15 ≤ a < 20 15
20 ≤ a < 50 15
10 20 30 40 50
Age
Freq
uenc
y 15
Pablo is hosting a party. He counts how many people are between 15 and 20, and 20 and 50.
Why is below graph somewhat unhelpful.How could we fix it?
Click to Start Bromanimation
Age (years) Frequency
15 ≤ a < 20 15
20 ≤ a < 50 15
10 20 30 40 50Age
Estim
ated
Fre
quen
cy
3
2
1
Let’s presume that within each age group, the ages are evenly spread.Then there would 3 people of each age in the 15-20 group, and 0.5 people of each age in the 20-50 group.
Click to Start Bromanimation
??
Freq
uenc
y D
ensi
ty 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?
Bar Charts vs Histograms
6 7 8 9
Shoe Size
Fre
quen
cy
Height
1.0m 1.2m 1.4m 1.6m 1.8m
Fre
quen
cy D
ensi
ty
Bar Charts• 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.
? ?
??
F.D.
Freq
Width
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
??
??
10 20 30 40 50Height (m)
5
4
3
2
1
Freq
uenc
y D
ensi
ty
Frequency = 15
Frequency = 30
Frequency = 40
Frequency = 25?
?
?
?
F.D.
Freq
Width
This triangle will help throughout.
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?
1
2
3
4
5
6
7
8
84
60??
40 20 = 2
18 30 = 0.6 ??
30 30 = 1 ?Frequency Density
Start by adding a Frequency Density column
4.2 (using graph)?6 (using graph)?
Determining the frequency density scaleFr
eque
ncy
Den
sity
0 10 20
Height (m)
Frequency
4
3
2
1
?
Copy the diagram and table, then work out the scale on the frequency density axis.
Freq
uenc
y D
ensi
ty
0 10 20
Height (m)
Frequency
16
12
8
4
?
Freq
uenc
y D
ensi
ty
20 28 36
Height (m)
Frequency
2
1?
Provided collection of past GCSE questions.
Questions
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:
F.D.
Freq
Width
? ?
?
?
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.
?
Proportion Histogram QuestionsSometimes 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.
Freq
uenc
y D
ensi
ty
Height (m)10 14 18 22 26
What proportion of people had a height:
Between 10 and 14m:
Between 14 and 18m:
Total area
Bro Tip: If the frequency density scale is missing, you can set it to what you like.
?
?
?
8
7
6
5
4
3
2
1
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 140-160g:(20 x 0.36) + (20 x 0.7) + (20 x 0.56) = 32.4
Proportion =
?
Solutions – Question 2
This is Q2 on your worksheet.
Provided collection of past GCSE questions.
Questions
Solutions – Question 1
Answer:
?
Solutions – Question 3
40605632
FD1642.44.8
16
12
8
4
0
Freq
uenc
y D
ensi
tyB1 for Frequency density label or appropriate unitsB2 for 4 correct histogram bars sq(B1 for 2 bars correct)
?
Solutions – Question 4
86?
0.05
0.04
0.03
0.02
0.01
?
Solutions – Question 5
