GCSE: Histograms Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 30 th August 2015...

GCSE: Histograms

Dr J Frost (jfrost@tiffin.kingston.sch.uk)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.

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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