GCSE StatisticsMore on Averages

4.5 Transforming Data GCSE Statistics only

Sometimes it is easier to calculate the mean by transforming the dataI have seen this referred to as using an assumed mean (yr. 9 impact maths book?)

Here’s some data find the mean

76 81 73 92 83

Find the difference using an assumed mean of 70

6 11 3 22 13

Find the mean of these numbers = 11

Add this to your assumed mean 70 + 11 = 81 this is the mean of the original data

Same data different assumed mean

Here’s some data find the mean

76 81 73 92 83

Find the difference using an assumed mean of 80

-4 1 -7 12 3

Find the mean of these numbers = = 1

Add this to your assumed mean 80 + 1 = 81 this is still the mean of the original data

Measure Advantages Disadvantages

MODEUse when the data are non numeric or when asked to find the most popular item

• Easy to find• Can be used with any type

of data• Unaffected by open-ended

or extreme values• The mode will be a data

value

• Mathematical properties are not useful

• There is not always a mode or sometimes there is more than one

MEDIANUse the median to describe the middle of a set of data that has an extreme value

• Easy to calculate• Unaffected by extreme

values• Mathematical properties

are not useful

MEANUse the mean to describe the middle of a set of data

that does not have an extreme value

• Uses all the data• Mathematical properties

are well known and useful

• Always affected by extreme values

• Can be distorted by open ended classes

4.6 Deciding which average to use

Page 130 stats book page 40 unit 1

Which average would you use for these sets of data?

1. Red, red, blue, green, blue yellow2. £10, £10, £10, £15, £15, £15, £20, £20, £223.

Wage (£) frequency

600 5

800 20

1000 100

1200 8

2500 2

6000 1

In each of the following questions, explain why you would use the MODE, MEDIAN or MEAN average

1) The GCSE results for a group C C C C D D D B D C C C

2) The wages of 10 people working in an office £150 £180 £190 £330 £120 £240 £450 £500 £125 £270

3) The average height of a group of people

4) The average amount of money spent by a Year 9 student during the weekend

5) The average number of days in a month

4.7 Weighted Mean GCSE Statistics only

When you sit an exam one paper can hold more importance than another and the results are weighted

The papers for your GCSE maths are weighted Unit 1 is 30% of the final markUnit 2 is 30% of the final markUnit 3 is 40% of the final mark

Your final result will be worked out using this weighting.

4.7 Weighted Mean GCSE Statistics only

Example

Example

Example

4.8 Measures of Spread

These are also known as measures of dispersion.

You have met the range = largest value – smallest value

You have may have met quartiles before in the context of cumulative frequency graphsand their best buddy the interquartile range

New to you may be percentiles and deciles variance (we will look at this after unit 1 is finished) standard deviation (we will look at this after unit 1 is finished)

The Range

A crude measure of spread as it only takes into account the largest and smallest of the data values

Example 1

find the range of: 12 6 18 24

Range = 24 – 6 = 18

The Range

Example 2

The speeds v, (to the nearest mile per hour), of cars on a motorway were recorded by the police.

estimate the range of the speeds

Speed v (mph) Frequency

20 < v ≤ 30 2

30 < v ≤ 40 14

40 < v ≤ 50 29

50 < v ≤ 60 22

60 < v ≤ 70 13

the speeds are given to the nearest mphlowest speed = 20.5 mphhighest speed = 70.5 mph

range = 70.5 – 20.5 = 50 mph

The Quartiles (Bob and Frank)

the lower quartile Q1 is the value such that one quarter (25%) of the values are less than or equal to it

the middle quartile Q2 is the median

the upper quartile Q3 is the value such that three quarters (75%) of the values are less than or equal to it

the median and quartiles split the data into four equal parts. That is why they are called quartiles!

A frequently used measure of spread is the inter-quartile range

inter-quartile range (IQR) = upper quartile – lower quartile (Q3 – Q1)

page 135 has the formulae for finding the quartiles

Example

7 9 13 5 6 12 3

Put the data in order

3 5 6 7 9 12 13

find out how many data items you have

n = 7

Q1 = ¼(7 + 1 ) = 2nd value which is 5

Q3 = ¾(7 + 1) = 6th value which is 12

inter-quartile range = Q3 - Q1

= 12 – 5 = 7

Turn to the book page 135 to look at finding quartiles in frequency tables

Turn to the book page 139 to find out about percentiles and deciles

there is no way I can get that graph on this screen until I buy the revision guide!

GCSE Statistics

Exercise 4D page 129 – assumed meanExercise 4E page 131 - choosing your averageExercise 4F page 133 - Weighted meanExercise 4G page 139 – Measures of Spread

GCSE Maths Unit 1

Exercise 2E page 41 – Using the three types of averageExercise 2J page 51 – range and interquartile range