GCSE Statistics More on Averages

4.5 Transforming Data GCSE Statistics only Sometimes it is easier to calculate the mean by transforming the data I 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 = −4+1−7+12+3 5 = 5 5 = 1 Add this to your assumed mean 80 + 1 = 81 this is still the mean of the original data

Measure Advantages Disadvantages 4.6 Deciding which average to use Measure Advantages Disadvantages MODE Use 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 MEDIAN Use the median to describe the middle of a set of data that has an extreme value Easy to calculate Unaffected by extreme values MEAN Use 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 Page 130 stats book page 40 unit 1

Which average would you use for these sets of data? Red, red, blue, green, blue yellow £10, £10, £10, £15, £15, £15, £20, £20, £22 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 The GCSE results for a group C C C C D D D B D C C C The wages of 10 people working in an office £150 £180 £190 £330 £120 £240 £450 £500 £125 £270 The average height of a group of people 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 mark Unit 2 is 30% of the final mark Unit 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 graphs and 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 mph lowest speed = 20.5 mph highest 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 mean Exercise 4E page 131 - choosing your average Exercise 4F page 133 - Weighted mean Exercise 4G page 139 – Measures of Spread GCSE Maths Unit 1 Exercise 2E page 41 – Using the three types of average Exercise 2J page 51 – range and interquartile range