| QCA Unit | Mathematics at KS 3, Unit : Mean, median, and modal averages. | |||
| Year Group | Year 8 | Number in class | 30 | |
| Time for lesson | 1 hour | No. of computers | 15 (minimum). | |
Aims/Learning Outcomes: Learners will be able to recognise the difference between different types of average, i.e. mean, median, and mode, and know how to calculate each variant accurately. They will be able to understand the statistical implications of the different types of average, and assess their usefulness/appropriateness in different situations.
All:. Will know the difference between the mean, median, and mode.
Will be able to calculate the mean, median, and mode.
Will be able to calculate the range.
Some: Will be able to make objective judgements as to the usefulness of different types of average in specific calculations.
Keywords: add, divide, mean, median, mode, range, subtract sum.
Resources:interactive whiteboard, pre-prepared presentation on computer networks (see below).
Differentiation: By learning outcome, and differentiated tasks activities. Group work at different levels where the tasks are simplified/extended in terms of content and/or language for specific groups. Those in this class who are in lower ability Maths groups to be aided by teaching support staff as available.
1. Introduction. On the interactive whiteboard, display the definitions of the three types of average in text, i.e.,
AVERAGE.
Mean is the average calculated by adding the numbers together and dividing by the amount of numbers, i.e.,
14, 7, 12, 3, 19, 6, 25, 3, 9, 2.
14 + 7 + 12 + 3 + 19 + 6 + 25 + 3 + 9 + 2 = 100
There are 10 numbers, so the mean is 100 divided by 10 = 10.
MEDIAN.
The Median is the middle value from a set of numbers, i.e. the number which is furthest away from the two ends of the range of data.
13, 13, 13, 13, 14, 14, 16, 18, 21.
As there are 9 numbers in the list, the middle number can be found by the following calculation, (9+1) divided by 2 =10 divided by 2 = the 5th number,
13, 13, 13, 13, 14, 14, 16, 18, 21.
So 14 is the median.
If there is no ‘middle’ number, then the median is the mean average of the two middle numbers. For example,
1, 3, 9, 14.
In the case of even numbers there can be no ‘middle’ number, so the two middle numbers are added together and divided by 2, i.e. the number of values being added.
3 + 9 = 12 divided by 2 = 6.
MODE
The mode is the number repeated most often in any set of numbers.
12, 13, 14, 15, 15, 15, 19, 27.
In this set of numbers, 15 is the value repeated most often, so this is the mode.
RANGE
The range of a set of data is the difference between the greatest and smallest value,\
So in this set of data, 12, 14, 19, 35, 43, the range is calculated by carrying out the operation 43 – 12 = 31.
So, the range is 31.
Main Activity. Set the class differentiated sets of problems in which they must find the mean, median, mode, and range of data sets. These should be differentiated for the various ability groups.
Premier League Clubs 2012-13 Season, Alphabetical Order.
| Club | Price |
| Arsenel | 49 |
| Aston Villa | 28 |
| Chelsea | 49.8 |
| Everton | 36 |
| Fulham | 35 |
| Liverpool | 39 |
| Man City | 32 |
| Man United | 35 |
| Newcastle United | 28 |
| Norwich City | 26 |
| Queens Park Rangers | 29 |
| Reading | 32 |
| Southampton | 27 |
| Stoke City | 28 |
| Sunderland | 25 |
| Swansea City | 26 |
| Tottenham Hotspur | 47 |
| West Bromwich Albion | 28 |
| West Ham United | 36 |
| Wigan Athletic | 23 |
Questions
1. What is the mean average of the Premier League Clubs 2012-13 Season match day price?
2. How many clubs actually charge that ticket price on a match day?
3. What is the Median ticket price?
4. What is the modal ticket price?
5. What is the range of this data?
Plenary: go through the solutions to the questions. Discuss any issues
1. What is the mean average of the Premier League Clubs 2012-13 Season match day price?
| Club | Price £ |
| Arsenal |
49 |
| Aston Villa |
28 |
| Chelsea |
49.8 |
| Everton |
36 |
| Fulham |
35 |
| Liverpool |
39 |
| Man City |
32 |
| Man Utd |
35 |
| Newcastle Utd |
28 |
| Norwich City |
26 |
| QPR |
29 |
| Reading |
32 |
| Southampton |
27 |
| Stoke City |
28 |
| Sunderland |
25 |
| Swansea City |
26 |
| Tottenham Hotspur |
47 |
| WBA |
28 |
| West Ham Utd |
36 |
| Wigan Athletic |
23 |
|
658.8 |
So, the average match day price is £658.8 divided by 20, i.e. = £32.94.
2. How many clubs actually charge that ticket price on a match day?
None.
3. What is the Median ticket price?
| Club | Price £ |
| Chelsea |
49.8 |
| Arsenal |
49 |
| Tottenham Hotspur |
47 |
| Liverpool |
39 |
| Everton |
36 |
| West Ham Utd |
36 |
| Fulham |
35 |
| Man Utd |
35 |
| Man City |
32 |
| Reading |
32 |
| QPR |
29 |
| Aston Villa |
28 |
| Newcastle Utd |
28 |
| Stoke City |
28 |
| WBA |
28 |
| Southampton |
27 |
| Norwich City |
26 |
| Swansea City |
26 |
| Sunderland |
25 |
| Wigan Athletic |
23 |
Arranged in numerical order, the ticket prices appear like this. As the total of 20 clubs is an even number, there is no median number in the genuine sense of the term. So, as explained earlier, the two middle values are added together and their ‘mean’ average becomes the media. The two middle values are..
Reading £32 and QPR £29. 32+29= 61, divided by 2 is 30.5.
So, the median value match day ticket price is £30.50.
4. What is the modal ticket price?
Aston Villa, Newcastle United, Stoke City and West Bromwich Albion all charge £28 for a match day ticket. This is the price which occurs most frequently and therefore this is the mode.
5. What is the range of this data?
The range of this data is the highest ticket price (Chelsea) minus the lowest ticket price (Wigan Athletic).
£49.80 – £23 = £26.80.
Questions for discussion.
How useful are the different kinds of averages? Ask the class these questions.
If we say that the (mean) average price of a match ticket in the Premier League is £32.94, how many of the Premiership Clubs might actually charge that to get in?
The answer is, none. The mean is the sum of all the ticket prices, divided by the number of values, i.e. the number of clubs.
So how helpful are the following categories of information, i.e.
Average prices of goods, such as food or fuel?
Average earnings or wages?
Average length of holidays?
Average exam results in a school?
Which do the class think is the most useful type of average model for describing these types of values?
Would it be better to try and use one of these models in all kinds of analysis, or to use different models in various situations?
How could ICT be used to support the calculation of averages, i.e. in a data handling program?
EdExcel or similar spreadsheets could be use to total a series of data and for division to calculate a mean average. Data could also be sorted to reveal the median and modal values.
Relevant NC Level Descriptors.
For assessment purposes, successful completion of this lesson will enable pupils to achieve the following aspects of the National Curriculum Level Descriptors in Mathematics.
Level 4: Pupils are developing their own strategies for solving problems and are using these strategies both in working within mathematics and in applying mathematics to practical contexts. In solving problems with or without a calculator, pupils check their results are reasonable by considering the context or the size of the numbers. Pupils look for patterns and relationships, presenting information and results in a clear and organised way. This requirement will be met when pupils learn to recognise the difference between different types of average, i.e. mean, median, and mode, and understand how to calculate each variant accurately.
Level 5: Pupils understand and use the mean of discrete data. They compare two simple distributions, using the range and one of the mode, median or mean. They interpret graphs and diagrams, including pie charts, and draw conclusions. This requirement will be met when pupils show that they are able to understand the statistical implications of the different types of average, and assess their usefulness/appropriateness in different situations.
Level 6: Pupils are beginning to give mathematical justifications, making connections between the current situation and ones they have met before. This requirement will be met when pupils show that they are able to judge the meaning of statistics presented in real world situations.
Level 7: Starting from problems or contexts that have been presented to them, pupils explore the effects of varying values…working with and without ICT. They progressively refine or extend the mathematics used, giving a reason for their choice of mathematical presentation and explaining features they have selected. Pupils justify their generalisations. This requirement will be met when pupils show that they are able to understand how statistical interpretations are constructed, assessing their usefulness/appropriateness in different situations.