Averages: mean, median, mode and range

Click on the image below to play the video

Recap of the ‘mean’

Mean……....add up the numbers and divide by the number of numbers………often called the average.

For example: the mean of 3, 3, 4, 5, 5, 18, 22 is 7.5

because 3 + 3 + 4 + 5 + 5 + 18 + 22 = 60 (add up the numbers)

and 60 divided by 8 is 7.5 (divide by the number of numbers)

Hint: to divide by 8; half it, half it again, & then half it again.

Recap of the ‘median’

Median…….this is the middle number when the numbers are in order of size.

For example: finding the median of 7, 3, 10, 5, 3, 8, 4 we first arrange them in order of size namely 3, 3, 4, 5, 7, 8, 10

Next: the median of 3, 3, 4, 5, 7, 8, 10 is 5

because 5 is in the middle of 3, 3, 4, 5, 7, 8, 10

Recap of the ‘median’

Median…….the middle number.

However, when the middle is shared between two values you need to ‘average’ the middle two values……

For example: the median of 2, 3, 3, 3, 4, 7, 8, 8 is 3.5

because 3 & 4 are the middle of 2, 3, 3, 3, 4, 7, 8, 8 and the ‘average’ of 3 & 4 is 3.5 (3+4 divided by 2).

Recap of the mode

Mode……..the number which occurs the most….the most frequently occurring number.

If there is no single number which occurs the most we say that there is no mode.

For example: There is no mode for the numbers 1, 1, 2, 2, 3, 3, 4, 7, 8 because no single number appears more often than any other.

(Be careful not to say that the mode is 0 or zero, use the expression ‘no mode’.)

Recap of the Range

Range = (highest value) minus (least value)

For example: the range of

3, 3, 4, 5, 6, 6, 6, 7, 8, 9, 9, 9, 11 is 11-3 = 8

3 and 11 represent the extremes of (3, 3, 4, 5, 6, 6, 6, 7, 8, 9, 9, 9, 11)

A common error or common misconception is to give the range as two values for example “3 to 11” or “3-11”. The range will always be a single number representing the width of the data.

Read through the question on the next slide.

There is a lot of information. Take a moment to familiarise yourself with the data in the table.

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Question:

A teacher wants to calculate the mean points score achieved by pupils in an end of term test. There were 44 pupils in the year group.

4 pupils got a Grade A

31 pupils got a Grade B

6 pupils got a Grade C

3 pupils got a Grade D

So we get the total of 44 pupils who took the test by adding:

4 +31 + 6 + 3 = 44

Grade Achieve

d

Number of

pupils

Points per

grade

Total points

per grade

A 4 21 84

B 31 27 837

C 6 33

D 3 39

Total 44

Question:

A teacher wants to calculate the mean points score achieved by pupils in an end of term test. There were 44 pupils in the year group.

You will notice that there are gaps for the total points for Grade C and Grade D.

For Grade A, 4 pupils have scored 21 points so therefore 4 x 21 = 84 which gives us the total number of points per grade.

For Grade B, 31 pupils have scored 27 points so 31 x 27 = 837

Now find the missing data for Grades C and D.

Grade Achieve

d

Number of

pupils

Points per

grade

Total points

per grade

A 4 21 84

B 31 27 837

C 6 33

D 3 39

Total 44

?

?

Click on the image below to play the video

Question:

A teacher wants to calculate the mean points score achieved by pupils in an end of term test. There were 44 pupils in the year group. The mean points score is given by the formula:

mean points score =

(total points for the year group)

(total number of pupils in the group)

What was the mean points score for the year group? (Give your answer to the nearest whole number.)

Note: the total points for grades A and B have been done for you. Try to fill in the total points for grade C and D without a calculator.

Grade Achieve

d

Number of

pupils

Points per

grade

Total points

per grade

A 4 21 84

B 31 27 837

C 6 33

D 3 39

Total 44

?

?

Answer for grade C and D total points:

6 x 33 = 198 (because 6 pupils scored 33 marks each)

3 x 39 = 117 (because 3 pupils scored 39 marks each)

Grade Achieve

d

Number of

pupils

Points per

grade

Total points

per grade

A 4 21 84

B 31 27 837

C 6 33 198

D 3 39 117

Total 44

Insert Video clip 07…….3.19 – 3.47

Click on the image below to play the video

Now work out the mean points score:

You can use a calculator for this.

Grade Achieve

d

Number of

pupils

Points per

grade

Total points

per grade

A 4 21 84

B 31 27 837

C 6 33 198

D 3 39 117

Total 44

Click on the image below to play the video

Answer:

Mean points score =

(total points for the year group)

(total number of pupils in the group)

Mean points score =

(84 + 837 + 198 + 117)

(4 + 31 + 6 + 3)

Grade Achieve

d

Number of

pupils

Points per

grade

Total points

per grade

A 4 21 84

B 31 27 837

C 6 33 198

D 3 39 117

Total 44

Final answer:

Mean points score =

(total points for the year group)

(total number of pupils in the group)

mean points score =

(84 + 837 + 198 + 117)

(4 + 31 + 6 + 3)

mean points score =

(1236) / (44) = 28.090909…..

mean points score = 28

(to the nearest whole number)

Grade Achieve

d

Number of

pupils

Points per

grade

Total points

per grade

A 4 21 84

B 31 27 837

C 6 33 198

D 3 39 117

Total 44

Read through the information on the next slide carefully. It will be used for several questions.

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Table of pass rates in exams:

The table above shows the percentage of your class achieving ‘pass’ and above, in English, Mathematics and Science between 2005 and 2009.

We are going to use this data for a number of questions.

Year English Maths Science

2005 75% 71% 87%

2006 75% 73% 86%

2007 75% 87%

2008 77% 74% 86%

2009 79% 75% 86%

Mean 73.2% 86.4%

Click on the image below to play the video

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Question on the ‘mean’:

Question:

What is the mean percentage of your class achieving ‘pass’ and above, for English for the five-year period 2005 to 2009?

Year English Maths Science

2005 75% 71% 87%

2006 75% 73% 86%

2007 75% 87%

2008 77% 74% 86%

2009 79% 75% 86%

Mean 73.2% 86.4%?

Click on the image below to play the video

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Answer:

Question: What is the mean percentage of your class achieving ‘pass’ and above, for English for the five-year period 2005 to 2009?

Answer: 76.2% (75+75+75+77+79)/5 or 381/5

Do you know what a mean is?

Year English Maths Science

2005 75% 71% 87%

2006 75% 73% 86%

2007 75% 87%

2008 77% 74% 86%

2009 79% 75% 86%

Mean 76.2% 73.2% 86.4%

Click on the image below to play the video

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Question using the ‘mean’:

Question:

What percentage achieved ‘pass’ in Maths in 2007?

Year English Maths Science

2005 75% 71% 87%

2006 75% 73% 86%

2007 75% 87%

2008 77% 74% 86%

2009 79% 75% 86%

Mean 73.2% 86.4%

?

Click on the image below to play the video

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Answer:

Question: What percentage achieved ‘pass’ in Maths in 2007?

Answer: 73% (73.2x5=366 71+73+74+75=293 & 366-293=73)

Year English Maths Science

2005 75 71 87

2006 75 73 86

2007 75 73 87

2008 77 74 86

2009 79 75 86

Mean 73.2 86.4

Click on the image below to play the video

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Question on the ‘mode’:

Question:

What is the modal Science mark for the five year period?

Year English Maths Science

2005 75% 71% 87%

2006 75% 73% 86%

2007 75% 87%

2008 77% 74% 86%

2009 79% 75% 86%

Mean 73.2% 86.4%

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Click on the image below to play the video

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Answer:

Question:

What is the modal Science mark for the five year period?

Answer 86% Because 86 occurs the most frequently

Year English Maths Science

2005 75% 71% 87%

2006 75% 73% 86%

2007 75% 87%

2008 77% 74% 86%

2009 79% 75% 86%

Mean 73.2% 86.4%

Insert Video clip called Question 3….0.01 – 0.20

Click on the image below to play the video

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Question on the ‘median’:

Question:

What is the median Science mark for the five year period?

Year English Maths Science

2005 75% 71% 87%

2006 75% 73% 86%

2007 75% 87%

2008 77% 74% 86%

2009 79% 75% 86%

Mean 73.2% 86.4%

Insert Video clip called Question 3….0.25 – 0.51

Click on the image below to play the video

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Answer:

Question: What is the median Science mark for the five year period?

Answer: 86% arrange 87,86,87,86,86 in order of size 86,86,86,87,87 (Find the middle ↑ data point.)

Year English Maths Science

2005 755 71% 87%

2006 75% 73% 86%

2007 75% 87%

2008 77% 74% 86%

2009 79% 75% 86%

Mean 73.2% 86.4%

For suggested resources on mean, median, mode and range, please see

the next slides.

Explanations of mean, median, mode and range

http://www.youtube.com/watch?v=81zcjULlh58

http://www.youtube.com/watch?v=uydzT_WiRz4 (set to music)

http://www.youtube.com/watch?v=MY66ImqDwcA (from start of video to 05.10 min point only)

https://www.youtube.com/watch?feature=player_embedded&v=k3aKKasOmIw

Need more explanations?

http://www.bbc.co.uk/schools/teachers/ks2_activities/maths/mode_median_mean.shtml

http://www.bbc.co.uk/bitesize/ks2/maths/data/mode_median_mean_range/play/ with practice activities!

http://www.cimt.plymouth.ac.uk/projects/mepres/book8/bk8i5/bk8_5i2.htm worksheet

http://www.mathgoodies.com/lessons/vol8/mean.html - mean

http://www.mathgoodies.com/lessons/vol8/median.html - median

http://www.mathgoodies.com/lessons/vol8/mode.html - mode

http://www.mathgoodies.com/lessons/vol8/range.html - range

https://www.khanacademy.org/math/probability/descriptive-statistics/central_tendency/e/mean_median_and_mode

http://www.quia.com/rr/51667.html

Need more practice?

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Thank you and good luck with ‘averages’

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