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Averages: mean, median, mode and range

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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.

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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

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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).

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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.)

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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.

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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. 8

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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 Achieved Number of pupils Points per grade Total points per grade A42184 B3127837 C633 D339 Total44

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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. 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 Achieved Number of pupils Points per grade Total points per grade A42184 B3127837 C633 D339 Total44 ? ?

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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. 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 Achieved Number of pupils Points per grade Total points per grade A42184 B3127837 C633 D339 Total44 ? ?

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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 Achieved Number of pupils Points per grade Total points per grade A42184 B3127837 C633198 D339117 Total44

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Insert Video clip 07…….3.19 – 3.47 Click on the image below to play the video

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Now work out the mean points score: You can use a calculator for this. Grade Achieved Number of pupils Points per grade Total points per grade A42184 B3127837 C633198 D339117 Total44

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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 Achieved Number of pupils Points per grade Total points per grade A42184 B3127837 C633198 D339117 Total44

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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 Achieved Number of pupils Points per grade Total points per grade A42184 B3127837 C633198 D339117 Total44

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Read through the information on the next slide carefully. It will be used for several questions. 19

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20 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. YearEnglishMathsScience 200575%71%87% 200675%73%86% 200775%87% 200877%74%86% 200979%75%86% Mean73.2%86.4%

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22 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? YearEnglishMathsScience 200575%71%87% 200675%73%86% 200775%87% 200877%74%86% 200979%75%86% Mean73.2%86.4% ?

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24 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? YearEnglishMathsScience 200575%71%87% 200675%73%86% 200775%87% 200877%74%86% 200979%75%86% Mean76.2%73.2%86.4%

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26 Question using the mean: Question: What percentage achieved pass in Maths in 2007? YearEnglishMathsScience 200575%71%87% 200675%73%86% 200775%87% 200877%74%86% 200979%75%86% Mean73.2%86.4% ?

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28 Answer: Question: What percentage achieved pass in Maths in 2007? Answer: 73% (73.2x5=366 71+73+74+75=293 & 366-293=73) YearEnglishMathsScience 2005757187 2006757386 2007757387 2008777486 2009797586 Mean73.286.4

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Click on the image below to play the video

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30 Question on the mode: Question: What is the modal Science mark for the five year period? YearEnglishMathsScience 200575%71%87% 200675%73%86% 200775%87% 200877%74%86% 200979%75%86% Mean73.2%86.4%

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31 Click on the image below to play the video

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32 Answer: Question: What is the modal Science mark for the five year period? Answer 86% Because 86 occurs the most frequently YearEnglishMathsScience 200575%71%87% 200675%73%86% 200775%87% 200877%74%86% 200979%75%86% Mean73.2%86.4%

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Insert Video clip called Question 3….0.01 – 0.20 Click on the image below to play the video

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34 Question on the median: Question: What is the median Science mark for the five year period? YearEnglishMathsScience 200575%71%87% 200675%73%86% 200775%87% 200877%74%86% 200979%75%86% Mean73.2%86.4%

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Insert Video clip called Question 3….0.25 – 0.51 Click on the image below to play the video

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36 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.) YearEnglishMathsScience 200575571%87% 200675%73%86% 200775%87% 200877%74%86% 200979%75%86% Mean73.2%86.4%

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For suggested resources on mean, median, mode and range, please see the next slides.

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Explanations of mean, median, mode and range http://www.youtube.com/watch?v=81zcjULlh58 http://www.youtube.com/watch?v=uydzT_WiRz4http://www.youtube.com/watch?v=uydzT_WiRz4 (set to music) http://www.youtube.com/watch?v=MY66ImqDwcAhttp://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?

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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/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.htmhttp://www.cimt.plymouth.ac.uk/projects/mepres/book8/bk8i5/bk8_5i2.htm worksheet http://www.mathgoodies.com/lessons/vol8/mean.htmlhttp://www.mathgoodies.com/lessons/vol8/mean.html - mean http://www.mathgoodies.com/lessons/vol8/median.htmlhttp://www.mathgoodies.com/lessons/vol8/median.html - median http://www.mathgoodies.com/lessons/vol8/mode.htmlhttp://www.mathgoodies.com/lessons/vol8/mode.html - mode http://www.mathgoodies.com/lessons/vol8/range.htmlhttp://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? 39

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Thank you and good luck with averages 40

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