1 Averages MENU Introduction Mean Median Mode Range Why use Median ? Why use Mode ? Mean from Freq table Mean from grouped Frequency table Median & Mode from A Frequency table Median and Mode from Grouped Freq table Interquartile Range Why use the Interquartile Range ? Cumulative Frequency Graphs Box and Whisker Diagrams Comparing Box and Whisker Diagrams. Starter problems Mean 1Mean 2 Mean & Range 1 Mean & Range 2Mean & Range 3 Mean, Median & Range Mean & Range 4 Mean Mode & Range Main MENU Mean questions Median questions Mean from Freq table Questions Mean from grouped Freq table questions Median & Mode from a Frequency table questions Median and Mode from Grouped Freq table questions Interquartile Range Questions Cumulative Frequency Graphs questions Box and Whisker Diagrams questions

2 Average A single number which FAIRLY represents a group of other numbers. There are 3 types of average ( 3 different ways of working out an average number ) Menu

3 MEAN MEDIAN MODE Add up all the numbers and then divide by how many numbers there are. Put the numbers in order of size and then find the middle number. The number which appears most often. Menu

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5 MEAN Add up all of the numbers and then divide by how many numbers there are. Maths % English % History % French % Science-- 86 % = 65.8 % Menu

6 Work out the Mean average for each of the following : 1) 8, 7, 2, 5, 9, 6. 2) 8.2, 7.9, 5.1, 6.3 3) 24, 17, 68, 59, 44, 51, 77 Work out the missing number : 4) 2, 7, 5) 7, 12, 11, 2, Mean = 5 Mean = 7 ? ? Mean = 6.2 to 1 d.p Mean = 6.9 to 1 d.p Mean = 48.6 to 1d.p 6 3 Menu Answers

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8 Median (Put in order of size then find the middle number) Median = = 7.5 What if there is an even number of numbers ? Menu

9 Another Method of Locating The Median From Raw Data 2, 3, 3, 3, 5, 7, 8, 9, 9, 11, 12, 14, 14, 15, 18 Firstly count how many numbers there are in the list. 15 Then ADD 1 and divide by = 16 2 = 8 th number. Now count to it. Median = 9 What if there is an even number of numbers ? Menu

10 What if there is an even number of numbers ? 3, 5, 7, 11, 11, 13, 15, 21, 22, 23, 26, 30 Count how many numbers there are: 12 Then ADD 1 and divide by = 13 2 = 6.5 th number Between the 6 th & 7 th numbers Median = 14 Menu

11 Work out the Median Average for the following. 1) 3, 6, 2, 6, 5, 1, 0, 1, 3, 4, 9, 1, 0 2) 12, 20, 9, 12, 10, 17, 9, 14 3) 16, 37, 90, 28, 72, 24, 95, 59 4) 0.32, 0.25, 0.7, 0.4, 0.52, 0.3, 0.9 5) 0.63, 0.124, 0.98, 0.2, 0.801, 0.64 Median = 3 Median = 12 Median = 48 Median = Median =0.4 Menu Answers

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13 Mode (Modal Average) (The number/data which appears most often) Ex1) 3, 5, 2, 5, 3, 2, 4, 1, 3 Ex2) 12, 14, 14, 15, 11, 15, 16, 10 Ex3) 1, 0, 2, 1, 0,2, 3, 3 Mode = 3 Mode = 14 & 15 No Mode Menu Answers

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15 So anybody with a small range must be clever ? The RANGE Highest number – Lowest number What does it show ? Paul’s exam marks: Maths 89% Science 90% English 90% History 88% French 91% Range: = 3 David’s exam marks: Maths Science English History French Range = 3 ? ? ? ? ? 4% 3% 6% 5% 4% Small Range Consistent Reliable Predictable David must be clever ! Obviously not ! Menu

16 Jane’s exam marks: Maths 87% Science 55% History 32% French 74% English 45% Range = 87 – 32 = 55 Big Range Inconsistent Unreliable Unpredictable Menu

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18 Why use the Median instead of the Mean ? Office wages: £12,000£11,000 £14,000 £10,500 £80,000 Mean : 12,000 11,00014,000 10,500 80, ,500 ÷ 5 = £25,500 12,000 11,00014,000 10,500 80,000 Median = £12,000 I have got average qualifications, average experience, average expertise and I will be doing an average job. I should be paid an average wage ! O.K. So £25,500 is not a fair reflection of the office wages. Perhaps I should ask for the Median wage. Fair enough. The Median is used when extreme values distort the Mean. Menu

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20 Why use the Mode ? (Modal Average) Popularity, Surveys. Car Colour Frequency Blue24 Red21 Green15 Other17 Shoe size Sold Menu

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22 Mean From A Frequency Table Number of children in families : 2, 3, 1, 1, 4, 0, 1, 3, 2, 5, 3, 2, 2, 1, 0, 1, 4, 6, 5, 2, 1, 0, 2, 1, 4, 5, 3, 3, 2, 1, 2 Number of children Frequency Mean : (3 x 0) + (8 x 1) + (8 x 2) + (5 x 3) + (3 x 4) + (3 x 5) + (1 x 6) = Mean = 2.3 to 1d.p. Menu

23 Work out the Mean average (to 1 d.p.) for each of the following. (Show all of your working out) Marks f Days off Number of people ) 2) Goals in match Number of matches ) 156 = = = Menu Answers

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25 Mean from a Grouped Frequency Table Marks in a test : 12, 23, 43, 6, 24, 33, 36, 41, 44, 13, 43, 36, 27, 15, 10, 14, 39, 31, 45, 19, 21, 34, 22, 14, 11, 2, 14, 34, 41, 41, 25, 30, 18, 22, 24, 33 Mark Freq Mean = (2 x 4.5) + (10 x 14.5) + (8 x 24.5) + (9 x 34.5) + (7 x 45) Mean = Mean = 27.1 to 1d.p. Menu

26 Work out the Mean average. Length of Stay (days) Frequency ) Number of correct answers f ) Number of days absent Frequency ) Mean = 9.9 days to 1 d.p. Mean = 26.7 correct answers Mean =22.3 days absent Menu Answers

27 Time taken (mins) Number of students ) Length of cars —6.0f ) Mean = 52.0 min to 1 d.p. Mean = 4.02m Work out the Mean Average Menu Answers

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29 Median and Mode from a Frequency TableMark f , 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5 Count how many numbers there are+ 1 2 Add up all the Frequencies = 27 2 = 13.5 th number 1 Mode (Highest Frequency)=1 Menu

30 Work out the Median and Mode. Mark f Days off Number of people ) 2) 3) Number of children Number of families Median = 4 Median = 1 Mode = 6 Mode = 3 Mode = 2 Menu Answers

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32 Median and Mode from a Grouped Frequency Table Time(Secs) f Median: Add up all of the frequencies = 42 2 = 21 st number Median is groupMode is group Menu

33 Median and Mode from a Grouped Frequency Table Time(Secs) f Length (cm) Leaves ) 2) Weight (Kg) 0 W<5 5W<10 10W<15 15W<20 20W<25 f ) Median is Modal group is Median is Modal group is Median is 5 W<10 Modal group is 5 W<10 Menu Answers

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35 Interquartile Range. Interquartile Range = Upper Quartile – Lower Quartile Median = 29 L.Q. = 14 U.Q. = 47 Interquartile Range = 47 – 14 = Median = 35 L.Q. = 14 U.Q. = 58 Interquartile Range = 58 – 14 = 44 What if there is an extra number ? Menu

36 Work out the MEDIAN and INTERQUARTILE RANGE for each of the following : 1) 18, 11, 30, 8, 31, 10, 11, 24, 12 2) 55, 36, 60, 32, 60, 83, 48, 21, 90, 72 3) 17, 31, 8, 37, 14, 45, 24, 38 Median = 12, Int Range = 27 – 10.5 = 16.5 Median = 57.5, Int Range = 72 – 36 = 36 Median = 27.5, Int Range = 37.5 – 15.5 = 22 Menu Answers

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38 Why do we bother with the Interquartile Range ? Why don’t we simply use the Range all of the time ? Sarah’s exam marks : 88 %, 90 %, 89%, 91%, 92%, 93%, 89%, 90% Range = 93 – 88 = 5% (Small Range) A Small Range means that Sarah is very consistent, predictable, reliable. Mark’s exam marks : Range = 94 – 32 = 62% (Big Range) A Big Range means that Mark is very inconsistent, unpredictable, unreliable. But Mark is predictable ! It was only the 32% that gave the impression that he is inconsistent ! I will try the Interquartile Range ! 32% 88%89% 90% 91%92% 94% Median = 90.5 L.Q. = 88.5U.Q. = 92 Interquartile Range = 92 – 88.5 = 3.5 (Small Interquartile Range) This small Interquartile Range shows that Mark is consistent. A small Interquartile Range also shows that the Middle half of the numbers are bunched together. Menu

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40 Cumulative Frequency (C.f.) Graph. Time (Secs) Freq (f) c.f Time taken for a class to answer a Maths question. c.f. Time (s) Cumulative Frequency is a ‘running total’ of the frequencies How many pupils were in the class ? You must plot the UPPER boundary against the c.f. xx x x x x x x The curves are usually ‘S’ shaped and they NEVER dip down ! We can use the curve to work out the Median and the Quartiles ! 28 pieces of data Median will be the 28 ÷ 2 = 14 th number Median = 58 seconds The Lower Quartile will be the ¼ of 28 th number = 7 th number. Lower Quartile = 47 seconds The Upper Quartile will be the ¾ of 28 th number = 21 st number. Upper Quartile = 66 seconds Interquartile Range = = 19 seconds Menu

41 Cumulative Frequency (C.f.) Graph. Time (Secs) Freq (f) c.f Time taken for a class to answer a Maths question. c.f. Time (s) xx x x x x x x How many pupils took less than 56 seconds ? 12 pupils How many pupils took longer than 64 seconds ? 28 – 18 = 10 pupils. Menu

42 Cumulative Frequency (C.f.) Graph. Time (Secs) Freq (f) c.f Time taken for a class to answer a Maths question. c.f. Time (s) xx x x x x x x What % took less than 44 secs ? x = 17.9 % 60 % of the pupils answered within the time limit. What was the time limit ? 60 X 28 = secs Menu

43 Draw Cumulative Frequency Graphs for the following and use them to answer the questions : ( Firstly, copy and complete the tables. ) Height (m) FrequencyC.f. 5—10 5—109 10— — — — —356 35—402 40— ) a)What is the Median height ? b)What is the Interquartile Range ? c)How many trees were less than 38 m tall ? d)How many trees were more than 26 m tall ? Weight (Kg) Frequencyc.f.60—701 70—803 80—908 90— — — — a)What is the Median weight ? b)What is the Interquartile Range ? c)What Percentage of people were under 84 Kg ? d)What percentage of people were over 106 Kg ? 2) Ans Menu

Height of Tree (m) C.f x x x x x x x x x Median = 21.5 m L.Q. = 14.5 U.Q. = 26.5 m Interquartile Range 26.5 – 14.5 = 12 m 69 trees under 38 m 20 trees over 26 m Qu Menu

Weight of Adults (Kg) c.f. x x x x x x x x Median = 101 KgL.Q. = 93 Kg U.Q.= 112 Kg Interquartile Range= = 19 Kg 6 57 x 100 = 10.5 % x 100 = 36.8 % Qu Menu

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47 Box and Whisker Diagrams. 12, 18, 32, 40, 45, 60, 70, 90, 94 Lowest Value = 12 Highest Value = 94 Median = 45Upper Quartile = 80 Lower Quartile = 25 Menu

48 Draw a Box And Whisker Diagram for each of the following: 1) 23, 20, 24, 22, 18, 21, 23, ) 1.9, 4.3, 2.6, 1.4, 4.4, 1.5, 3.1, 2.8, Menu Answers

49 Comparing Box and Whisker Diagrams The following will show a forest of red trees and a forest of green trees : Menu

Smallest = 3Biggest = 9Median = 7.5 L.Q. = 6 U.Q. = 8.5 Smallest = 5 Big = 11.5 Med = 8 L.Q. = 7 U.Q. =11 Positive Skew Negative Skew Long whiskers can indicate outliers Long boxes show large spread of data Height of Trees (metres) Menu

51 A competition has three different games. Sarah has played two of the games. GAME A GAME B GAME C SCORE5880 To win, Sarah needs a mean score of 70. How many points does she need to score in game C ? ? 72 Menu Answers

52 Paul has these 4 cards. The mean is Paul takes another card. The mean of the 5 cards is still 6. What number is on his new card ? ?6 Menu Answers

53 Mike has 6 cards : 8888 ?? Mean = 8 Range = Menu Answers

54 Bob and Helen play three games. Their scores have the SAME MEAN. The RANGE of Bob’s score is TREBLE the range of Helen’s scores. What are the two missing scores ? Bob50 Helen ??3565 Menu Answers

55 Joe has three darts scores with a Mean of 24 and a Range of 26. His first dart scored 18. What were his other two scores ? 4014 Menu Answers

56 Three people have a Median age of 40 and a Mean age of 52. The Range of their ages is 64. How old is each person ? Menu Answers

57 Three people have a Mean age of 20. The Range of their ages is 8. What is the LOWEST possible age of : a) The youngest person ? b) The oldest person ? Menu Answers

58 Jenny has five cards numbered in the range of 0 to 20. She says: ‘The Range of my cards is 10, the Mode is 7 and the Mean is 10.’ Is this possible ? ‘The Range of my cards is 10, the Mode is 7 and the Mean is 10.’ Is this possible ? Menu Answers

59 End of Averages Presentation. Return to previous slide.