Mean. A Fair s fair. Ann s group. Ken s group. Which group do you think did better? Ken. Ann. Name Number of cans. Ann 2 Joe 6 Tina 3 Chinny 9

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1 33 Mean his work will help you find the mean for a set of data. A Fair s fair Ann s group Name Number of cans Ann 2 Joe 6 ina 3 Chinny Ken s group Name Number of cans Ken Asif 2 Siobhan 7 My group did better than yours because we collected more cans. I think we did better. Ann Ken Which group do you think did better? 151

2 A1 his table shows how many cans Sharon s group collected. (a) How many cans did Sharon s group collect in total? (b) Find the mean number of cans for Sharon s group. Sharon s group Sharon 6 Amy 7 Rajit 8 Mark 7 A2 hese tables show some results for other groups. Find the mean number of cans for each group. Amit s group Amit Prakash 1 Jane 2 Debbie 1 Kay 1 Jason s group Jason 3 Isha Purva 5 Nina 3 Rick 3 Helen 1 A3 Class J also collect cans for recycling. hese tables show how many they collected one week. Sarah s group Sarah 2 John 5 Paul 3 Mary 12 Vijay Fay 3 (a) Find the mean number of cans for each group. (b) Which group had the highest mean? (c) Which group had the lowest mean? Gopal 4 Wayne s group Wayne 4 Irvin 0 Stephen Alan 4 Jo 2 Harriet 5 Sohan s group Sohan Diana 4 Louise 1 Jim 6 Kim 2 2 Ravi s group Ravi Joseph 11 Jamie 4 152

3 A4 Class N collects bottles for recycling. hese tables show how many bottles they collected one week. Group 2 Group 3 Adam 12 Andrew Group 1 Patrick 13 Sam 12 Shane 13 Ryan 16 Jamie 2 Chris Jenny Lucy Wing Yang 5 Lee 0 Sara 1 Martin 21 Lynn 6 Samantha 8 Jacob 14 Group 5 Barry 16 Leighanne 12 Vicky 13 Michael Katie 26 Delroy 11 Sean eresa Jason Clare 12 Sarah 22 Group 4 Which group do you think each person belongs to? Everyone in my group collected roughly the same number of bottles. My group had the highest mean. (a) (b) I didn t collect any bottles. We collected the same number of bottles as the group with the highest mean but our mean was lower. (d) 1 (c) I collected the most bottles. (e) (f) he mean for my group was bottles. 153

4 A5 he table shows group 1 s results for the week before. Group 1 Shane 5 Chris 2 Wing Yang 5 Sara Martin he total number of bottles is 60. So the mean number is 60 8, which is 7.5. But you can t have 7.5 bottles. 13 Lynn 7 Samantha 7 Jacob A mean of 7.5 is OK. It is just a number you can use to compare. 12 Here are the results for the other groups. Group 2 Adam Patrick 2 Group 3 Andrew 7 Ryan 21 Sam Lee 15 Jamie 8 Lucy Jenny Group 5 Barry Group 4 Leigh 14 Vicky 8 Michael 8 Katie 6 Delroy 13 Sean 13 eresa 1 Jason 0 7 Sarah 4 Clare (a) Find the mean number of bottles for each group. (b) Which group had the highest mean? (c) Which group had the lowest mean? Pack it in Do the makers always put exactly the same number of sweets in a packet? Count the number of sweets in some packets. You see average contents on some packets. What does it mean? What would be a good number to use for average contents on the packets you used? 154

5 B Comparing means his activity is described in the teachers guide. B1 Some classes do a memory experiment with ten pictures. (a) his shows how many pictures each pupil in class 7Y remembered. What is the mean number of pictures remembered by 7Y? (b) hese are the results for class. What is the mean number of pictures remembered by? (c) Which class do you think did better at remembering these pictures? B2 he classes do a memory experiment with ten words. his shows how many words each pupil in the two classes remembered. 7Y (a) Find the mean number of words remembered for each class. (b) Which class did better at remembering the words? *B3 Were the classes better at remembering pictures or words? Give a reason for your answer. C Decimal means C1 his table shows the weights of some new-born baby girls. (a) Who weighed 3.61 kg at birth? (b) List the babies that weighed less than 3.5 kg at birth. (c) What is the mean weight of these baby girls? Girls Rosanna 3.14 Ruth 3.2 Poppy 2.84 Ellen 3.61 Jenny 3.02 Faye

6 C2 his table shows the weights of some new-born baby boys. (a) What is the mean weight of these baby boys? (b) he mean birth weight of a baby boy in the UK is about 3.4 kg. Which of these baby boys weighed more than this? Boys Girls Daniel 3.81 Andrew 4.22 David 2.6 Ben 3.26 Peter 3.85 C3 hese tables shows the birth weights of some babies that were born two weeks earlier than expected. Find the mean weight of (a) the baby girls (b) the baby boys C4 What can you say about the mean weights of the boys compared to the girls in question C3? here is a National Seal Sanctuary in Cornwall. It looks after rescued seals until they are well enough to be released back to the sea. his table shows information about some young seals the sanctuary rescued. Girls Kim 1.84 Estelle 2.76 Keisha 2.2 Jane 2.0 Boys Girls Clara 3.22 Phuong 4.27 Ravi 2.68 Sue 3.42 Adam 2.0 Mira 2.41 Rory 3.15 Chris 3.16 Stephen 2.38 Simon 2.8 Philip 4.13 David 2.4 Name Reason for Weight at Weight at rescue rescue (kg) release (kg) Lion abandoned Peace malnourished Yorkie malnourished Gurnard trauma by nets Shanny malnourished Montague malnourished Bass malnourished Mako malnourished Allis malnourished Mackerel eye infection

7 Use the table on page 156 to answer these questions. C5 What did Mako weigh when she was released? C6 Which seals weighed less than 15 kg at rescue? C7 Which seal was the lightest at rescue? C8 How much weight did Bass gain between rescue and release? C Which seals gained over 50 kg between rescue and release? C What was the mean weight for the seals at rescue? C11 What was the mean weight at release? *C12 Some seals are not fit enough to survive on their own. hey stay at the seal sanctuary. he tables show the approximate weights of these seals. Calculate the mean weight for (a) the male seals (b) the female seals Male Girls seals Benny 2 Spitfire 220 Charlie 225 Flipper 2 Scooby 200 Magnus 250 Female Girlseals Fatima 160 Honey 150 Anneka 150 Silky 175 Sheba 165 Jenny 180 wiggy 145 Aiming high a game for two players You need A set of game cards (sheet 234) he game 15 fifteen12 twelve he winner Shuffle the cards and deal two cards to each player. Place the rest of the cards face down. ake turns to pick a card if you want to. Each player can stop at any time. Once one player has stopped, the other player can continue until he or she wants to stop or has 6 cards. he maximum number of cards for one player is 6. Once both players have stopped picking up cards the game is over. he player holding the set of cards with the highest mean at the end of the game 6 six nine 157

8 What progress have you made? Statement I can calculate the mean when the 1 data are whole numbers. Evidence Jen s group Jen 7 Fiona 8 Halim Don 4 Jim s group Jim 2 Dawn 8 Kay Jeff 8 Pete 3 What is the mean number of cans for (a) Jen s group (b) Jim s group 2 he table below shows the number of matches in some boxes. Box A B C D E Number of matches Find the mean number of matches for these boxes. I can calculate the mean from data with decimals. 3 hese tables show the weights of some one-year-old children. Boys Girls Joe.8 Jake 13.6 Noah.8 Leroy.7 ony.3 Anil 11.1 Halim 8.4 Kit.5 Girls Razia.8 Linda.7 Shirley 11.3 Mel.5 Victoria.1 ara 8.7 Mira. Find the mean weight of (a) the boys (b) the girls 158

9 33 Mean he concept is introduced through the idea of equal sharing of objects. Practical work is included. Pupils work with decimals but in general the questions avoid rounding or truncating. A spreadsheet or graphic calculator can be used to calculate the mean of a list of data. Pupils could complete part of this unit using a spreadsheet or graphic calculator. p 151 A Fair s fair p 155 B Comparing means p 155 C Decimal means he mean of whole-number data, where the mean is a whole number or has one decimal place A practical activity involving larger sets of data Means of data involving decimals Essential Optional 5 or packets of Smarties, M&Ms etc. iles, counters or multilink OHP transparencies of sheets 232 and 233 Sheet 234 on thick card (one per pair) Practice booklet pages 66 to 6 A Fair s fair (p 151) he objective of the teacher-led discussion is to introduce the mean for a set of discrete data through the idea of equal sharing. 5 or packets each of Smarties, M&Ms etc. for Pack it in Optional: iles, counters or multilink to represent the cans/bottles Mean Cans are used so pupils can visualise them being shared out equally between members of the groups. Introduce the mean number of cans as the result of sharing the cans equally between members of the group. Using tiles, counters or multilink to represent the cans may help. he point of the discussion is to arrive at a mean of 20 4 = 5 cans for Ann s group and 18 3 = 6 cans for Ken s group. Ken s group has the highest mean number of cans per person and so can be judged to have done better. his may be judged to be unfair by some pupils who still judge that Ann s group did better because they collected more cans. Pupils could try questions A1 to A4 without a calculator to consolidate number skills. Use of a calculator is assumed in all subsequent work.

10 A5 Decimals are avoided in questions A1 to A4. You might want to mention them in your introduction or wait until question A5. Note that group 5 has a mean which is a recurring decimal. You may need to discuss how to deal with this. he pupils enjoyed this and we had a really good discussion. Pack it in (p 154) You will need 5 or small packets of M&Ms, 5 or tubes of Smarties and 5 or packets of other sweets. It is less expensive (but boring) to count the contents of reuseable packets such as paper clips, drawing pins etc. Discuss what average contents means the mean number of sweets in the packets. Divide the class into groups, and give each group one of the packets to count the contents. When all groups have finished, results can be recorded on the board, and the mean contents of the packets worked out. (Counting the contents of 5 or packets avoids the complication of non-terminating decimals.) Pupils can now discuss what a sensible average contents label for the packets of M&Ms might be. he activity can be repeated with the different types of sweets. B Comparing means (p 155) Pupils find the mean for longer lists of discrete data where they are not explicitly asked to find the totals before calculating the mean.hey use means to compare sets of data. he context is memory experiments. Pupils are usually highly motivated by calculating means about themselves. Items were attached to the board and we compared the average short-term memory of boys against girls. hey really enjoyed this so much so that they wanted to test how well they d remembered the items in the next lesson so we did their average longterm memory too! OHP transparencies of sheets 232 and 233 Explain to pupils that they are going to carry out an experiment to compare the ability to remember pictures and words. Start with a transparency of sheet 232. Show it to pupils for 12 seconds and then give them approximately 45 seconds to write down as many objects as they can remember. (Order and spelling do not matter.) Now do the same with sheet 233. For each experiment, find the mean number of items remembered. hese means may be non-terminating decimals which will need rounding to one decimal place. Pupils may need reminding about rounding. B1, B2 Mistakes are less likely if pupils work in pairs with one reading the list of numbers to the other. All figures in questions B1 and B2 are based on results of actual experiments. 33 Mean 11

11 C Decimal means (p 155) C4 Pupils find means for sets of continuous data where both the data and the mean are decimals (one or two decimal places). he context for this section is weights (of babies and seals). he information on baby weights is based on authentic data. he UK means for 14/5 were kg (boys) and kg (girls) based on all deliveries (single births, twins, triplets etc). All names and weights for the seals are based on authentic data obtained from he National Seal Sanctuary in Gweek, Cornwall. Information on he National Seal Sanctuary including recent rescues can be found at You may like to point out that although the mean weight for the boys is higher, it is a girl (Phuong) who is the heaviest baby in these tables. *C12 hese figures give non-terminating decimals as mean weights. Aiming high (p 157) his game consolidates finding the mean for discrete data where the mean can be a whole number or a decimal. Sheet 234 (copied on to thick card one for each pair) his game really helped emphasise that you need to divide by the number of pieces of data. I thought pupils would have problems deciding when one decimal was greater than another but they coped pretty well. he cards can be placed in a pile and the top card picked each time or they can be spread out for pupils to choose from. Either way, they need to be copied on to card that is thick enough so that the numbers do not show through. Alternatively, the game can be played in groups of three, with one player acting as the dealer (as in Pontoon with players twisting or sticking ). In one school, pupils organised a class tournament where 2 points were awarded for a win, 1 for a draw and 0 for a loss. Pupils can be encouraged to think about a strategy for maximising their chances of winning each time. A Fair s fair (p 151) A1 (a) 28 cans A2 Amit s group: 3 cans Jason s group: 4 cans (b) 7 cans A3 (a) Sarah s group: 6 cans Wayne s group: 4 cans Sohan s group: 3 cans Ravi s group: 8 cans (b) Ravi s group (c) Sohan s group Mean

12 A4 (a) Group 5 (b) Group 2 (c) Group 4 (d) Group 1 (e) Group 2 (f) Group 3 A5 (a) Group 2: 11.4 bottles Group 3: 8.5 bottles Group 4: 12.2 bottles Group 5: 6.6 or 6.7 bottles (b) Group 4 (c) Group 5 B Comparing means (p 155) B1 (a) = 7.1 pictures (b) = 7.8 pictures (c) Class has the higher mean. B2 (a) 7Y: = 6.3 words : = 7.2 words (b) Class has the higher mean. *B3 For each class, the mean number of pictures is higher than the mean number of words. his suggests both classes are better at remembering pictures. C Decimal means (p 155) C1 (a) Ellen (b) Rosanna, Ruth, Poppy, Jenny (c) = 3.26 kg C2 (a) = 3.62 kg (b) Daniel, Andrew, Peter C3 (a) = 2.7 kg (b) = 3.04 kg C4 he pupil s comments, for example: he mean weight for the boys is higher than the mean weight for the girls in C3. he mean weights for the baby girls in C3 (born two weeks early) is less than the mean weight in C1 (c). he mean weights for the baby boys in C3 (born two weeks early) is less than the mean weight in C2 (a). C5 65 kg C6 Peace, Shanny, Montague, Allis C7 Montague C kg C Yorkie, Gurnard, Montague, Bass C = kg C11 60 = 6 kg *C12 (a) kg or 21( ) kg (b) kg or 160( ) kg What progress have you made? (p 158) 1 (a) 28 4 = 7 cans (b) 30 5 = 6 cans = 0.4 matches 3 (a) =.4 kg (b) 70 7 = kg Practice booklet Section A (p 66) 1 (a) 25 children (b) 5 2 (a) (i) 5 (ii) 4 (iii) 3 (b) he Lings had the highest mean. (c) he Cassells had the lowest mean. 3 A Scruton B Ling C Instone D Scruton E Cassell F Shilton Section B (p 68) 1 (a) Event otal Mean Fun run Sponsored silence Sponsored spell Sponsored walk (b) he mean was lowest for the sponsored walk. 33 Mean 121

13 2 he pupil s answer explained, for example: Sponsored spell as it has the highest mean Sponsored silence as the numbers taking part are going up. Section C (p 6) *2 (a) Mean = 3 = = 3.33 to the nearest penny (b) he mean for High Street was higher than the means for Gresham Street and Bassett Cresent, but lower than those for Sidcup Street and Nonesuch Road. 1 Street otal ( ) Mean ( ) Gresham Street Nonesuch Road Sidcup Street Bassett Crescent (a) Nonesuch Road (b) Sidcup Street (c) he pupil s answer explained Mean