2. Urgent Memo From:MD (Managing Director) To: TB (The Board) Date:1 August 2014 We have a problem. Our sales in the children’s cereals market are down. Our rival’s sales in the same sector are up. Please look at their website before attending an urgent Board meeting at 2pm tomorrow in the conference room. MD 2

3. Urgent Memo From:MD (Managing Director) To: DM ( Director of Marketing ) Date:2 August 2014 Our share of the children’s cereal market is falling. Our rival’s new Chocopops cereal seems to be all the rage and their sales are soaring. Our research suggests that kids think our brands are boring and our cereal tastes awful. We need a new approach immediately. Report to me within 7 days or you are all fired. MD 3

4. Urgent Memo From:DM (Director of Marketing) To: MD (Managing Director) Date:6 August 2014 We have an idea that we think you will like. It is called SHOCKWHEAT We were inspired by your research that suggested our cereals taste awful and we will be emphasising that on the packaging. And to encourage brand loyalty we have gone retro by putting a free monster in each packet. There are six different monsters to collect and that should guarantee significant sales. I have enclosed a mock-up of the box for your thoughts. DM 4

5. 5

6. Urgent Memo From:MD (Managing Director) To: DM (Director of Marketing) Date:8 August 2014 Well done. The board think the idea is so stupid it will probably work! But there are a few things we need to know. For instance … On average, how many packets will a customer have to buy to get a full set of monsters? We need an answer within 7 days before we make a final decision. MD 6

7. Urgent Memo From:MD (Managing Director) To: DM (Director of Marketing) Date:8 August 2014 Thank you. We are nearly ready to proceed. But there is one last thing to we need to know. To avoid annoying our customers unnecessarily we want to offer a free set of monsters to anyone who has bought a certain number of packets. Can you recommend a suitable figure and explain your reasons? MD 7

8. How many packets will you need to buy? First thoughts – gut reaction 8

9. How many packets will you need to buy? First thoughts – gut reaction Dot plot: 9

10. How many packets will you need to buy? First thoughts – gut reaction Stem and leaf plot: 10

11. How many packets will you need to buy? First thoughts – gut reaction Stem and leaf plot: 11

12. How many packets will you need to buy? First thoughts – gut reaction Stem and leaf plot: 12

13. Sum of all guesses = 291 20 guesses Mean (fair shares for all) = 291 ÷ 20 = 14.5 Calculating the mean 13

14. Some new notation Σx (sigma x) stands for the sum of all of our guesses x ( x bar) represents the mean n represents the number of guesses x = Σx n Calculating the mean 14

15. Collecting the data 15

16. How many packets will you need to buy? 16

17. How many packets will you need to buy? Experimental data Dot plot: 17

18. How many packets will you need to buy? Experimental data Stem and leaf plot: 18

19. How many packets will you need to buy? Experimental data Stem and leaf plot: 19

20. How many packets will you need to buy? Experimental data Stem and leaf plot: 20

21. Calculating the mean 21

22. Urgent Memo From:MD (Managing Director) To: DM (Director of Marketing) Date:10 August 2014 Thank you for that information but there are a few other things we need to know. Please fill in the blanks below: 25% of our customers should get a set after buying ?? packets. 75% of our customers should get a set after buying ?? packets. 90% of our customers should get a set after buying ?? packets. Answer within 24 hours please, our sales are still falling. MD 22

23. The median effectively divides the data set into two subsets. In our case, there will be 30 data points in each subset. We now find the medians of each of these subsets. Finding the key points in our data set Totals 23

24. This means splitting the ordered data into four equal sets. We have 60 results which means 15 in each group, as shown below. Finding the quartiles Totals 24

25. Lower quartile, Q 1, is mid-way between 10 and 10. Q 1 = 10 Median, Q 2, is mid-way between 13 and 13. Q 2 = 13 Upper quartile, Q 3, is mid-way between 16 and 17. Q 3 = 16.5 The minimum value, Q 0 is 7 The maximum value Q 4 is 31 These values are often referred to as the five key points and are useful for describing the data. Finding quartiles 25

26. Comparing our original estimates with our experimental data 26

27. The median effectively divides the data set into two subsets. In our case, there will be 30 data points in each subset. We now find the medians of each of these subsets. Finding the key points in our data set Totals 27

28. The MD and the board also want some idea of how many packets are likely to be bought by 90% of their customers in order to get a full set of monsters. This means splitting the ordered data into 10 equal sets. We have 60 results which means 6 in each group as shown below. This means the 90th percentile is 19 and so we can report to the board that 90% of their customers are likely to get a full set if they buy 19 packets or fewer. Finding percentiles 28

29. Urgent Memo From:MD (Managing Director) To: DM (Director of Marketing) Date:15 August 2014 Thank you. We are nearly ready to proceed. But there is one last thing to we need to know. To avoid annoying our customers unnecessarily we want to offer a free set of monsters to anyone who has bought a certain number of packets. Can you recommend a suitable figure and explain your reasons? MD 29

30. For any data set, the IQR (interquartile range) = Q3 – Q1 In our case the IQR = 16.5 – 10 = 6.5 We define an outlier to be any results that are more than 1.5 x IQR beyond the upper and lower quartiles. For high outliers, this means the results are greater than 16.5 + 1.5 x 6.5 = 26.25 This means we have one high outlier, 31 Clearly there can be no low outliers in this example. Finding outliers 30

31. Core Maths Support Programme 60 Queens Road Reading RG1 4BS E-mail cmsp@cfbt.com Call 0118 902 1243