Mathematics Higher Tier

1. Mathematics Higher Tier Handling data GCSE Revision

2. Higher Tier – Handling Data revision Contents : Questionnaires Sampling Scatter diagrams Pie charts Frequency polygons Histograms Averages Moving averages Mean from frequency table Estimating the mean Cumulative frequency curves Box and whisker plots Theoretical probability Experimental probability Probability tree diagrams

3. Be careful when deciding what questions to ask in a survey or questionnaire Questionnaires What is your age? Don’t be personal Burning fossil fuels is dangerous for the earth’s future, don’t you agree? Don’t be leading Do you buy lemonade when you are at Tescos? Don’t reduce the number of people who can answer the question Do you never eat non-polysaturate margarines or not? Yes or no? Don’t be complicated Here is an alternative set of well constructed questions. They require yes/no or tick-box answers. How old are you? 0-20, 21-30, 31-40, 41-50, 0ver 50 Do you agree with burning fossil fuels? Do you like lemonade? Which margarine do you eat? Flora, Stork, Other brand, Don’t eat margarine This last question is very good since all of the possible answers are covered. Always design your questionnaire to get the data you want.

4. The larger the sample the better When it is impossible to ask a whole population to take part in a survey or a questionnaire, you have to sample a smaller part of the population. Sampling Therefore the sample has to be representative of the population and not be biased. RANDOM SAMPLING Here every member of a population has an equal chance of being chosen: Names out of a bag, random numbers on a calculator, etc. STRATIFIED SAMPLING Here the population is firstly divided into categories and the number of people in each category is found out. The sample is then made up of these categories in the same proportions as they are in the population using % or a scaling down factor. The required numbers in each category are then selected randomly. The whole population = 215 Lets say the sample is 80 so we divide each amount by 215/80 =2.6875

5. Here are 4 scatter diagrams and some questions that may be asked about them C A B D G E F H Scatter diagrams Strong positive correlation Weak positive correlation No correlation Strong negative correlation As B increases so does A As D increases so does C No link between variables No relationship between E and F As H increases G decreases Describe what each diagram shows Describe the type of correlation in each diagram Give examples of what variables A  H could be Draw a line to best show the link between the two variables A = No. of ice creams sold , B = Temperature C = No. of cans of coke sold , D = Temperature E = No. of crisps sold , F = Temperature G = No. of cups of coffee sold , H = Temperature

6. 350 160 1220 210 1660 Pie charts Draw a pie chart for the following information Step 1 Find total Step 2 Divide 360 by total to find multiplier Step 3 Multiply up all values to make angles 900 360  900 = 0.4 3600 Step 4 Check they add up to 3600 and draw the Pie Chart C5 C4 BBC 1 ITV Pie Chart to show the favourite TV channels at Saint Aidan’s BBC 2

7. Frequency polygons can be used to represent grouped and ungrouped data 32 24 16 8 0 0 10 20 30 40 50 Freq. Weekly tips over the year 32 Boy 2 24 16 Boy 1 8 0 £ 0 10 20 30 40 50 Frequency polygons Frequency Step 1 Draw bar chart Step 2 Place co-ordinates at top of each bar X X X Step 3 Join up these co-ordinates with straight lines to form the frequency polgon X X £ You may be asked to compare 2 frequency polygons Which boy has been tipped most over the year ? Explain your answer.

8. 5 4 FD 3 2 1 0 10 20 30 40 50 60 70 Length (cm) Histograms Histograms • A histogram looks similar to a bar chart but there are 4 differences: • No gaps between the bars and bars can be different widths. • x-axis has continuous data (time, weight, length etc.). • The area of each bar represents the frequency. • The y-axis is always labelled “Frequency density” where • Frequency density = Frequency/width of class interval Example 1 : Draw a histogram for this data 2 more rows need to be added

9. Sometimes the upper and lower bounds of each class interval are not as obvious: 15 12 FD 9 6 3 12 13 14 15 16 17 18 19 20 21 Time (min) Example 2 : Draw a histogram for this data Example 3: Draw your own histogram for this data

10. “It’s mean coz U av 2 work it out” Mean “Mode is the Most common number” Mode “The difference between the highest and lowest values” “Median is the Middle value after they have been put in order of size” Median Range Mean = Total No. of items Averages M , M , M , R Calculate the mean, median, mode and range for these sets of data 4 , 1 , 1 1 , 2 , 3 , 4 , 5 4 , 9 , 2 3 , 5 , 6 , 6 10 , 6 , 5 6 , 10 , 8 , 6 8 , 8 , 8 , 4 1 , 2 , 2 , 3 1 , 4 , 4 7 , 7 , 6 , 4

11. Moving averages are calculated and plotted to show the underlying trend. They smooth out the peaks and troughs. Freq. Weekly sales 48 x 40 x x x 32 x x x x x x 24 x x x x x 16 8 0 0 2 4 6 8 10 week Moving averages Calculate the 4 week moving average for these weekly umbrella sales and plot it on the graph below 1st average = (34+45+26+32)/4 = 34.25 plotted at mid-point 2.5 2nd average = (45+26+32+17)/4 = 30 plotted at mid-point 3.5 etc. Last average = (28+18+26+20)/4 = 23 plotted at mid-point 7.5 Explain what the moving average graph shows Estimate the next week’s sales having first predicted the next 4 week average

12. Mean from frequency table 50 pupils were asked how many coins they had in their pockets - Here are the results x x x x x x x = 50 Total no. = 0 + 9 + 20 +39 +32+15+ 0 = 115 of coins = = = = = = = Median at 25/26 pupil (50 in total) 000000011111111122222222223333333333333.. Median = 2 coins 25th 26th Calculate the mean no. of coins per pupil Mean = Total coins = 115 No. of pupils 50 = 2.3 coins per pupil Calculate the median, mode and range Mode (from table) = 3 coins Range = 5 - 0 = 5 Now work out the Mean , Median , Mode , Range for this set of pupils 2.17 2 2 4

13. Mid Points Totals x 4.5 9 = x = 159.5 14.5 x 24.5 343 = x = 34.5 897 x 756.5 44.5 = 2165 70 Estimating the mean In the Barnsley Education Authority the number of teachers in each school were counted. Here are the results: Step 1 Find mid-points Calculate an estimate of the mean number of teachers per school Step 2 Estimate totals and overall number of teachers Step 3 Divide overall total by no. of schools Now work out an estimate of the mean no. of teachers per school here: Est. mean = Est. no. of teachers No. of schools = 2165 = 30.9 70 = 31 teachers per school 14.17 teachers per school

14. The cumulative frequency is found by adding up as you go along (a running total) Cumulative frequency curves The number of houses in each village in Essex were counted Cumulative freq. Step 1 Work out cumulative frequencies 7 31 Step 2 Write down the co-ordinates you are going to plot 60 78 90 Step 3 Draw the cumulative frequency curve Co-ordinates: (50, 0) , (100, 7) , (150, 31) , (200, 60) , (250, 78) , (300, 90) The graph will need the Cumulative Frequency on the y-axis 0  90 and No. of houses on the x-axis 0  300 All points must be joined using a smooth curve

15. c.f. 100 100th percentile 90 80 UQ 70 60 50 Median 40 30 LQ 20 10 0 0 50 100 150 200 250 300 No. of houses 175 215 140 Cumulative frequency curves • From your curve calculate the : • Median • Lower quartile • Upper quartile • Inter quartile range • No. of villages with • more than 260 • houses in Answers: Median = 175 houses LQ = 140 houses UQ = 215 houses IQR = 215 – 140 = 75 houses >260 hs = 9 villages

16. Another way of showing the readings from a cumulative frequency curve is drawing a box and whisker plot (or box plot for short) 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 Box and whisker plots Box and whisker plots Box plots are good for comparing 2 sets of data Work out how this box and whisker plot has been drawn for yourself Explain which part is the box and which parts are the whiskers Comment upon 2 differences between the 2 box plots

17. Theoretical probability To calculate a probability write a fraction of: NO. OF EVENTS YOU WANT TOTAL NO. OF POSSIBLE EVENTS Here some counters are placed in a bag and one is picked out at random. Find these probabilities: P(number 6) = P(orange) = 4 P(number 1) = 1 P(not number 1) = P(number from 1 to 4) = 1 4 1 P(purple) = 2 P(counter) = 2 1 P(number 3 or 4) = 3 3 P(white or number 4) = P(yellow or number 1) =

18. Of course in real life probabilities do not follow the theory of the last slide. The probability calculated from an experiment is called the RELATIVE FREQUENCY Experimental probability If the result of tossing a coin 100 times was 53 heads and 47 tails, the relative frequency of heads would be 53/100 or 0.53 A dice is thrown 60 times. Here are the results. • What is the relative frequency (as a decimal)of shaking a 4 ? • What, in theory, is the probability of shaking a 4 ? (as a decimal) • Is the dice biased ? • Explain your answer. • How can the experiment be improved ? 16/60 = 0.266 1/6 = 0.166 No Only thrown 60 times Throw 600 times

19. Spin 2 Spin 1 Probability tree diagrams A five sided spinner has 2 blue and 3 red outcomes. It is spun twice ! Find the probability of getting two different colours P(bb)  2/5 x 2/5 = 4/25 P(blue) 2/5 P(blue) = 2/5 P(br)  2/5 x 3/5 = 6/25 P(red) = 3/5 In this example the probabilities are not affected after each spin 6/25 + 6/25 = 12/25 P(rb)  3/5 x 2/5 = 6/25 P(blue) = 2/5 P(red) = 3/5 P(red) = 3/5 P(rr)  3/5 x 3/5 = 9/25

20. A sweet jar holds 5 blue sweets and 4 red sweets. 2 sweets are picked at random ! Pick 2 Pick 1 Probability tree diagrams Find the probability of getting two sweets the same colour P(bb)  5/9 x 4/8 = 20/72 P(blue) = 4/8 20/72 + 12/72 = 32/72 In this example the probabilities are affected after each sweet is picked P(blue) = 5/9 P(br)  5/9 x 4/8 = 20/72 P(red) = 4/8 P(rb)  4/9 x 5/8 = 20/72 P(blue) = 5/8 P(red) = 4/9 P(red) = 3/8 P(rr)  4/9 x 3/8 = 12/72