3. 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

4. Questionnaires Be careful when deciding what questions to ask in a survey or questionnaire What is your age? Burning fossil fuels is dangerous for the earth’s future, don’t you agree? Do you buy lemonade when you are at Tescos? Do you never eat non- polysaturate margarines or not? Yes or no? Don’t be personal Don’t be leading Don’t reduce the number of people who can answer the question 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.

5. Sampling 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. Therefore the sample has to be representative of the population and not be biased. The larger the sample the better 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. AgeMaleFemale 0-293426 30-594654 60-1936 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 whole population = 215 AgeMaleFemale 0-291310 30-591720 60-713 Lets say the sample is 80 so we divide each amount by 215/80 =2.6875 The required numbers in each category are then selected randomly.

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

7. 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 = 0.4 360 0 Step 4 Check they add up to 360 0 and draw the Pie Chart 166 0 21 0 122 0 35 0 16 0 BBC 1 BBC 2 ITV C4 C5 Pie Chart to show the favourite TV channels at Saint Aidan’s

8. Frequency polygons Frequency polygons can be used to represent grouped and ungrouped data Step 1 Draw bar chart Step 2 Place co-ordinates at top of each bar Step 3 Join up these co- ordinates with straight lines to form the frequency polgon 01020304050 0 8 16 24 32 Frequency £ X X X X X You may be asked to compare 2 frequency polygons Boy 1 Boy 2 010203040 50 0 8 16 24 32 Freq. £ Weekly tips over the year Which boy has been tipped most over the year ? Explain your answer.

9. 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 Length (cm)0<L<1010<L<4040<L<6060<L<65 Frequency451207015 Example 1 : Draw a histogram for this data 2 more rows need to be added Class width1030205 Freq. Den.4.543.53 010204030506070 1 2 3 4 5 Length (cm) FD

10. Sometimes the upper and lower bounds of each class interval are not as obvious: Time (T, nearest minute) 13 -14151617 - 2021 Frequency 261517203 Example 2 : Draw a histogram for this data Lower bound 12.514.515.516.520.5 Upper bound 14.515.516.520.521.5 Class width 21141 Freq. Den. 13151753 1213141615171819 3 6 9 12 15 Time (min) FD 2021 Weight (W, nearest Kg) 24 -27282930 - 3536 Frequency 40131412018 Example 3: Draw your own histogram for this data

11. Averages 1, 2, 3, 4, 5 1, 2, 2, 31, 4, 4 3, 5, 6, 6 8, 8, 8, 46, 10, 8, 6 4, 1, 14, 9, 210, 6, 5 7, 7, 6, 4 M, M, M, R “The difference between the highest and lowest values” Range “Mode is the Most common number” Mode “Median is the Middle value after they have been put in order of size” Median “It’s mean coz U av 2 work it out” Mean = Total No. of items Mean Calculate the mean, median, mode and range for these sets of data

12. Moving averages Moving averages are calculated and plotted to show the underlying trend. They smooth out the peaks and troughs. Calculate the 4 week moving average for these weekly umbrella sales and plot it on the graph below 1 st average = (34+45+26+32)/4 = 34.25 plotted at mid-point 2.5 2 nd 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 x x x x x x Weekly sales 02468 10 0 16 24 32 Freq. week 40 48 8 x x x x x x x x x Explain what the moving average graph shows Estimate the next week’s sales having first predicted the next 4 week average

13. Mean from frequency table 50 pupils were asked how many coins they had in their pockets - Here are the results xxxxxxx ======= = 50 Total no. = 0 + 9 + 20 +39 +32+15+ 0 = 115 of coins 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 Median at 25/26 pupil (50 in total) 000000011111111122222222223333333333333.. Median = 2 coins 25 th 26 th Mode (from table) = 3 coins Range = 5 - 0 = 5 Now work out the Mean, Median, Mode, Range for this set of pupils 2.17224

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

15. Cumulative frequency curves No. of housesNo. of villages 50 < P < 1007 100 < P < 15024 150 < P < 20029 200 < P < 25018 250 < P < 30012 The cumulative frequency is found by adding up as you go along (a running total) The number of houses in each village in Essex were counted Cumulative freq. 7 31 60 78 90 Step 1 Work out cumulative frequencies Step 2 Write down the co-ordinates you are going to plot 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

16. Cumulative frequency curves 90 80 70 60 50 40 20 30 10 0 050100150200250300 100 c.f. No. of houses From your curve calculate the : Median Lower quartile Upper quartile Inter quartile range No. of villages with more than 260 houses in 100 th percentile Median LQ UQ 140175215 Answers: Median = 175 houses LQ = 140 houses UQ = 215 houses IQR = 215 – 140 = 75 houses >260 hs = 9 villages

17. Box and whisker plots Another way of showing the readings from a cumulative frequency curve is drawing a box and whisker plot (or box plot for short) Box plots are good for comparing 2 sets of data SexLowest age Lower quartile Median age Upper quartile Highest age Male715364865 Female915243254 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 Work out how this box and whisker plot has been drawn for yourself Comment upon 2 differences between the 2 box plots Explain which part is the box and which parts are the whiskers

18. Theoretical probability 1 2 3 4 1 1 2 3 1 4 P(counter) = P(number 3 or 4) = P(white or number 4) = P(yellow or number 1) = P(orange) = P(number 1) = P(not number 1) = P(purple) = P(number 6) = P(number from 1 to 4) = 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:

19. 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 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 No. on dice123456 No. of times761516610 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 = 0.2661/6 = 0.166NoOnly thrown 60 timesThrow 600 times

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

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