Mathematics Higher Tier Handling data GCSE Revision

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

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.

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 Age Male Female 0-29 34 26 30-59 46 54 60- 19 36 Age Male Female 0-29 13 10 30-59 17 20 60- 7

Here are 4 scatter diagrams and some questions that may be asked about them F H G 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

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 160 C5 C4 350 BBC 1 1660 1220 ITV Pie Chart to show the favourite TV channels at Saint Aidan’s 210 BBC 2

Frequency polygons can be used to represent grouped and ungrouped data Step 1 Draw bar chart 10 20 30 40 50 8 16 24 32 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 10 20 30 40 50 8 16 24 32 Freq. £ Weekly tips over the year Boy 2 Which boy has been tipped most over the year ? Explain your answer. Boy 1

Histograms Histograms 10 20 40 30 50 60 70 Length (cm) FD 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 Length (cm) 0<L<10 10<L<40 40<L<60 60<L<65 Frequency 45 120 70 15 Class width 10 30 20 5 Freq. Den. 4.5 4 3.5 3 2 more rows need to be added 10 20 40 30 50 60 70 1 2 3 4 5 Length (cm) FD

Sometimes the upper and lower bounds of each class interval are not as obvious: Example 2 : Draw a histogram for this data Time (T, nearest minute) 13 -14 15 16 17 - 20 21 Frequency 26 17 20 3 Lower bound 12.5 14.5 15.5 16.5 20.5 Upper bound 21.5 Class width 2 1 4 Freq. Den. 13 15 17 5 3 12 13 14 16 15 17 18 19 3 6 9 Time (min) FD 20 21 Example 3: Draw your own histogram for this data Weight (W, nearest Kg) 24 -27 28 29 30 - 35 36 Frequency 40 13 14 120 18

M , M , M , R Averages Mean Mode Range Median “It’s mean coz U av 2 work it out” Mean = Total No. of items Mean “Mode is the Most common number” Mode “The difference between the highest and lowest values” Range “Median is the Middle value after they have been put in order of size” Median 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

1st average = (34+45+26+32)/4 = 34.25 plotted at mid-point 2.5 Moving averages are calculated and plotted to show the underlying trend. They smooth out the peaks and troughs. 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 Weekly sales 2 4 6 8 10 16 24 32 Freq. week 40 48 x Explain what the moving average graph shows Estimate the next week’s sales having first predicted the next 4 week average x

Mean from frequency table x = 50 = 50 pupils were asked how many coins they had in their pockets - Here are the results Calculate the mean no. of coins per pupil x = = 50 Total no. = 0 + 9 + 20 +39 +32+15+ 0 = 115 of coins 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 25th 26th 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

Estimating the mean x = In the Barnsley Education Authority the number of teachers in each school were counted. Here are the results: x = Totals 9 159.5 343 897 756.5 2165 Mid Points 4.5 14.5 24.5 34.5 44.5 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 70 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

The number of houses in each village in Essex were counted 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 No. of houses No. of villages 50 < P < 100 7 100 < P < 150 24 150 < P < 200 29 200 < P < 250 18 250 < P < 300 12 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

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 90 80 70 60 50 40 20 30 10 100 150 200 250 300 c.f. No. of houses 100th percentile UQ Median LQ Answers: Median = 175 houses LQ = 140 houses UQ = 215 houses IQR = 215 – 140 = 75 houses >260 hs = 9 villages 140 175 215

Box and whisker plots 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 and whisker plots Box and whisker plots Box plots are good for comparing 2 sets of data Sex Lowest age Lower quartile Median Upper Highest Male 7 15 36 48 65 Female 9 24 32 54 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 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 Comment upon 2 differences between the 2 box plots

4 1 1 4 1 2 2 1 3 3 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) =

Experimental probability 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. No. on dice 1 2 3 4 5 6 No. of times 7 15 16 10 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

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 Spin 2 P(bb)  2/5 x 2/5 = 4/25 P(blue) 2/5 Spin 1 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(blue) = 2/5 P(rb)  3/5 x 2/5 = 6/25 P(red) = 3/5 P(red) = 3/5 P(rr)  3/5 x 3/5 = 9/25

Probability tree diagrams A sweet jar holds 5 blue sweets and 4 red sweets. 2 sweets are picked at random ! Probability tree diagrams Find the probability of getting two sweets the same colour Pick 2 P(bb)  5/9 x 4/8 = 20/72 P(blue) = 4/8 Pick 1 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