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
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

5. Here are 4 scatter diagrams and some questions that may be asked about themF 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

6. 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

7. 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

8. 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

9. 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 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 36
Frequency 40 13 14
120 18

10. 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

11. 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 = ( )/4 = plotted at mid-point 2.5
2nd average = ( )/4 = 30 plotted at mid-point 3.5 etc.
Last average = ( )/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

12. 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 = = 115
of coins
Mean = Total coins = 115
No. of pupils
= 2.3 coins per pupil
Calculate the median,
mode and range
Median at 25/26 pupil (50 in total)
Median = 2 coins th 26th
Mode (from table) = 3 coins
Range = = 5
Now work out the Mean , Median , Mode , Range for this set of pupils
2.17 2 2 4

13. 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
= = 30.9 70
= 31 teachers per school
14.17 teachers per school

14. 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

15. 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

16. 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
Comment upon 2 differences
between the
2 box plots

17. 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) =

18. 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

19. 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

20. 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 = 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