Mathematics

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Mathematics Higher Tier
Handling data
GCSE Revision
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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 Experim
ental probability Probability tree diagrams
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Be careful when deciding what questions to ask in
a survey or questionnaire
Questionnaires
What is your age?
Dont be personal
Burning fossil fuels is dangerous for the earths
future, dont you agree?
Dont be leading
Do you buy lemonade when you are at Tescos?
Dont reduce the number of people who can answer
the question
Do you never eat non-polysaturate margarines or
not? Yes or no?
Dont 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, Dont 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.
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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.
The larger the sample the better
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
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Here are 4 scatter diagrams and some questions
that may be asked about them
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 relationship between E and F
As H increases G decreases
No link between variables
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
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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 Aidans
BBC 2
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Frequency polygons can be used to represent
grouped and ungrouped data
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.
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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
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Sometimes the upper and lower bounds of each
class interval are not as obvious
Example 2 Draw a histogram for this data
Example 3 Draw your own histogram for this data
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Averages
M , M , M , R
Calculate the mean, median, mode and range for
these sets of data
1 , 2 , 3 , 4 , 5
3 , 5 , 6 , 6
4 , 1 , 1
4 , 9 , 2
10 , 6 , 5
1 , 2 , 2 , 3
1 , 4 , 4
8 , 8 , 8 , 4
6 , 10 , 8 , 6
7 , 7 , 6 , 4
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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 (34452632)/4 34.25 plotted at
mid-point 2.5
2nd average (45263217)/4 30 plotted at
mid-point 3.5 etc.
Last average (28182620)/4 23 plotted at
mid-point 7.5
Explain what the moving average graph shows
Estimate the next weeks sales having first
predicted the next 4 week average
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Mean from frequency table
50 pupils were asked how many coins they had in
their pockets - Here are the results
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
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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
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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
60
Step 2 Write down the co-ordinates you are
going to plot
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
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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 gt260 hs 9 villages
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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
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
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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)
1
2
P(number 3 or 4)
3
3
P(white or number 4)
P(yellow or number 1)
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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
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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(br) ? 2/5 x 3/5 6/25
P(blue) 2/5
P(red) 3/5
6/25 6/25 12/25
In this example the probabilities are not
affected after each spin
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
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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
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