Higher Tier – Handling Data revision

Contents : QuestionnairesSamplingScatter diagramsPie chartsFrequency polygonsHistogramsAveragesMoving averagesMean from frequency tableEstimating the meanCumulative frequency curvesBox and whisker plots Theoretical probabilityExperimental probability Probability tree diagrams

QuestionnairesBe 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.

SamplingWhen 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 SAMPLINGHere every member of a population has an equal chance of being chosen:Names out of a bag, random numbers on a calculator, etc.

STRATIFIED SAMPLINGHere the population is firstly divided into categories and the number of people in each category is found out.

Age Male Female

0-29 34 26

30-59 46 54

60- 19 36

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 Age Male Female

0-29 13 10

30-59 17 20

60- 7 13

Lets say the sample is 80 so we divide eachamount by 215/80

=2.6875

The required numbers in each category are then selected randomly.

Scatter diagrams

E

F

C

D

A

B H

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

Strongpositive

correlation

Weakpositive

correlation

Nocorrelation

Strongnegative

correlation

Describe what each diagram showsDescribe 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

Pie charts

Favourite Channel

No. of Pupils

BBC 1 415

BBC 2 54

ITV 304 C4 87

C5 40

Multiplier

Angle

x 0.4 = 1660

x 0.4 = 210

x 0.4 = 1220 x 0.4 = 350

x 0.4 = 160

Draw a pie chart for the following information

Step 1Find total

Step 2Divide 360 by total to find multiplier

Step 3Multiply up all values to make angles

900 360 900 = 0.4 3600

Step 4Check they add up to 3600 and draw the Pie Chart

1660

210

1220

350

160

BBC 1

BBC 2

ITV

C4C5

Pie Chart to show the favourite TV channels at Saint Aidan’s

Frequency polygonsFrequency polygons can be used to

represent grouped and ungrouped data

Step 1Draw bar chart

Step 2Place co-ordinates at top of each bar

Step 3Join up these co-ordinates with straight lines to form the frequency polgon0 10 20 30 40 50

0

8

16

24

32

Frequency

£

X

XX

XX

You may be asked to compare 2 frequency polygons

Boy 1

Boy 2

0 10 20 30 40 500

8

162432

Freq.

£

Weekly tips over the yearWhich boy has been tipped most over the year ? Explain your answer.

HistogramsHistograms

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<10

10<L<40

40<L<60

60<L<65

Frequency 45 120 70 15

Example 1 : Draw a histogram for this data

2 more rows need to be added

Class width 10 30 20 5

Freq. Den. 4.5 4 3.5 3

0 10 20 4030 50 60 70

12345

Length (cm)

FD

Sometimes the upper and lower bounds of each class interval are not as obvious:

Time (T, nearest minute) 13 -14 15 16 17 - 20 21

Frequency 26 15 17 20 3

Example 2 : Draw a histogram for this data

Lower bound 12.5 14.5 15.5 16.5 20.5

Upper bound 14.5 15.5 16.5 20.5 21.5

Class width 2 1 1 4 1

Freq. Den. 13 15 17 5 3

12 13 14 1615 17 18 19

369

1215

Time (min)

FD

20 21

Weight (W, nearest Kg) 24 -27 28 29 30 - 35 36

Frequency 40 13 14 120 18

Example 3: Draw your own histogram for this data

Averages

1 , 2 , 3 , 4 , 5

1 , 2 , 2 , 3 1 , 4 , 4

3 , 5 , 6 , 6

8 , 8 , 8 , 46 , 10 , 8 , 6

4 , 1 , 1 4 , 9 , 2 10 , 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

Moving averagesMoving 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

Week 1 2 3 4 5 6 7 8 9

No. of umbrellas

34 45 26 32 17 28 18 26 20

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

x

x xx

xx

Weekly sales

0 2 4 6 8 100

162432

Freq.

week

4048

8

x

x

xx

xx

x x

xExplain what the moving average graph shows Estimate the next week’s sales having first predicted the next 4 week average

Mean from frequency table

50 pupils were asked how many coins they had in their pockets - Here are the results

No. COINS 0 1 2 3 4 5 6

No. PUPILS 7 9 10 13 8 3 0xx x xx xx

= = = == == = 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 pupilCalculate the median,

mode and range Median at 25/26 pupil (50 in total)000000011111111122222222223333333333333..

Median = 2 coins 25th 26th

Mode (from table) = 3 coinsRange = 5 - 0 = 5

Now work out the Mean , Median , Mode , Range for this set of pupils

No. COINS 0 1 2 3 4

No. PUPILS 2 5 10 8 5

2.17 2 2 4

Estimating the mean

No. ofTeachers

No. ofSchools

0 < T < 9 2

10 < T < 19 11

20 < T < 29 1430 < T < 39 26

40 < T < 49 17

No. ofTeachers

No. ofSchools

0 < T < 4 1

5 < T < 9 4

10 < T < 14 915 < T < 19 13

20 < T < 24 3

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 1Find mid-points

MidPoints

4.5

14.524.5

34.544.5

Step 2Estimate totals and overall number of teachers

Step 3Divide overall total by no. of schools

xx

xx

x

=====

Totals

9

159.5343

897756.5

216570

Now work out an estimate of the mean no. of teachers per school

here:

14.17 teachers per school

Est. mean = Est. no. of teachersNo. of schools

= 2165 = 30.9 70

= 31 teachers per school

Cumulative frequency curves

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

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 1Work out cumulative frequencies

Step 2Write down the co-ordinates you are going to plot

Step 3Draw 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 90and No. of houses on the x-axis 0 300All points must be joined using a smooth curve

Cumulative frequency curves

90

80

70

60

50

40

20

30

100

0 50 100 150 200 250 300

100c.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

100th percentile

Median

LQ

UQ

140 175 215

Answers:Median = 175 housesLQ = 140 housesUQ = 215 housesIQR = 215 – 140 = 75 houses>260 hs = 9 villages

Box and whisker plotsBox and whisker plotsAnother 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

Sex Lowest

age

Lower

quartile

Median

age

Upper

quartile

Highest

age

Male 7 15 36 48 65

Female 9 15 24 32 54

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

Theoretical probability

1 23

4

11

23

14

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 WANTTOTAL NO. OF POSSIBLE EVENTS

Here some counters are placed in a bag and one is picked out at random. Find these probabilities:

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 dice 1 2 3 4 5 6

No. of times 7 6 15 16 6 10

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

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(blu

e) =

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

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(blu

e) =

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