Level 3 Award in Mathematics for numeracy

Teaching:

Session 5: Handling data

Gail Lydon & Jo Byrne

Log into the online page for data handling

• https://ccpathways.co.uk/level-3-maths-online/

• You'll need the password L3Maths16

• Click on the Data Handling button

• Have you had a chance to look at HO1 Data Handling?

If not have a look now and note on your ILP anything you

need to look at after the session.

Session aim

• To review & extend participants’

personal mathematics relating to

statistics and probability

• To apply concepts of statistics and

probability to solve problems

What do you need to know about statistics?

What do you need to know about statistics?

• Mean, median and mode

• Sampling techniques

• Data transformation (measures of

dispersion, curve fitting, spread, upper and

lower quartiles, interquartile range,

standard deviation)

• Statistical diagrams

• Regression

• Correlations

• Probability

Flipped activity review

• HO1

Frequency distributions

What’s the difference?

Measures of dispersion

• Range

• Interquartile range

• Variance

• Standard deviation

Box & whisker plots

Group A

Group B

Variance and standard deviation

Normal distribution

Standard deviation

Standard deviation is used to assess if a data point is

OR

Standard and

expected (ie within expected variation)

Unexpected

and unusual (significantly above or

below the average)

Standard deviation is represented by lower case

sigma

𝝈

68 – 95 – 99.7 rule

Σ

Upper case sigma

• What does Σ mean?

• It is the summation operator (so have a go at

calculating the following)

Upper case sigma

Standard deviation

Use the population

standard deviation (σ)

when data is relates to

the entire population:

The standard deviation

of a sample of the

population (s) can be

estimated using:

Note:

• xi represents the individual values of x;

• is the mean of all the values of x

(sometimes μ is used)

• N is the number of values

Let’s have a go at a standard

deviation problem

• Have a look at HO2 (under R5)

Grouped and cumulative

frequency distributions

Frequency diagram

Grouped frequency distribution

450

400

350

300

250

200

150

100

50

0

Score 0-10 11-20 21-30 31-40 41-50 51-60 61-70

Frequen

cy

10 60 230 420 250 28 2

Cumulative frequency table

Score 0-10 11-20 21-30 31-40 41-50 51-60 61-70

Frequency 10 60 230 420 250 28 2

Score 10 20 30 40 50 60 70

Cumulative

frequency

10 70 300 720 1000 998 970

Cumulative frequency graph

Score 10 20 30 40 50 60 70

Cumulative

frequency

10 70 300 720 970 998 1000

Box & whisker plot

Lower quartile

Upper quartile

Median

What’s misleading?

45

40

35

30

25

20

15

10

5

0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Score

Fre

qu

en

cy

Score 0-10 11-20 21-30 31-50 51-100 101-150

Frequency 20 38 42 18 6 1

Frequencies proportional to area of bars

45

40

35

30

25

20

15

10

5

0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Score

Fre

qu

en

cy

Score 0-10 11-20 21-30 31-50 51-100 101-150

Frequency 20 38 42 18 6 1

Bivariate analysis

Correlation

Negative correlation

Probability

Probability

P(a) = No. successful outcomes

Total possible outcomes

Combined probabilities

When two coins are tossed there are

three possible outcomes:

• Two heads

• One head

• No heads

The probability of two heads is

therefore ⅓

Train punctuality data

On-time 84%

Up to 15 min late 7%

15-30 min. late 4%

More than 30 min. late 2%

Cancelled 3%

Scheduled services running in the last month:

Probability & frequency data

450

400

350

300

250

200

150

100

50

0 0 10 20 30 40 50 60 70 80 90 100

Score

Fre

qu

en

cy

Score 0-10 11-20 21-30 31-40 41-50 51-60 61-70

Frequency 10 60 230 420 250 28 2

Pearson’s correlation coefficient

The Pearson product-moment correlation

coefficient (or Pearson correlation coefficient, for

short)

is a measure of the strength of a linear association (the

relationship) between two variables and is denoted by r.

Basically, a Pearson product-moment correlation

attempts to draw a line of best fit through the data of two

variables, and the Pearson correlation coefficient, r,

indicates how far away all these data points are to this

line of best fit (i.e., how well the data points fit this new

model/line of best fit).

Pearson correlation coefficient – working it out

There are a number of variations of Pearson’s.

Pearson correlation coefficient cont

Pearson’s r is always between -1 and 1

Pearson correlation coefficient cont

Example 1 – a perfect positive relationship between x and

y.

As x increases y increases exactly the same.

Pearson correlation coefficient cont

As x increases y decreases – perfect negative relationship

Pearson correlation coefficient cont

So what r=0 look like?

Pearson correlation coefficient cont

So what r=0 look like?

• Let’s look back at the equation we saw earlier

• What does Σ mean? Remember?

• It is the summation operator and you calculated

Pearson correlation coefficient an example

• What does √ mean?

• So this isn’t a complicated equation – its just a long

winded one – so you need to write down all of your

steps.

Pearson correlation coefficient an example

Pearson correlation coefficient an example

Subject Age (x) Yearly income

(y)

1 18 15,000

2 25 29,000

3 57 26,000

4 45 52,000

5 26 32,000

6 64 80,000

7 37 41,000

8 40 45,000

9 24 26,000

10 33 33,000

In this case what does

n=?

n=10 so

Pearson correlation coefficient

an example

Subject Age (x) Yearly

income

(y)

1 18 324 15,000

2 25 625 29,000

3 57 3249 26,000

4 45 2025 52,000

5 26 676 32,000

6 64 4096 80,000

7 37 1369 41,000

8 40 1600 45,000

9 24 576 26,000

10 33 1089 33,000

3692

=

136,161

15,629

Pearson correlation coefficient

an example

Subject Age (x) Yearly

income (y)

1 18 324 15,000

2 25 625 29,000

3 57 3249 26,000

4 45 2025 52,000

5 26 676 32,000

6 64 4096 80,000

7 37 1369 41,000

8 40 1600 45,000

9 24 576 26,000

10 33 1089 33,000

136,161

15,629

Pearson correlation coefficient

an example

Subject Age (x) Yearly

income (y)

1 18 324 15,000 15,0002

2 25 625 29,000 29,0002

3 57 3249 26,000 26,0002

4 45 2025 52,000 52,0002

5 26 676 32,000 32,0002

6 64 4096 80,000 80,0002

7 37 1369 41,000 41,0002

8 40 1600 45,000 45,0002

9 24 576 26,000 26,0002

10 33 1089 33,000 33,0002

136,161

15,629

421,0002

=177,241,000,000

Pearson correlation coefficient

an example

Pearson

correlation

coefficient

Subject Age

(x)

Yearly

income (y)

(x)(y)

1 18 15,000 270,000

2 25 29,000 725,000

3 57 68,000 3,876,000

4 45 52,000 2,340,000

5 26 32,000 832,000

6 64 80,000 5,120,000

7 37 41,000 1,517,000

8 40 45,000 1,800,000

9 24 26,000 624,000

10 33 33,000 1,089,000

Sum 369 421,000 18,193,000

Pearson correlation coefficient

So what is the 0.99 indicating?

That x (age) and y (yearly income) have a

strong positive relationship.

Pearson correlation coefficient cont

Pearson’s r is always between -1 and 1

https://statistics.laerd.com/statistical-guides/pearson-

correlation-coefficient-statistical-guide.php

http://study.com/academy/lesson/pearson-correlation-

coefficient-formula-example-significance.html

• https://www.youtube.com/watch?v=BXXtkYOqAfM

• https://www.youtube.com/watch?v=SC1kvvoH10Y

• https://www.youtube.com/watch?v=2SCg8Kuh0tE -

this is my favourite

• Estimating from a scatterplot

• https://www.youtube.com/watch?v=372iaWfH-Dg

• https://www.youtube.com/watch?v=2B_UW-RweSE

• This example is age and yearly income but can just

as easily be used to consider whether there is a

correlation between maths and music

Pearson Product Moment Correlation

Coefficient

Questions for you to work on:

• Probability

• Statistics

• Statistical distributions

• Correlation and regression

Probability:

Matthew is an A-level student studying English, History and

French.

He plans to go to University and has received a conditional

offer from the University of Central Lancashire of ‘B C C’

in any combination of subjects

a) What are the chances that Matthew exactly meets the

offer from the University of Central Lancashire?

b) What are the chances that Matthew meets or exceeds

the University of Central Lancashire’s offer?

c) How would it affect your conclusions if your assumptions

are wrong?

Matthew’s homework grades English B C C C D C D B

History B C C B C D C B

French C D C B C B C C

Statistics

In April 2011 a survey of customer transaction values was carried out by a

retail business. The following data was collected

a) i) Plot the histogram and the Cumulative Frequency Curve for this data

ii) Estimate the median transaction value and its Inter-quartile range from

the graphs.

b) For the above data calculate

i) The mean transaction value (stating clearly any assumptions made)

ii) The standard deviation

c) A similar survey was carried out in October 2010. The resulting data gave a

mean value of £27.90 and a standard deviation of £14.30.

Comment on how the spending pattern of customers has changed between the

two months.

Transaction value (£’s) No. of customers Below 10.00 137 10.00 to 19.99 259 20.00 to 29.99 297 30.00 to 39.99 378 40.00 to 49.99 193 50.00 to 59.99 84 60.00 to 69.99 52

Statistical distributions

Applicants for a certain job are given an aptitude test. Past-

experience shows that the scores from the test are

normally distributed with a mean of 60 and standard

deviation of 12 marks.

What percentage of applicants would be expected to pass

the test if the minimum score required was 75?

What would the pass mark need to be if the company

wanted only 4% of applicants to pass?

What would be the percentage failing if the standard

deviation were 20 points?

What practical use could the company make of this

information?

Correlation and regression

The number of customers in different regions and the corresponding

monthly sales of a product are:

1. Plot a scatter diagram of ‘Sales Volume’ against ‘Number of customers’

2. Calculate the regression line and the coefficient of determination r2

3. Check your answers to b) in ONE way by using Excel functions or its Regression

facility

4. Plot the regression line on the same graph as the scatter diagram

What information does the coefficient of determination give you about

the regression line?

Comment on the match between the data-points and the regression

line. Is this consistent with the r2 value? 5. Use the regression line to predict ‘Sales Volume’ for Regions having 1000, 4000 and

7000 customers. In each case comment critically on the reliability of these predictions

Sales Region 1 2 3 4 5 6 7 8 9 10

Number of customers (00’s) 26 22 50 43 48 32 30 34 40 50

Sales volume ((£000s) 146 149 325 252 312 188 195 196 260 298