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5.4 – The product moment correlation coefficient

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FORMULA

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PMCC varies between -1 and 1.

 

means

Perfect positive correlation.

 

means

No correlation

 

means

Perfect negative correlation.

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PMCC varies between -1 and 1.

 

means

Perfect positive correlation.

 

means

No correlation

 

means

Perfect negative correlation.

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Perfect negative

Strong negative

No correlation

Weak positive

Perfect positive

Maths score

English score

Strong correlation

Strong correlation

 

 

 

 

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Interpreting the PMCC

 

 

 

 

 

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Interpreting the PMCC

“Interpret” vs “State”

In general in Statistics exams, the word ‘interpret’ means “explain in context using non-statistical language”.

 

A bad answer (that may or may not be accepted):

“Strong negative correlation” (this is stating the correlation not interpreting it)

A good answer:

“As the waiting time increases, the customer satisfaction tends to decrease”.

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Interpreting the PMCC

“Interpret” vs “State”

In general in Statistics exams, the word ‘interpret’ means “explain in context using non-statistical language”.

 

A bad answer (that may or may not be accepted):

“Strong negative correlation” (this is stating the correlation not interpreting it)

A good answer:

“As the waiting time increases, the customer satisfaction tends to decrease”.

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Item age (years) t

85

256

142

120

Value (£) P

350

1200

425

300

 

Example:

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Item age (years) t

85

256

142

120

Value (£) P

350

1200

425

300

 

Example:

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Time

Population of Tanzania

The data points for population over time doesn’t fit a straight line very well, so there is not a strong correlation between time and population.

Comment on this claim.

Exponential model

Linear model

Important Point 1

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Time

Population of South Korea

The data shows that as time increases, population increases.

Comment on this claim.

Correlation does not imply causation.

As time increases, the population tends to increase, so there is a positive correlation before time and population.

​

But there are examples, e.g. the highlighted data points above, where this is not true: time increased but the population decreased.

We say there is not a causal relationship between time and population, meaning that a (positive) change in one quantity does not directly cause a (positive) change in the other.

Important Point 2

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Time

Population of South Korea

There is strong positive correlation.

Comment on the student’s answer.

The question asks to interpret the relationship, not to state the type of correlation.

​

Interpret means to give a worded description in the context of the problem.

​

Her answer could have been:

“As the time increases, the population tends to increase.”

Question: Interpret the relationship between time and population.

Important Point 3

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EFFECT OF CODING ON PMCC

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Does the equation of the regression line change?

Does the PMCC change?

Yes. The regression translates right so will have a different equation.

No. As the data points and regression line moved together, the extent to which the data fits the line doesn’t change.

Effect of Coding on PMCC

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Effect of Coding on PMCC

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Effect of Coding on PMCC

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The Product Moment Correlation Coefficient (PMCC), which is a measure of the strength of linear relationship between two statistical variables.

 

 

 

Perfect negative

Strong negative

No correlation

Weak positive

Perfect positive

Maths score

English score

Time

Population

Recall that PMCC only measures the strength of linear correlation, i.e. how well the data fits to a straight line. For example, population growth tends to be exponential not linear, so the PMCC may be low even though population and time are strongly (exponentially) correlated.

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Item age (years) t

85

256

142

120

Value (£) P

350

1200

425

300

 

 

We will use our calculator’s statistics mode to calculate this directly.

Calculating PMCC using a Calculator

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Let’s do it on our calculators!

Baby

A

B

C

D

E

F

Head Circumference (x)

31.1

33.3

30.0

31.5

35.0

30.2

Gestation Period (y)

36

37

38

38

40

40

 

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Calculating PMCC using a Calculator

These are instructions for the Casio fx-CG50

Enter the values, pressing EXE after each and using the arrow keys to move around.

Choose 2 for Statistics.

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2

 

3

 

4

Item age (years) t

85

256

142

120

Value (£) P

350

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425

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Calculating PMCC using a Calculator

These are instructions for the Casio fx-570/991CW

Choose 2-Variable.

Use the arrows and OK to select Statistics.

1

2

Enter the values, pressing EXE after each value. Use the arrow keys to move around.

3

 

4

Item age (years) t

85

256

142

120

Value (£) P

350

1200

425

300

 

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IQ (i)

105

108

115

122

147

165

Chess rating (c)

1450

1610

1820

1570

1480

2700

 

 

English score

78

92

89

62

36

Maths score

67

100

53

72

54

 

1

2

a

b

a

b

Your Turn

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IQ (i)

105

108

115

122

147

165

Chess rating (c)

1450

1610

1820

1570

1480

2700

 

 

English score

78

92

89

62

36

Maths score

67

100

53

72

54

 

1

2

a

b

a

b

Your Turn

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Recall that variance, which is a measure of the ‘variability’ of data, is defined to be the average squared distance from the mean.

 

 

Summing these and dividing by the number of data points then averages these squared distances.

 

 

Spec note: In the UK, this skill is only in Edexcel Further Maths FS2, Edexcel IAL, OCR FM Statistics and CCEA (Northern Ireland).

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This can be simplified into a more easily calculatable form, in a similar way to the variance formula being simplified.

PMCC can then be calculated as follows:

 

 

 

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?

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Quickfire!

?

?

?

?

?

?

?

?

Quite often the values are given to you in an exam.

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Quickfire!

Quite often the values are given to you in an exam.

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CODING BIVARIATE DATA

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EXTRA THINGS USED TO SOLVE QUESTIONS : A MUST KNOW

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EXTRA THINGS USED TO SOLVE QUESTIONS : A MUST KNOW