Product moment correlation Starter:. Product moment correlation.

Product moment correlationStarter:

Product moment correlationLearning objectives:Understand the purpose of a scatter graph,

the type of data it is used to represent and be able to describe what it shows using both mathematical and context-based vocabulary

Know what the product moment correlation coefficient, , represents and know how to calculate it from raw data

Appreciate the limitation of when interpreting data

Scatter graphsScatter Graphs enable us to examine the relationship between two variables, x and y.

Scatter graphs are used with ‘bivariate data’ – this is data where we have two variables connected to one individual/object, hence ‘paired’ data.Average Temperature (oC) 13 16 18 21 14 25 11 24 15 27 20 19

Rainfall (mm) 40 36 43 25 44 28 50 17 39 7 28 20

Average Temperature vs Rainfall

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Average Temperature (oC)

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a) What kind of graph is this?

b) Which point has been incorrectly plotted?

c) Why do we draw such a graph?

d) Using Mathematical vocabulary, explain what the graph shows you.

e) Relate your answer to part (d) to the context of the situation.

Product moment correlationLearning objectives:Understand the purpose of a scatter graph,

the type of data it is used to represent and be able to describe what it shows using both mathematical and context-based vocabulary

Know what the product moment correlation coefficient, , represents and know how to calculate it from raw data

Appreciate the limitation of when interpreting data

What does correlation mean?Correlation means there is a linear relationship between two variables – i.e. we can draw a line of best fit.

As height increases, weight increases.

What does this scatter graph show about the relationship between the height and weight of twenty Year 10 boys?

This is called a positive correlation.This is called a positive correlation.

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What does this scatter graph show?

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It shows that life expectancy decreases as the number of cigarettes smoked increases.

This is called a negative correlation.This is called a negative correlation.

What does correlation mean?Correlation means there is a linear relationship between two variables – i.e. we can draw a line of best fit.

What types of correlation exist?

Positive correlation: as one variable increases, so does the other variable

Negative correlation: as one variable increases, the other variable decreases

Zero correlation: no linear relationship between the variables

Average Temperature vs Rainfall

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Comment on the two

examples of negative

correlation shown here.

What does correlation mean?Correlation means there is a linear relationship between two variables – i.e. we can draw a line of best fit.

What types of correlation exist?

Positive correlation: as one variable increases, so does the other variable

Negative correlation: as one variable increases, the other variable decreases

Zero correlation: no linear relationship between the variables

Correlation can be strong or weak

Correlation: issue to considerWhat kind of correlation is there?

How strong is the correlation?

Product moment correlationLearning objectives:Understand the purpose of a scatter graph,

the type of data it is used to represent and be able to describe what it shows using both mathematical and context-based vocabulary

Know what the product moment correlation coefficient, , represents and know how to calculate it from raw data

Appreciate the limitation of when interpreting data

Product moment correlation coefficient

This is a way to measure the strength of the correlation numerically.It is denoted by

perfect positive correlationzero correlation

perfect negative correlation

Product moment correlation coefficient

Product moment correlation is calculated using the following formula…

Where:

Using your calculatorIn the stat menu there is a very useful mode called ‘reg’.

It can be used to calculate values and for calculating the equation of the least squares regression line.

It can also calculate for us too!

TaskExercise A – Page 141Questions 1 & 3

Limits of correlation: non-linear relationships

Here, measures linear relationships only!It is no use for analysing non-linear relationships.0

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Note that clear non-linear relationships identified on scatter diagrams should always be commented

upon but you should also note that the evaluation of r is not appropriate.

Limits of correlation: cause and effect

Here,

Does this mean stretching a child’s foot will make them better at maths?14

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The correlation found between foot length and score in maths is often called SPURIOUS and should be

treated with caution.

Any suggestion that correlation may indicate cause and effect in the relationship between two variables

should be considered very carefully!!!

Limits of correlation: ‘freak’ results

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𝒓=𝟎 𝒓=𝟎 .𝟕𝟏

An unusual result can drastically alter the value of r. Unexpected results (outliers) should be commented

on and it may be best to exclude them from the analysis.

TaskExercise C – Page 144All questions