s1: chapter 6 correlation

S1 :: Chapter 6Correlation

www.drfrostmaths.com Dr J Frost (jfrost@tiffin.kingston.sch.uk)

Last modified: 20th January 2016

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Type of correlation:Weak positive correlation? ?

strength type

Weak negative correlation??

Strong positive correlation? ?

No correlation?

Recap of correlationCorrelation gives the strength of the relationship (and the type of relationship) between two variables.

Recall that variance is defined as “the average squared distance from the mean”. We could therefore express in terms of :

! represents the total squared distance from the mean.

Simplified formula

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Formula based on definition

?Bro Exam Tip: Given in formula booklet, but useful to memorise.

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Covariance

We understand variance as ‘how much a variable varies’.

We can extend variance to two variables.We might be interested in how one variable varies with another.

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We can say that as distance (say ) increases, the cost (say ) increases. Thus the covariance of and is positive.

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(this won’t be tested in an exam but is intended to provide background)

Covariance

Comment on the covariance between the variables.

As increases, doesn’t change very much. So the covariance is small (but positive)

As increases, doesn’t change very much. So the covariance is small (but positive)

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𝑥

𝑦 𝑦

𝑥

(this won’t be tested in an exam but is intended to provide background)

Covariance

Comment on the covariance between the variables.

As varies, doesn’t vary at all. So we say that variables are independent, and the covariance is 0.

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𝑥

𝑦

As increases, decreases. So the covariance is negative.?

𝑥

𝑦

(this won’t be tested in an exam but is intended to provide background)

Simplified formula

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Interesting things to note (but not examined):• Just as , • How could be expressed in terms of covariance?

i.e. variance is the extent to which a variable varies with itself!

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Just as gave a measure of how much a variable varies, gives a measure of how two variables and vary with each other.

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Product Moment Correlation Coefficient (PMCC)We saw that gives a measure of how two variables vary with each other. That sounds like correlation!Wouldn’t it be nice if we could somehow ‘normalise’ it so we end up with just a number between -1 and 1…

𝑟=𝑆𝑥𝑦

√𝑆𝑥𝑥𝑆𝑦𝑦We’ll interpret what that means in a second.

is known as the Product Moment Correlation Coefficient (PMCC).

Have an intelligent guess based on the discussion

above.

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Interpreting the PMCC

We’ve seen the PMCC varies between -1 and 1.

𝑟=1means

Perfect positive correlation.

𝑟=0means

No correlation

𝑟=−1means

Perfect negative correlation.

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Interpreting the PMCC

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𝑟=0.8

𝑟=0

𝑟=−0.4

𝑟=0.96

Match the value to each scatter diagram.

Example

Baby A B C D E FHead Circumference () 31.1 33.3 30.0 31.5 35.0 30.2

Gestation Period () 36 37 38 38 40 40

Σ𝑥=191.1 Σ 𝑦=229Σ𝑥2=6105.39Σ 𝑦2=8753

Σ𝑥𝑦=7296.7

𝑆𝑥𝑥=Σ𝑥2−

(Σ 𝑥 )2

𝑛=18.855

𝑆 𝑦𝑦=Σ 𝑦2−

(Σ 𝑦 )2

𝑛=12.833

𝑆𝑥𝑦=Σ 𝑥𝑦−(Σ 𝑥 ) (Σ 𝑦 )

𝑛=3.05

𝑟=𝑆𝑥𝑦

√𝑆𝑥𝑥𝑆𝑦𝑦

=0.196

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? 𝑛=6?

Let’s do it on our calculators!

Baby A B C D E FHead Circumference () 31.1 33.3 30.0 31.5 35.0 30.2

Gestation Period () 36 37 38 38 40 40

• Put in Stats mode: MODE • Select for (i.e. calculations to do with linear

relationships)• Insert the data into your table. Use the arrow keys and

‘=‘ to add the values.• Once done, press the button. This goes to normal

calculation input.• We want to insert into your calculation. Press , and

choose for REGRESSION.• Select for . is now in your calculation, so press =.

Test Your UnderstandingJune 2013 Q1

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Further Practice

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

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

Bob wants to establish if there’s a connection between waiting time () at the post office and customer satisfaction (). He calculates as -0.81. Interpret this correlation coefficient.

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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Exam Questions (on provided sheet)

Q1

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(Before you go on to Q2) Effects of coding

We know that and

Therefore, if all our data values get k times bigger in size and values become times bigger, what happens to…

(Recap) Variance of : times as big: times as big: times as big: times as big: Unaffected!

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Bro Exam Note: For the purposes of the S1 exam, you just need to remember that:!• Coding affects in the

same way that the variance is affected. i.e. If the variance becomes 9 times larger, so does .

• PMCC is completely unaffected by (linear) coding.

𝑟=0.655

Example

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𝑝=𝑥−10201

𝑞=𝑦−3005

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We can now just find the PMCC of this new data set, and no further adjustment is needed.

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Exam Questions (on provided sheet)

Q2

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Q3

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Q4

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Q5

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Q6

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Q7

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Exam Questions (on provided sheet)

Q8

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Exam Questions (on provided sheet)

Q9

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Limitations of correlationOften there’s a 3rd variable that explains two others, but the two variables themselves are not connected.

Q1: The number of cars on the road has increased, and the number of DVD recorders bought has decreased. Is there a correlation between the two variables?

Buying a car does not necessarily mean that you will not buy a DVD recorder, so we cannot say there is a correlation between the two.

Q2: Over the past 10 years the memory capacity of personal computers has increased, and so has the average life expectancy of people in the western world. Is there are correlation between these two variables?

The two are not connected, but both are due to scientific development over time (i.e. a third variable!)

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