Correlation – PMCC

Monday 18th March 2013

Learning objective:To be confident finding the Product Moment Correlation Coefficient and using it to interpret data.

• Accuracy

• Interpretation

Starter: Can you find my mistakes?

Product moment correlation coefficient

Weight, x Calories, y x2 y2 xy

A 85 250

B 74 222

C 57 185

D 62 190

E 81 239

6th

6th

n

yxxySxy

nx

xSxx

2

2 )(

ny

ySyy

2

2 )(

(∑x)2 (∑y)2 ∑x∑y

7th

7th

yyxx

xy

SS

Sr

∑x = ∑y2 = ∑x2 = ∑y = ∑xy =

( When two sets of random variables bivariate) ; data are displayed on a scatter graph we

are used to describing the correlation but how

?do you measure it

( ) Two sets of random variables bivariate data we can describe correlation but how do you

?measure it x - = -y - = +- x + = -

x - = +

y - = +

+ x + = +

x

x

y

x - = -

y - = -

- x - = +

yx

xy

x - = +y - = - + x - = -yx

– ?Covariance how do you interpret it

When the covariance is positive it suggests positive correlation

When covariance is negative it suggests negative correlation

When the covariance is close to zero .it suggests no correlation

covariance

xySn

yyxx

– Covariance can you see any potential ?problems with this method alone

– When the covariance is positive it suggest positive correlation

– When covariance is negative it suggest negativecorrelation

– When the covariance is close to zero it suggests no.correlation

:You guessed it– ( ’ )you don t know the range

covariance

xySn

yyxx

Pearson Moment Correlation Coefficient

Karl Pearson

1857 - 1936

Is to standardise the covariance so that it can interpreted easily. It converts the covariance to a number between -1 to 1, where:

• -1 is a perfect negative correlation

• 1 is a perfect positive correlation

• 0 is no correlation

2 2

x x y y

nrx x y y

n n

The effect of scaling

If you work out the correlation coefficient for

- & ( ) sales of ice cream temperature t in

. Fahrenheit Would you expect the correlation to

change if you worked on the same data but in

?Celsius

– . No scaling has no effect on correlation

Be aware of correlationclaims

Some things may look like they are connected :but they are not

– :General knowledge and height 7 13 Children in a school from year to year are asked . general knowledge questions The correlation is worked

. out using height and their score In your opinion does ? height have any effect on their score If not can you suggest what is the explanatory factor that is connected

?to both

Outliners– As all data items are used outliners will effect the

. correlation coefficient When outliners are obvious it is .worth ignoring them altogether

- .Non linear relationships– . . . . Pearson's p m c c is only suitable for linear

relationships