Practical Statistics

Correlation

There are six statistics that willanswer 90% of all questions!1. Descriptive2. Chi-square3. Z-tests4. t-tests5. Correlation6. Regression

Correlation tests the degree of association between measures if you haveinterval and ratio measures.

Darwin

Galton

Karl Pearson

Correlation is based on a very simple idea thatKarl Pearson saw….

If you take two measures of thesame person or object, multiple them, and then add

the products across person or objects… such as:

Person M1 M2 Product

1 5 5 25 2 4 4 16 3 3 3 9 4 2 2 4 5 1 1 1

Sum = 55

The largest possible sum will occur if M1 and M2are in perfect ordinal order.

Person M1 M2 Product

1 5 4 20 2 4 5 20 3 3 3 9 4 2 2 4 5 1 1 1

Sum = 54

Note what happens when only pair is out of order.

The smallest possible sum will occur if M1 and M2are in perfect inverse order.

Person M1 M2 Product

1 5 1 5 2 4 2 8 3 3 3 9 4 2 4 8 5 1 5 5

Sum = 35

But Pearson wanted anIndex of Association, not

just a measure of associationthat would change with

every n and scale value.

The “normal score” is equal to:

Z

x x

Sii

x

Converting the measures to Z scores…

Person M1 M2 Product

1 -1.41 -1.41 2.0 2 -0.71 -0.71 0.5 3 0 0 0 4 0.71 1.71 0.5 5 1.41 1.41 2.0

Sum = 5.0

Converting the measures to Z scores…And arranging for perfect inverse order:

Person M1 M2 Product

1 -1.41 1.41 -2.0 2 -0.71 0.71 -0.5 3 0 0 0 4 0.71 -1.71 -0.5 5 1.41 -1.41 -2.0

Sum = -5.0

So:

Z Z M axxi

n

y 1

When X and Y are in ranked order.

Z Z Zxi

n

yi

n

1

02

1

Then X = Y, and :

This is called “the sum of the cross products”

When M1 and M2 are in perfect ordinal order:

Z Z Zxi

n

yi

n

1

02

1

In perfect order, this value will always = n - 1

The average of the sum of cross products isthe correlation.

rZ Z

nxy

x yi

n

1

1

How to do this on Excel:

http://www.gifted.uconn.edu/siegle/research/Correlation/excel.htm

The correlation can be graphically shown by using a scatter plot:

The correlation is related to the shape of the scatter plot:

http://en.wikipedia.org/wiki/Scatter_plot

http://www.itl.nist.gov/div898/handbook/eda/section3/eda33q.htm

The correlation is an INDEX of association. Itcontains three pieces of information:

1. How much association is present?2. Is that a “significant” association?3. And, what is the magnitude of that association?

The correlation is an INDEX of association. Itcontains three pieces of information:

The correlation is r…..

R-squared is the amount of variation accountedfor in Y by knowing X.

The correlation is an INDEX of association. Itcontains three pieces of information:

R-squared is the amount of variation accountedfor in Y by knowing X.

s s sy y y x2 2 2 ' . 2

2

2 '

y

y

s

sr

Be careful!!

The correlation does not tell you that X isthe cause of Y.

It is a necessary condition for cause, but it doesnot prove cause…

Correlation proves causation: Is a logical fallacy by which two events that

occur together are claimed to have a cause-and-effect relationship.

The fallacy is also known as cum hoc ergo propter hoc

(Latin for "with this, therefore because of this")

Since the 1950s, both the atmospheric CO2 level

and crime levels have increased sharply.Hence, atmospheric CO2 causes crime.

Young children who sleep with the light on are much more likely to develop myopia in later life.

Service Encounter

Are the summative measures of the service encounter related to each other?

Service Encounter

Are the summative measures of the service encounter related to each other?

Service Encounter

Are the summative measures of the service encounter related to personality?