Practical Statistics Correlation

There are six statistics that will answer 90% of all questions! 1. Descriptive 2. Chi-square 3. Z-tests 4. t-tests 5. Correlation 6. Regression

Correlation tests the degree of association between interval and ratio measures.

Darwin had a problem. How can the associations and degree of differences between observations be expressed in meaningful mathematics?

Darwin Galton Karl Pearson

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

If you take two measures of the same person or object, multiple them, and then add the products across persons or objects… such as: Person M1 M2Product Sum = 55

The largest possible sum will occur if M1 and M2 are in perfect ordinal order. Note what happens when only one measure changes. Person M1 M2Product Sum = 54

The smallest possible sum will occur if M1 and M2 are in perfect inverse order. Person M1 M2Product Sum = 35

The “normal score” or “standardized score” is equal to:

Converting the measures to Z scores… Person M1 M2Product Sum = 5.0

Note, that the sum is equal to the number of people in the sample, i.e., 5. Person M1 M2Product Sum = 5.0

Note, what happens when the measure in Z scores are arranging for perfect inverse order: Person M1 M2Product Sum = -5.0

Then X = Y, and :

This is called “the sum of the cross products”

So: When X and Y are in ranked order the max will be equal to n, and when ranked in perfect negative order, the max will be -n. (Or, n-1 and –(n-1) if taken from a sample).

The average of the sum of cross products is the correlation.

This means that a perfect association always has a value of 1.0 when in positive order and a value of -1.0 when in negative order. A value of zero would indicate a random relationship between the two variables. r = 1.0 or r = -1.0

How to do this on Excel:

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

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

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

The correlation is an INDEX of association. It contains three pieces of information: 1.How much association is present,

The correlation is an INDEX of association. It contains three pieces of information: 1.How much association is present, 2.Is that a “significant” association, That is, can we reject the H 0 that the true association is zero (there is no association).

The correlation is an INDEX of association. It contains 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. That is, how much of the change in one variable can be known by looking at the change in the other variable.

The correlation is an INDEX of association. It contains three pieces of information: The correlation is r….. and r-squared is the amount of variation accounted for in Y by knowing X.

The correlation is r….. r-squared is the amount of variation accounted for in Y by knowing X.

Be careful!! The correlation does not tell you that X is the cause of Y. It is a necessary condition for cause, but it does not prove cause…

That 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 ") and false cause.

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