# Correlation

Correlation is basically whether or not there is any relationship between two sets of data.

Correlation is a measure of association.

Is there is any kind of relationship, then a change in one variable can be associated with a change in the other.

A lot of relationships are infact linear but very few are actually perfect because there are normally other factors involved

Correlation describes the strength of the relationship between two variables

This relationship can be described by a __Regression Equation__

After calculating any correlation coefficient it is normal to state the number of pairs used and to give a probability value to your coefficient.

Also sometimes called co-movement

this describes the relationship between two groups of numbers

When you are talking about correlation you are always dealing with paired scores

Rather than using vaque words like "More than" or "a little" we can quantify exactly using numbers

The numbers used to quantify exactly how much association there is are called correlation coefficients

### A Simple Example

For example "the higher the income the more expenditure there is".

Rather than use words like "more" to describe an association this can actually be qualified by using a correlation coefficient.

A correlation defines precisely the extent to which two things (or variables) are associated.

The number used to express this correlation is called a correlation coefficient.

Most variables will typically be one of the following:

"a little bit associated" -

"fairly associated" -

"very well associated" -

Perfect correlations are obviously very unusual and scatter diagrams of the two variables will typically has a less than perfect association.

In these situations a line is calculated and drawn to as to come closest to any many points as possible

This type of line is called a "regression line" or "a line of best fit".

When you have a correlation coefficient of less than 1 it means that some of the pairs do not quite fit the general pattern.

This technique of correlation measures the strength of association between the variables.

There are two widely used measures of correlation.

1) Spearmans Rank Correlation Coefficient

2) Pearsons Product Moment Correlation Coefficient

### How can you tell if there is a Correlation ?

There are several ways to tell if there is a correlation between two variables:

1) Scatter Chart

2) Correlation Table

3) Calculate the "product moment" coefficient

4) Calculate the coefficient of rank (examples are Spearmans and Kendalls Tau)

5) Using Regression

### What information can you obtain about the Correlation ?

The following information can be obtained:

1) Whether there is a correlation (ie a relationship) between the two variables

2) The direction of the correlation

3) The strength of the correlation

4) The proportion of a variability in one variable that can be accounted for by its relationship with the other variable ??

### Calculating the "product moment" coefficient

This is a measure of the linear correlation and is often referred to as the Pearson's coefficient.

This coefficient attempts to show the closeness of the relationship between the two variables.

This coefficient can be calculated by dividing the mean product of the deviations from the mean, by the product of the standard deviations. This has the following formula:

This coefficient is not only an attempt to show whether a correlation exists but is also an attempt to show how closely it exists.

This formula can be re-written by substituing the basic formula for standard deviation:

EQ

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