PPA 696 RESEARCH METHODS
CORRELATION
Correlation
Scatterplots
CORRELATION
Correlation coefficients are statistics which can help to describe datasets which contain variables measured at the interval and ratio levels.Correlation coefficients are measures of association between two (or more)variables.
Correlation is a measure of association that tests whether a relationshipexists between two variables. It indicates both the strength of the associationand its direction (direct or inverse). The Pearson product-moment correlationcoefficient, written as r, can describe a linear relationship betweentwo variables.
For example is there a relationship between:
the budget of the police department and the crime rate?
the hours of batting practice and a player's batting average?
The value of r can range from 0.0, indicating no relationshipbetween the two variables, to positive or negative 1.0, indicating a stronglinear relationship between the two variables.
| Value of r | Indications |
| 0.0 | No linear relationship between the two variables |
| +1.0 | Strong positive linear relationship; as X increases in value, Y increasesin value also; or as X decreases in value, Y decreases also. |
| -1.0 | Strong inverse linear relationship; as X increases in value, Y decreasesin value; or as X decreases in value, Y increases in value. |
SCATTERPLOTS
It is useful to obtain a plot of the joint distribution of the valuesof the two variables, X and Y. These are called scatterplots. The valuesof X are displayed on the lower, or horizontal axis (called the X-axis)and the values of Y are displayed on the upper or vertical axis (calledthe Y-axis).
If small values of X are associated with small values for Y, and largevalues of X are associated with large values of Y, then the data will stretchfrom the lower left hand corner of the plot to the upper right hand cornerof the plot. This indicates a positive relationship.
If small values of X are associated with large values for Y, and largevalues of X are associated with small values of Y, then the data will stretchfrom the upper left hand corner of the plot to the lower right hand cornerof the plot. This indicates an inverse relationship.
If there is no discernible pattern to the distribution, then the twovariables probably are not related in a linear fashion. There may be astrong, non-linear relationship between the two variables (for example,think of the normal curve) but it cannot be detected by r.
When there are only a few data points, it is fairly easy to estimatethe strength of the relationship by eyeballing the data. However, withmany data points statistics are needed to summarize the strength and directionof the relationship.
The Pearson r assumes that the variables are measured at theinterval or ratio level. If the variables are measured at the ordinal level,however (for example, a Likert-type scale), then the Spearman rank correlationcan be used. Neither Pearson nor Spearman are designed for use with variablesmeasured at the nominal level; instead, use the point-biserial correlation(for one nominal variable) or phi (for two nominal variables).
The formula for r is as follows:
