Correlation

● Correlation describes the linear relationship

between observed variables.

Computing correlation

● One way to compute a correlation coefficient is to

convert all the raw scores (both variables) to z

scores.

● This will give us two sets of z scores.

● To calculate the correlation, cross-multiply the

two sets of z scores and normalize by n.

The sign of the correlation coefficient

● The denominator of r must be positive, so the sign of r will be determined by the numerator (the

cross-product of the z scores).

The sign and range of the correlation.

● For a particular observation (a pair of scores), when the z score of x has the same sign as the z score of y, r will be positive.

● When the z score of x has the same sign as the z

score of y, r will be positive.

● The value of r can never be greater than 1.0 nor

less than -1.0.

● The sign of the correlation is given by the slope of the scatterplot.

● When the slope is positive, 0 >= r <= 1.

● When the slope is negative, 0 >= r <= -1.

The shape of the scattergram

● When the scatterplot is elongated, the correlation

will tend to be “larger” (nearer 1 or -1).

● When the scatterplot is blobby the correlation will

tend to be nearer 0.

The influence of sample means and SDs

● Because correlation is a function of the z scores of the variables, r is not affected by the mean or the standard deviation of the original observations.

The influence of variability

● As variability decreases, the correlation increases.

● As variability increases, the correlation decreases.

The influence of the range of scores

● All else being equal, the smaller the range of x and y scores, the lower the correlation coefficient.

● If each score reflects both the true value of

the variable and additive random variability,

then as the range of scores decrease the

apparent influence of variability increases.

The influence of the slope

● For the same reason, the absolute slope of the

original variables is also unimportant.

The influence of linearity

● The correlation coefficient only measures the

linear relationship between the variables.

● Therefore, if the slope isn't constant

throughout the range of scores, r will decline.

Factors that influence the correlation

● The strength of the linear relationship between x

and y.

● The strength of any nonlinear relationship between x and y.

● The range of the scores across the variables.

● Measurement error (and other uncontrolled sources of variability).

Potential problems with correlation

● Non-normally distributed variables.

● Restriction of range.

● Nonlinear relationship between variables.

● Poor measurement reliability.

Factors influencing correlation

● The true underlying relationship between x and y.

● The range of scores in x and y.

● All else being equal, as the range of x or y

becomes more restricted, the value of the

correlation will decrease.

● When samples are pooled, the correlation of

the aggregated data will depend on where the

samples lie relative to one another on both the

x and y variables.