We use correlation to describe the strength and direction of the relationship between variables.
Although we call it correlation, the full name for the technique we are going to use is Pearson's Product-Moment Correlational
Coefficient for Linear Correlation. We will call it Pearson's r for short.
The correlational coefficient, the r sub xy, for the relationship between x and y,
measures how closely the data fit the model of a straight line on a scatter plot diagram
The concept of correlation was invented by genius and polymath, Sir Francis Galton.
Galten used correlation to demonstrate the principles of heredity and intelligence in addition to proving that
fingerprints are unique.,
pioneering the use of the questionnaire
experimenting with a normal distribution,
inventing other statistics, and doing plenty of other geniusy things.
The correlational coefficient was developed by statistician Karl Pearson who described a
mathematical property of two variables that divides their covariance together by their variance separately.
This is what we call Pearson's r, and the r does not stand for
relationship. The r actually stands for regression.
Linear correlation means that a correlation can be graphed in a straight line called a regression line/
Linear correlation implies that two variables share something in common
changes in one are reliably associated with predictable changes in the other.
The correlational coefficient therefore
expresses the degree of linear relatedness between the two variables.
Scattered about the page you see some examples of
relational coefficients all of the coefficients that we work with will share some things in common.
A correlational coefficient is a two-digit number. It varies between
-1 and +1, and it has a plus or minus sign and each part tells us something important.
The r tells Us that this is a correlation.
The sign, positive or negative, indicates the direction of the linear relationship.
A positive sign indicates a positive relationship a, negative sign indicates a negative relationship.
The absolute value of the correlation indicates the strength of the relationship.
Correlations close to either -1 or +1 are very strong correlations. Close to 0 are weak.
The direction of relationship is indicated by a positive or negative sign in a positive relationship,
also known as a direct relationship.
Both of the variables move in the same direction, as
one variable increases the other increases, or as one variable decreases the other decreases.
For example, height and weight are correlated, especially for children. As
you grow taller, you will likely weigh more.
If we are running the concession stand at the football game, as temperature increases
the sales of cold soda will also increase.
As the temperature decreases,
we will sell less cold soda in
a negative relationship, also called an inverse relationship.
The variables move in opposite directions. As one variable increases the other decreases.
For example, as you study more, the fewer silly mistakes you will make.
At the concession stand, as temperature decreases, the sales of hot chocolate increases.
As temperature increases, as it gets hot, we will sell less hot chocolate.
The strength of the relationship is indicated by the absolute value of the number in the correlational coefficient.
The strength of the relationship is the extent to which any given x-value is paired with one and only one y-value.
When determining the strength of the relationship,
ignore the sign and focus only on the number.
A perfect correlation has a value of either
+1 or -1. There is a perfectly consistent, predictable
relationship between the two variables. In a perfect correlation, the scatter dots form a straight line.
In this perfect negative correlation you can see that every dot is exactly on the regression line.
Values close to 0 are weak correlations.
The correlations get stronger as they move closer to either
+1 or -1, either direction.
strong but, less than perfect, correlation would be close to positive or negative 1,
indicating a consistent relationship
between the two variables.
When data points on the scatterplot are closer to a straight line,
that correlation is stronger.
Correlations get weaker as the correlational coefficient approaches 0.
Moderate or weak correlations would be closer to 0, indicating an
inconsistent relationship between the two variables.
You can see this pattern on the scatterplot.
Notice how the dots in the moderate correlation are farther from the regression line than they were for the stronger correlation.
And when there is no correlation r = 0,
and there's no relationship between the two variables, the scatterplot is elliptical or circular.
As you look at this scatterplot you can see there's no clear place where you would draw a regression line.
There's no relationship between these variables.
Every x-value could be associated with any of the y-values.
There is a relationship between
variability and the strength of the relationship between two variables. As variability increases,
differences in the y scores at each x become larger.
the relationship or correlation becomes weaker
Notice that in a strong correlation the dots are very close to the regression line.
In a moderate correlation the dots move further from the regression line in a more football shaped or elliptical.
When there is no relationship or no linear trend, there is no place to even draw the regression line.
Greater variability equals weaker correlations
Greater consistency, or low variability equals stronger correlations.
Interpreting a correlation coefficient is sometimes done with what is called the
Eyeball Method. Just look at it as a
general rule, if the correlations are between 0.01 and .24, they are weak.
.25 to .49, moderate.
.50 to .74, large.
And above .74, very strong.
A perfect correlation, a deterministic function, would be either +1 or -1.
Remember that these are general rules of thumb, so for instance, the difference between a
Correlation of a .49 and and a .51 Is still a small difference,
even though the correlation could be said to have moved from moderate to large.
Increasing the correlation from a .25 to a .49 is a big
increase, even though both correlations could still be said to be moderate.
Finally, these values can be squared to calculate an effect size
for the correlation, and we will talk about that next.