Basic Statistics Correlation

Relationships Associations Var Relationships Associations Var Var Var Var

Independent variables In Research Information Dependent variable X1 ? Y X2 COvary X3 Independent variables

The Concept of Correlation Association or relationship between two variables Co-relate? r relation X Y Covary---Go together

Patterns of Covariation Zero or no correlation X Y Correlation Covary Go together X Y X Y Negative correlation Positive correlation

Scatter plots allow us to visualize the relationships The chief purpose of the scatter diagram is to study the nature of the relationship between two variables Linear/curvilinear relationship Direction of relationship Magnitude (size) of relationship Scatter Plots

Scatter Plot A Variable Y Variable X high low low high Represents both the X and Y scores Variable Y Exact value low low high Variable X An illustration of a perfect positive correlation

An illustration of a positive correlation Scatter Plot B high Variable Y Estimated Y value low low high Variable X An illustration of a positive correlation

Scatter Plot C Variable Y Variable X high low low high Exact value low low high Variable X An illustration of a perfect negative correlation

An illustration of a negative correlation Scatter Plot D high Variable Y Estimated Y value low low high Variable X An illustration of a negative correlation

Scatter Plot E Variable Y Variable X high low low high An illustration of a zero correlation

An illustration of a curvilinear relationship Scatter Plot F high Variable Y low low high Variable X An illustration of a curvilinear relationship

The Measurement of Correlation The Correlation Coefficient The degree of correlation between two variables can be described by such terms as “strong,” ”low,” ”positive,” or “moderate,” but these terms are not very precise. If a correlation coefficient is computed between two sets of scores, the relationship can be described more accurately. A statistical summary of the degree and direction of relationship or association between two variables can be computed

Pearson’s Product-Moment Correlation Coefficient r No Relationship Negative correlation Positive correlation -1.00 -.50 0 + .50 1.00 Direction of relationship: Sign (+ or –) Magnitude: 0 through +1 or 0 through -1

The Pearson Product-Moment Correlation Coefficient Recall that the formula for a variance is: If we replaced the second X that was squared with a second variable, Y, it would be: This is called a co-variance and is an index of the relationship between X and Y.

Conceptual Formula for Pearson r This formula may be rewritten to reflect the actual method of calculation

Calculation of Pearson r You should notice that this formula is merely the sum of squares for covariance divided by the square root of the product of the sum of squares for X and Y

Formulae for Sums of Squares Therefore, the formula for calculating r may be rewritten as:

Calculation of r Using Sums of Squares

An Example Suppose that a college statistics professor is interested in how the number of hours that a student spends studying is related to how many errors students make on the mid-term examination. To determine the relationship the professor collects the following data:

The Stats Professor’s Data Student Hours Studied (X) Errors (Y) X2 Y2 XY 1 4 15 16 225 60 2 12 144 48 3 5 9 25 81 45 6 10 36 100 7 8 49 64 56 28 42 18 Total X = 70 Y = 73  X2 =546 Y2=695 XY=429

The Data Needed to Calculate the Sum of Squares X Y X2 Y2 XY Total X = 70 Y = 73  X2 =546 Y2=695 XY=429 = 546 - 702/10 = 546 - 490 = 56 = 695 - 732/10 = 695 - 523.9 = 162.1 = 429 – (70)(73)/10 = 429 – 511 = -82

Calculating the Correlation Coefficient = -82 / √(56)(162.1) = - 0.86 Thus, the correlation between hours studied and errors made on the mid-term examination is -0.86; indicating that more time spend studying is related to fewer errors on the mid-term examination. Hopefully an obvious, but now a statistical conclusion!

Pearson Product-Moment Correlation Coefficient r perfect negative correlation Zero correlation Perfect positive correlation -1 +1 Negative correlation Positive correlation

Numerical values .73 - .35 Negative correlation Zero correlation Positive correlation Perfect Strong Moderate

The Pearson r and Marginal Distribution The marginal distribution of X is simply the distribution of the X’s; the marginal distribution of Y is the frequency distribution of the Y’s. Y Bivariate relationship Bivariate Normal Distribution X

Marginal distribution of X and Y are precisely the same shape. Y variable X variable

Interpreting r, the Correlation Coefficient Recall that r includes two types of information: The direction of the relationship (+ or -) The magnitude of the relationship (0 to 1) However, there is a more precise way to use the correlation coefficient, r, to interpret the magnitude of a relationship. That is, the square of the correlation coefficient or r2. The square of r tells us what proportion of the variance of Y can be explained by X or vice versa.

How does correlation explain variance? Suppose you wish to estimate Y for a given value of X. high How does correlation explain variance? Explained Variable Y Free to Vary 49% of variance is explained Explained low low high Variable X An illustration of how the squared correlation accounts for variance in X, r = .7, r2 = .49

Now, let’s look at some correlation coefficients and their corresponding scatter plots.

What is your estimate of r?

Y X What is your estimate of r? r = -1.00 r2 = 1.00 = 100%

Y X What is your estimate of r? r = +1.00 r2 = 1.00 = 100%

What is your estimate of r?

What is your estimate of r?