3 Independent variables In ResearchInformationDependent variableX1?YX2COvaryX3Independent variables
4 The Concept of Correlation Association or relationship between two variablesCo-relate?rrelationXYCovary---Go together
5 Patterns of Covariation Zero or no correlationXYCorrelationCovaryGo togetherXYXYNegative correlationPositive correlation
6 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 variablesLinear/curvilinear relationshipDirection of relationshipMagnitude (size) of relationshipScatter Plots
7 Scatter Plot A Variable Y Variable X high low low high Represents both the X and Y scoresVariable YExact valuelowlowhighVariable XAn illustration of a perfect positive correlation
8 An illustration of a positive correlation Scatter Plot BhighVariable YEstimated Y valuelowlowhighVariable XAn illustration of a positive correlation
9 Scatter Plot C Variable Y Variable X high low low high Exact valuelowlowhighVariable XAn illustration of a perfect negative correlation
10 An illustration of a negative correlation Scatter Plot DhighVariable YEstimated Y valuelowlowhighVariable XAn illustration of a negative correlation
11 Scatter Plot E Variable Y Variable X high low low high An illustration of a zero correlation
12 An illustration of a curvilinear relationship Scatter Plot FhighVariable YlowlowhighVariable XAn illustration of a curvilinear relationship
13 The Measurement of Correlation The Correlation CoefficientThe 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
14 Pearson’s Product-Moment Correlation Coefficient r No RelationshipNegative correlationPositive correlationDirection of relationship: Sign (+ or –)Magnitude: 0 through +1 or 0 through -1
15 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.
16 Conceptual Formula for Pearson r This formula may be rewritten to reflect the actual method of calculation
17 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
18 Formulae for Sums of Squares Therefore, the formula for calculating r may be rewritten as:
20 An ExampleSuppose 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:
21 The Stats Professor’s Data StudentHours Studied (X)Errors (Y)X2Y2XY14151622560212144483592581456103610078496456284218TotalX = 70Y = 73 X2 =546Y2=695XY=429
22 The Data Needed to Calculate the Sum of Squares XYX2Y2XYTotalX = 70Y = 73 X2 =546Y2=695XY=429= /10 = = 56= /10 = = 162.1= 429 – (70)(73)/10 = 429 – 511 = -82
23 Calculating the Correlation Coefficient = -82 / √(56)(162.1)=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!
26 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.YBivariate relationshipBivariate Normal DistributionX
27 Marginal distribution of X and Y are precisely the same shape. Y variableX variable
28 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.
29 How does correlation explain variance? Suppose you wish to estimate Y for a given value of X.highHow does correlation explain variance?ExplainedVariable YFree to Vary49% of variance is explainedExplainedlowlowhighVariable XAn illustration of how the squared correlation accounts for variance in X, r = .7, r2 = .49
30 Now, let’s look at some correlation coefficients and their corresponding scatter plots.