

Practice Problems (Answers)
Homework
Note that in scatterplots, the X and Y axes are equal in length and thus this type of graph does not obey the 3/4 high rule.
Is best illustrated with examples:
As the number of hours studied increased so did the grade. This is also called a "direct" relationship.
More realistic example
As the number of beers drank increased, the grade decreased. This is also called an "inverse" or "indirect" relationship.
More realistic example
So, basicially, there is no relationship between toe size and grade.
More realistic example
r =  ±  (0 ↔ 1) 

Sign  Magnitude  
Gives direction  Gives strength 
X  Z_{X}  Z_{X}^{2}  Y  Z_{Y}  Z_{X}Z_{Y}  

3  1.42  2.02  1  1.42  2.02  
5  .71  .50  2  .71  .50  
7  0  0  3  0  0  
9  .71  .50  4  .71  .50  
11  1.42  2.02  5  1.42  2.02  
σ=2.82 
∑Z_{X}^{2}=5=N 
σ=1.41 
∑Z_{X}Z_{Y}=∑Z_{X}^{2} 

μ=7  μ=3  
N=5 
If the relative position of the scores on the two variables is the same (as in the present case), then the z scores of each of the variables will be the same and ∑(Z_{X}Z_{Y}) would be equal to ∑Z_{X}^{2}. As we saw above, ∑Z_{X}^{2} is equal to N and thus r would equal N/N or 1.
Now for the perfect negative relationship between X & W.
X  Z_{X}  Z_{X}^{2}  W  Z_{W}  Z_{X}Z_{W}  

3  1.42  2.02  5  1.42  2.02  
5  .71  .50  4  .71  .5  
7  0  0  3  0  0  
9  .71  .50  2  .71  .5  
11  1.42  2.02  1  1.42  2.02  
σ=2.82  ∑Z_{X}^{2}=5=N 
∑Z_{X}Z_{W}=5 

μ=7  
N=5 
The scores again have the same relative position, but this time the relationship is indirect. In this case, ∑(Z_{X}Z_{W}) would be equal to N and r would be equal to N/N or 1.
Example: Scores on 20 point math and science quizzes. [Minitab]
Person  Math (X)  Science (Y) 

A  11  11 
B  13  10 
C  18  17 
D  12  13 
E  16  14 
N=5 
First step would be to create a scatterplot:
Since the scatterplot looks promising (suggests a strong positive relationship), create the necessary grid for the computations.
Person  Math (X)  Science (Y)  XY  X^{2}  Y^{2} 

A  11  11  121  121  121 
B  13  10  130  169  100 
C  18  17  306  324  289 
D  12  13  156  144  169 
E  16  14  224  256  196 
N=5  ∑X=70  ∑Y=65  ∑XY=937  ∑X^{2}=1014  ∑Y^{2}=875 
As was suggested by the scatterplot, there is indeed a strong positive correlation between the math and science scores.
A variant of Pearson’s r which is used with rank data is called Spearman’s Rho (r_{s}). This correlation coefficient is appropriate when either of the following two conditions are met:
In either case, both scales must be converted to ranks. And if we computed Pearson's r on the ranked data, it would give Spearman's Rho. However, for computations by hand, there is a simpler formula:
Where D= Rank of X – Rank of Y (i.e., a Difference score).
Person  Beauty  Sociability 

A  3  3 
B  1=most  2 
C  2  1=most 
D  5  4 
E  4  5 
N=5 
First step would be to create a scatterplot.
Since the scatterplot looks promising (suggests a strong positive relationship), create the necessary grid for the computations.
Person  Beauty  Sociability  D  D^{2} 

A  3  3  0  0 
B  1=most  2  1  1 
C  2  1=most  1  1 
D  5  4  1  1 
E  4  5  1  1 
N=5  ∑D=0  ∑D^{2}=4 
Then perform the computations:
Person  Beauty  Beauty (reranked) 
Science  Science (ranked) 

A  3  3  11  2 
B  1=most  5=most  10  1 
C  2  4  17  5=most 
D  5  1  13  3 
E  4  2  14  4 
N=5 
Then we would create a scatterplot of the ranked scores.
The data do not look very promising, but let's prepare the grid for the computations anyway.
Person  Beauty (reranked) 
Science (ranked) 
D  D^{2} 

A  3  2  1  1 
B  5=most  1  4  16 
C  4  5=most  1  1 
D  1  3  2  4 
E  2  4  2  4 
N=5  ∑D=0  ∑D^{2}=26 
Then perform the computations:
So as the scatter plot indicated, there wasn't much of a correlation.
Note: Tied ranks would get the average of the tie(s). Examples:


These are the reasons why it is important to create a scatterplot.
Example of a curvilinear (or nonmonotonic) relationship:
In general, curvilinearity in a relationship will result in an r that underestimates the true relationship.
Example of a Restricted Range  Foot size and age in 6 year olds:
Example of a Truncated Range  ACT scores and GPA in college students:
Possible causal relationships between X (television violence) and Y (violence in the real world) if they are correlated include:
Possibility Symbols Explanation Meaning a.X → Y X causes Y watching TV violence causes
real world violence b.X ← Y Y causes X real world violence is the reason
there is violence on TV c.X ← A → Y A causes both X & Y stress (A) causes both
real world & TV violence etc.B → C → X
B → YEtc. there are still other
complicating variablesMain point is that correlation doesn’t tell us much about causality. It should be noted that inferring causality from a correlation is an error that is extremely common amoung students, journalists, and even scientists themselves..
3. Some Specific Uses of Correlation
 Determining Reliabilities
Compare two raters (interobserver) or the same raters (intraobserver) observations of behavior to see if they agree. There is a problem like this in the homework for this section. Determining Validities
If ACT scores are highly correlated with GPA's then we can say that ACT scores are a valid predictor of GPA's. For Prediction
A set of procedures similar to correlation called regression is used for predicting one variable from one or more other variables.