Psychology 320: Psychological Statistics

Professor: Howard B. Lee

Lecture Notes

Week 4 : Chapter 6

Lecture 9

Correlations

In order to compute a correlation:
2 scores must exist in pairs for each person.
Data must exist in pairs and be identifiable to a specific person.

Pearson Correlation: A correlation coefficient computed on two variables measured on a continuous scale.

r = correlation or "Pearson Product Moment Correlation"
(-1 is less than or equal to r and r is less than or equal to +1 )
M = mean
SD = standard deviation

A positive correlation is one where if x (one variable) increases in value, y ( the second variable ) also increases in value, or if x decreases in value, y decreases in value.
x and y "follow each other".

A negative correlation is one where as x increases, y decreases, or as x decreases, y increases.
For ex., as you get older (increase in age) the number of live brain cells gets fewer (decrease in live brain cells ).

Linear correlation: A straight line function in data.

The line is called a "regression line."

Regression line: The straight line that best fits the given data. See also least squares.

Regression analysis: A statistical method for determining the type of relationship that exists among two or more variables and for using that relationship to make estimates or predictions.

In order to have a perfect positive correlation, all points must lie on a straight line.

students # correct on test (x) # incorrect on test (y)

1 9 1

2 5 5

3 6 4

4 1 9

5 2 8

6 8 2

students	# correct on test (x)	# incorrect on test (y)
1	9	1
2	5	5
3	6	4
4	1	9
5	2	8
6	8	2

What is the correlation between x and y?
r = -1, a perfect negative correlation.

r = 0, no linear relationship.

As x increases, y stays the same.

Lecture 10

Correlations:

Data Table ( Matrix )

Correlations between pairs of variables.
Ex. What is the relationship between quiz 1 and the final exam? (quiz 1 and the final exam is a pair of variables ).

N = the number of pairs (complete ones).

Formulas for correlations: (p. 139 in your book )

Computational formula: ( p. 147 in your book ).

person X Y X2 Y2 XY

1 7 4 49 16 (7)(4) = 28

2 10 6 100 36 (10)(6) = 60

3 8 7 64 49 (8)(7) = 56

4 5 5 25 25 (5)(5) = 25

5 4 9 16 81 (4)(9) = 36

6 3 6 9 36 (3)(6) = 18

total 37 37 263 243 223

person	X	Y	X2	Y2	XY
1	7	4	49	16	(7)(4) = 28
2	10	6	100	36	(10)(6) = 60
3	8	7	64	49	(8)(7) = 56
4	5	5	25	25	(5)(5) = 25
5	4	9	16	81	(4)(9) = 36
6	3	6	9	36	(3)(6) = 18
total	37	37	263	243	223

SumX = 37 SumY = 37 SumX2 = 263 SumY2 = 243 SumXY = 223

Now plug into the formula,

Remember this:
Correlations are reported to two decimal places.
All prior calculations require carrying to three decimal places.

Negative correlations mean that there is an inverse relationship, i.e. as one goes up, the other one goes down.

How do you evaluate a correlation?
1) Square the correlation => r squared.
The "coefficient of determination" tells how much variance is shared by the two variables

Venn Diagram

Lecture 11

person X Y XY

1 -3 117 -351

2 8 100 800

3 -12 224 -2688

4 16 356 5696

5 14 102 1428

6 -22 119 -2618

7 5 124 620

8 0 212 0

9 10 89 890

10 -2 310 -620

SumXY = 3157 Mx=1.4 My = 175.3 SDx = 11.236 SDy = 90.668

person	X	Y	XY
1	-3	117	-351
2	8	100	800
3	-12	224	-2688
4	16	356	5696
5	14	102	1428
6	-22	119	-2618
7	5	124	620
8	0	212	0
9	10	89	890
10	-2	310	-620
SumXY = 3157	Mx=1.4	My = 175.3	SDx = 11.236	SDy = 90.668

Plugging into the formula:

Regression Analysis

The constant will be different for both formulas and so will the slope value.
Predicted success in graduate school = (slope)( GPA )+ constant.
Both "success in grad. school" and "GPA" are variables.
The slope and the constant are determined by empirical data.

Add these overlaps to better predict success in graduate school.