statistics for the social sciences psychology 340 fall 2006 relationships between variables

Statistics for the Social Sciences

Psychology 340Fall 2006

Relationships between variables

Statistics for the Social Sciences

Correlation

• Write down what (you think) a correlation is.

• Write down an example of a correlation

• Association between scores on two variables– Age and coordination skills in children, as kids get older their motor coordination tends to improve

– Price and quality, generally the more expensive something is the higher in quality it is

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Correlation and Causality

• Correlational research design– Correlation as a kind of research design (observational designs)

– Correlation as a statistical procedure

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Another thing to consider about correlation

• Correlations describe relationships between two variables, but DO NOT explain why the variables are related

Suppose that Dr. Steward finds that rates of spilled coffee and severity of plane turbulents are strongly positively correlated.

One might argue that turbulents cause coffee spills

One might argue that spilling coffee causes turbulents

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Another thing to consider about correlation

• Correlations describe relationships between two variables, but DO NOT explain why the variables are related

Suppose that Dr. Cranium finds a positive correlation between head size and digit span (roughly the number of digits you can remember).

One might argue that bigger your head, the larger your digit span

1

2124

1537

One might argue that head size and digit span both increase with age (but head size and digit span aren’t directly related)

Statistics for the Social Sciences

Another thing to consider about correlation

• Correlations describe relationships between two variables, but DO NOT explain why the variables are related

For many years instructors have noted that the reported fatality rate of

grandparents increases during midterm and final exam periods. One might argue that college exams cause grandparent death

Statistics for the Social Sciences

Relationships between variables

• Properties of a correlation– Form (linear or non-linear)– Direction (positive or negative)– Strength (none, weak, strong, perfect)

• To examine this relationship you should:– Make a scatterplot - a picture of the relationship

– Compute the Correlation Coefficient - a numerical description of the relationship

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Graphing Correlations

• Steps for making a scatterplot (scatter diagram)1. Draw axes and assign variables to them2. Determine range of values for each

variable and mark on axes3. Mark a dot for each person’s pair of

scores

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Scatterplot

Y

X1

2

34

5

6

1 2 3 4 5 6

• Plots one variable against the other• Each point corresponds to a different individual

A 6 6

X Y

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Scatterplot

Y

X1

2

34

5

6

1 2 3 4 5 6

• Plots one variable against the other• Each point corresponds to a different individual

A 6 6B 1 2

X Y

Statistics for the Social Sciences

Scatterplot

Y

X1

2

34

5

6

1 2 3 4 5 6

• Plots one variable against the other• Each point corresponds to a different individual

A 6 6B 1 2C 5 6

X Y

Statistics for the Social Sciences

Scatterplot

Y

X1

2

34

5

6

1 2 3 4 5 6

• Plots one variable against the other• Each point corresponds to a different individual

A 6 6B 1 2C 5 6

D 3 4

X Y

Statistics for the Social Sciences

Scatterplot

Y

X1

2

34

5

6

1 2 3 4 5 6

• Plots one variable against the other• Each point corresponds to a different individual

A 6 6B 1 2C 5 6

D 3 4

E 3 2

X Y

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Scatterplot

Y

X1

2

34

5

6

1 2 3 4 5 6

• Imagine a line through the data points

• Plots one variable against the other• Each point corresponds to a different individual

A 6 6B 1 2C 5 6

D 3 4

E 3 2

X Y

• Useful for “seeing” the relationship– Form, Direction, and Strength

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Form

Non-linearLinear

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NegativePositive

Direction

• X & Y vary in the same direction

• As X goes up, Y goes up

• Positive Pearson’s r

• X & Y vary in opposite directions

• As X goes up, Y goes down

• Negative Pearson’s r

Y

X

Y

X

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Strength

• The strength of the relationship– Spread around the line (note the axis scales)

– Correlation coefficient will range from -1 to +1• Zero means “no relationship”• The farther the r is from zero, the stronger the relationship

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Strength

r = 1.0“perfect positive corr.”r2 = 100%

r = -1.0“perfect negative corr.”r2 = 100%

r = 0.0“no relationship”r2 = 0.0

-1.0 0.0 +1.0

The farther from zero, the stronger the relationship

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The Correlation Coefficient

• Formulas for the correlation coefficient:

r = XZ YZ∑N

€

r =SP

SSX SSY

€

SP = X − X ( ) Y −Y ( )∑

Used this one in PSY138 Common alternative

Statistics for the Social Sciences

The Correlation Coefficient

• Formulas for the correlation coefficient:

r = XZ YZ∑N

€

r =SP

SSX SSY

€

SP = X − X ( ) Y −Y ( )∑

Used this one in PSY138 Common alternative

Statistics for the Social Sciences

Computing Pearson’s r (using SP)

• Step 1: SP (Sum of the Products)

€

SP = X − X ( ) Y −Y ( )∑

mean3.64.0

6 61 25 6

3 4

3 2

X Y

€

X − X ( )

€

Y −Y ( )

€

X − X ( ) Y −Y ( )

Statistics for the Social Sciences

Computing Pearson’s r (using SP)

• Step 1: SP (Sum of the Products)

€

SP = X − X ( ) Y −Y ( )∑

mean3.64.0

2.4

0.0

6 61 25 6

3 4

3 2

X Y

€

X − X ( )

€

Y −Y ( )

€

X − X ( ) Y −Y ( )= 6 - 3.6

-2.6= 1 - 3.6

1.4= 5 - 3.6

-0.6= 3 - 3.6

-0.6= 3 - 3.6Quick check

Statistics for the Social Sciences

Computing Pearson’s r (using SP)

• Step 1: SP (Sum of the Products)

€

SP = X − X ( ) Y −Y ( )∑

mean3.64.0

2.4-2.6

1.4

-0.6

-0.6

0.0 0.0

6 61 25 6

3 4

3 2

X Y

€

X − X ( )

€

Y −Y ( )

€

X − X ( ) Y −Y ( )2.0= 6 - 4.0-2.0= 2 - 4.0

2.0= 6 - 4.0

0.0= 4 - 4.0

-2.0= 2 - 4.0Quick check

Statistics for the Social Sciences

Computing Pearson’s r (using SP)

• Step 1: SP (Sum of the Products)

€

SP = X − X ( ) Y −Y ( )∑

mean3.64.0

2.4-2.6

1.4

-0.6

-0.6

0.0

2.0-2.0

2.0

0.0

-2.0

0.0 14.0 SP

6 61 25 6

3 4

3 2

X Y

€

X − X ( )

€

Y −Y ( )

€

X − X ( ) Y −Y ( )4.8* =

5.2* =

2.8* =

0.0* =

1.2* =

Statistics for the Social Sciences

Computing Pearson’s r (using SP)

• Step 2: SSX & SSY

Statistics for the Social Sciences

Computing Pearson’s r (using SP)

• Step 2: SSX & SSY

mean3.64.0

2.4-2.6

1.4

-0.6

-0.6

0.0

2.0-2.0

2.0

0.0

-2.0

0.0 14.0

6 61 25 6

3 4

3 2

X Y

€

X − X ( )

€

Y −Y ( )

€

X − X ( ) Y −Y ( )4.85.2

2.8

0.0

1.2

€

X − X ( )2

5.76

15.20

SSX

2 =6.762 =

1.962 =

0.362 =

0.362 =

Statistics for the Social Sciences

Computing Pearson’s r (using SP)

• Step 2: SSX & SSY

mean3.64.0

2.4-2.6

1.4

-0.6

-0.6

0.0

2.0-2.0

2.0

0.0

-2.0

0.0 14.0

6 61 25 6

3 4

3 2

X Y

€

X − X ( )

€

Y −Y ( )

€

X − X ( ) Y −Y ( )4.85.2

2.8

0.0

1.2

€

X − X ( )2

5.766.76

1.96

0.36

0.36

15.20

€

Y −Y ( )2

2 =4.02 =4.02 =4.02 =0.02 =4.0

16.0

SSY

Statistics for the Social Sciences

Computing Pearson’s r (using SP)

• Step 3: compute r

€

r =SP

SSX SSY

Statistics for the Social Sciences

Computing Pearson’s r (using SP)

• Step 3: compute r

mean3.64.0

2.4-2.6

1.4

-0.6

-0.6

0.0

2.0-2.0

2.0

0.0

-2.0

0.0 14.0

6 61 25 6

3 4

3 2

X Y

€

X − X ( )

€

Y −Y ( )

€

X − X ( ) Y −Y ( )4.85.2

2.8

0.0

1.2

€

X − X ( )2

5.766.76

1.96

0.36

0.36

15.20

€

Y −Y ( )2

4.04.0

4.0

0.0

4.0

16.0

SSYSSX

SP

€

r =SP

SSX SSY

Statistics for the Social Sciences

Computing Pearson’s r

• Step 3: compute r

14.015.20 16.0

SSYSSX

SP

€

r =SP

SSX SSY

Statistics for the Social Sciences

Computing Pearson’s r

• Step 3: compute r

15.20 16.0

SSYSSX

r =14

SSXSSY

Statistics for the Social Sciences

Computing Pearson’s r

• Step 3: compute r

15.20

SSX

r =14

SSX * 16

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Computing Pearson’s r

• Step 3: compute r

€

r =14

15.2 *16

Statistics for the Social Sciences

Computing Pearson’s r

• Step 3: compute rr =

1415.2 * 16

=0.89

Y

X1

2

34

5

6

1 2 3 4 5 6

• Appears linear• Positive relationship• Fairly strong relationship• .89 is far from 0, near +1

Statistics for the Social Sciences

The Correlation Coefficient

• Formulas for the correlation coefficient:

r = XZ YZ∑N

r =SPSSXSSY

SP = X−X( ) Y −Y( )∑

Used this one in PSY138 Common alternative

Statistics for the Social Sciences

Computing Pearson’s r

(using z-scores)

• Step 1: compute standard deviation for X and Y (note: keep track of sample or population)

6 61 25 6

3 4

3 2

X Y

• For this example we will assume the data is from a population

Statistics for the Social Sciences

Computing Pearson’s r

(using z-scores)

• Step 1: compute standard deviation for X and Y (note: keep track of sample or population)

Mean 3.6

2.4-2.6

1.4

-0.6

-0.6

0.0

6 61 25 6

3 4

3 2

X Y

€

X − X ( )

€

X − X ( )2

5.766.76

1.96

0.36

0.36

15.20

SSXStd dev1.74

σ =SSX

N=

15.2

5= 1.74

• For this example we will assume the data is from a population

Statistics for the Social Sciences

Computing Pearson’s r

(using z-scores)

• Step 1: compute standard deviation for X and Y (note: keep track of sample or population)

Mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

2.0-2.0

2.0

0.0

-2.0

0.0

6 61 25 6

3 4

3 2

X YX −X( )

€

Y −Y ( )X −X( )2

5.766.76

1.96

0.36

0.36

15.20

€

Y −Y ( )2

4.04.0

4.0

0.0

4.0

16.0

SSYStd dev1.741.79

• For this example we will assume the data is from a population

σ =SSY

N

=16.0

5= 1.79

Statistics for the Social Sciences

Computing Pearson’s r

(using z-scores)

• Step 2: compute z-scores

Mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

2.0-2.0

2.0

0.0

-2.0

6 61 25 6

3 4

3 2

X Y

€

X − X ( ) Y −Y( )X −X( )2

5.766.76

1.96

0.36

0.36

15.20

Y −Y( )2

4.04.0

4.0

0.0

4.0

16.0Std dev

ZX

1.741.79

1.38=2.4

1.74

X −X( )sX

Statistics for the Social Sciences

Computing Pearson’s r

(using z-scores)

• Step 2: compute z-scores

Mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

2.0-2.0

2.0

0.0

-2.0

6 61 25 6

3 4

3 2

X Y

€

X − X ( ) Y −Y( )X −X( )2

5.766.76

1.96

0.36

0.36

15.20

Y −Y( )2

4.04.0

4.0

0.0

4.0

16.0Std dev

ZX

X −X( )sX

1.741.79

1.38-1.49

0.8

- 0.34

- 0.34

0.0 Quick check

Statistics for the Social Sciences

Computing Pearson’s r

(using z-scores)

• Step 2: compute z-scores

Mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

2.0-2.0

2.0

0.0

-2.0

6 61 25 6

3 4

3 2

X YX −X( )

€

Y −Y ( )X −X( )2

5.766.76

1.96

0.36

0.36

15.20

€

Y −Y ( )2

4.04.0

4.0

0.0

4.0

16.0Std dev

ZX ZY

1.741.79

1.1

Y −Y( )sY

=2.0

1.791.38-1.49

0.8

- 0.34

- 0.34

Statistics for the Social Sciences

Computing Pearson’s r

(using z-scores)

• Step 2: compute z-scores

Mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

2.0-2.0

2.0

0.0

-2.0

6 61 25 6

3 4

3 2

X YX −X( )

€

Y −Y ( )X −X( )2

5.766.76

1.96

0.36

0.36

15.20

€

Y −Y ( )2

4.04.0

4.0

0.0

4.0

16.0Std dev

ZX ZY

Y −Y( )sY

1.741.79

1.1-1.1

0.0

-1.1

1.1

0.0

1.38-1.49

0.8

- 0.34

- 0.34

Quick check

Statistics for the Social Sciences

Computing Pearson’s r

(using z-scores)

• Step 3: compute r

Mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

0.0

2.0-2.0

2.0

0.0

-2.0

0.0

6 61 25 6

3 4

3 2

X Y ZX ZY

5.766.76

1.96

0.36

0.36

15.20

€

Y −Y ( )2

4.04.0

4.0

0.0

4.0

16.0Std dev

ZX ZY

1.741.790.0

1.1-1.1

0.0

-1.1

1.1

0.0

1.52

X −X( ) X −X( )2

r =ZXZY∑N

Y −Y( )

1.38-1.49

0.8

- 0.34

- 0.34

* =

Statistics for the Social Sciences

Computing Pearson’s r

(using z-scores)

• Step 3: compute r

Mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

0.0

2.0-2.0

2.0

0.0

-2.0

0.0

6 61 25 6

3 4

3 2

X Y ZX ZY

5.766.76

1.96

0.36

0.36

15.20

€

Y −Y ( )2

4.04.0

4.0

0.0

4.0

16.0Std dev

ZX ZY

1.741.790.0

1.1-1.1

0.0

-1.1

1.1

0.0

1.521.64

0.88

0.0

0.37

X −X( ) X −X( )2

r =ZXZY∑N

=4.41

5

Y −Y( )

1.38-1.49

0.8

- 0.34

- 0.34

=0.89

4.41

Statistics for the Social Sciences

Computing Pearson’s r

(using z-scores)

• Step 3: compute r

Y

X1

2

34

5

6

1 2 3 4 5 6

• Appears linear• Positive relationship• Fairly strong relationship• .89 is far from 0, near +1

r =ZXZY∑N

=0.89

Statistics for the Social Sciences

A few more things to consider about correlation

• Correlations are greatly affected by the range of scores in the data– Consider height and age relationship

• Extreme scores can have dramatic effects on correlations – A single extreme score can radically change r

• When considering "how good" a relationship is, we really should consider r2 (coefficient of determination), not just r.

Statistics for the Social Sciences

Correlation in Research Articles

• Correlation matrix– A display of the correlations between more than two variables

Acculturation

• Why have a “-”?

• Why only half the table filled with numbers?

Statistics for the Social Sciences

Next time

• Predicting a variable based on other variables