Correlation

1. 1
Chapter 12: Bivariate Statistics
and Statistical Inference
“Figures don’t lie, but liars figure.”

2. 2
Hypothesis Testing
 Testing the relationship between two or more
variables.
 Statistical tests are used to find the probability
that the relationship between variables is due
to sampling error or to chance.
Type Example
Null hypothesis (Ho) – No relationship There is no relationship between
income and mental health.
Two-tailed hypothesis (H1) – There is
a relationship
There is a relationship between income
and mental health.
One-tailed hypothesis (H1) –
Directional relationship
The greater the income the greater the
mental health.

3. 3
Statistical Inference (cont’d)
 p-value
 p =.05 means there is a 5% chance that the
relationship found in the sample is a result of
sample error.
 p =.05 means there is a 95% that the relationship
is NOT due to sample error, and actually reflects
the differences in the population.
 Rejection level: If the p value is <.05, we reject
the null hypothesis and accept the alternative
hypothesis. (Why .05? – Convention).

4. 4
Types of Error
 Type I error
 We reject the null hypothesis, but no
relationship actually exists in the population.
 This will happen 5% of the time if the rejection
level is .05.
 We say there is a relationship, but we’re wrong.
 Type II error
 We don’t reject the null hypothesis, but the
relationship actually exists in the population.
 Could be due to sample error or low rejection level.
 We say there is not a relationship, but we’re
wrong.

5. Bivariate Statistics
The relationship between two variables
 Linear Correlation – Pearson’s r
 How do two interval or ratio level variables co-vary
(correlate).
 Ranges from 1 (positive) to -1 (negative or inverse)
 What is the relationship between two ratio
or interval level variables (scale)?
 Is there a relationship between age and final
exam score?
 Excel: Data>Data Analysis>Correlation
 Pearson as correlation coefficient
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6. 6
Bivariate Statistics
The relationship between two variables
 Positive correlation
 The greater one variable, the greater the other
 E.g., education and income (r =.86)
 Negative or Inverse correlation
 The greater one variable, the less the other
 E.g., Life satisfaction and illness (r = -.74)
 No correlation
 No relationship between variables
 E.g., IQ and shoe size (r = .02)
 Correlation does not imply cause and effect.

7. 7
Correlation (con’t.)
 Scatterplot – visually shows the
relationship between two variables.
No Correlation
0
5
10
15
20
25
30
0 2 4 6 8 10 12
Marital Satisfaction
Self-esteem

8. 8
Correlation (con.)
Size of the Correlation Description
Less than .20 Slight, almost negligible
.20 - .40 Low correlation; weak relationship
.40 - .70 Moderate correlation; substantial relationship
.70 - .90 High correlation; marked relationship
.90 – 1.00 Very high correlation; strong relationship
Coefficient of determination (r²) :
The amount of variance in one variable explained by
the other.
• Correlation of self-esteem and GPA: r = .60 then r² = .36.
• Self-esteem explains 36% of the variance in GPA.

9. 9
Hypothesis Testing (r)
Correlation
Probability
Attitude Scale
r = .048, p=.39
H0: r = 0 There is no relationship between Age and Attitudes
H1: r = 0 There is a relationship between Age and Attitudes
Accept the
null hypothesis

10. Reporting Correlation Results
 Correlations are reported with the degrees of
freedom (which is N-2) in parentheses and the
significance level
 r=_____ n= ______ p= ______
 r
 strength of relationship
 P-value
 Significant level
 n
 Sample size
 R-squared
 Coefficient of determination 10

11. Reporting Correlation Results
 There is a moderate negative correlation
between income and level of depression
 r(118) = -.068, p < 0.01
 r(118) = -.068, p = 0.001
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N= 120 Age Income Depression Level
Age r
p
1.00
Income r
p
0.384
0.043
1.00
Depression
level
r
p
0.025
0.913
-0.684
0.001
1.00

12. Reporting Correlation Results
 “A Pearson product-moment correlation coefficient
was computed to assess the relationship between
income and the level of depression. There was a
negative correlation between the two variables,
r(118) = -.068, p <.01. A scatterplot summarizes
the results (Figure 1) Overall, there was a
moderate, negative correlation between income
and level of depression. Increases in levels of
depression were correlated with decreases in
income.
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13. Rejection of Null Hypothesis
 Table B.4 lists the critical values for the
correlation coefficient
 First task is to determine degrees of
freedom
 N-2
 N= number of pairs used to compute the
correlation coefficient
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14. Table B.4
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15. Table B.4 Instructions
 Compute the values of the correlation coefficient
 Compare the value of the correlation coefficient with
the critical value listed in Table B.4
 If the obtained value is greater than the critical value
or tabled value, the null hypothesis (that the
correlation coefficient is equal to 0) is not the most
attractive explanation for any observed differences
 If the obtained value is less than the critical or tabled
value, the null hypothesis is the most attractive
explanation for any observed differences
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16. Helpful Links
 Which statistical test to use
 http://www.ats.ucla.edu/stat/mult_pkg/whatstat/
 http://www.csun.edu/~amarenco/Fcs%20682/When%20to%20use%20w
hat%20test.pdf
 Sample Size
 http://www.danielsoper.com/statcalc/calculator.aspx?id=47
 http://www.surveysystem.com/sscalc.htm#two
 Sampling chapter p. 232
 Effect size
 http://psych.wisc.edu/henriques/power.html
 Reporting Results
 http://my.ilstu.edu/~jhkahn/apastats.html
 https://web2.uconn.edu/writingcenter/pdf/Reporting_Statistics.pdf 16