This document discusses bivariate statistics and statistical inference. It defines hypothesis testing and the different types of hypotheses. It explains statistical inference including p-values and types of errors. It then describes measuring the relationship between two variables through correlation, including calculating Pearson's r correlation coefficient, interpreting the strength of correlations, and how to report correlation results including significance levels. Scatterplots are discussed as a way to visually depict relationships between variables. Hypothesis testing is also applied to correlations between variables.

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
5

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
11
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.
12

13. 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 13