Comparing 2 Means

Comparing Two Means (Part 1)

Ray Block, Jr.
PS #585 (Research Methods)
Fall 2003

Today’s Blueprint
Last Class: Correlation (Part 2)

Measuring Correlation (A Recap)
Interpreting Correlation (Contd.)

Today’s Class: Correlation (A Recap)

Comparing Two Means
Means Difference Tests

Correlation (A Recap)

Recap: Testing for Correlation:

Step #1: Arrange the scores of X and Y in a table listing the following:

The squares of X and Y (X2 and Y2)
The products of X and Y (XY)
The sums of X and Y (Sigma X and Sigma Y)
The sums of squares for X and Y (Sigma X squared and Sigma Y squared)
The sum of products of X and Y (Sigma XY)
The mean of X and Y (X-Bar and Y-Bar)

Step #2: Using the sums, calculate Pearson’s correlation:

Step #3: Find the degrees of freedom

The general formula is: df = N –K
For bivariate correlations: df = N –2

Step #4: Based on the information gained in the above three steps:

Compare Pearson’s r you obtain with the critical value from Table [see Appendix in most Sstatistics textbooks]
If the obtained value > critical value, reject H0
If the obtained value < critical value, retain H0

Comparing Two Means

The Big Picture:

For this class, we will move from bivariate correlation analysis to bivariate experimental analysis
The goal is to examine the relationship between two variables, specifically:

How to determine if 2 variables are related
(If variables relate), how are they related
How to make predictions from one variable to another

Correlational Analysis:

Correlation looks at the relationship between 2 variables
It tells us whether 2 variables share something in common with each other (relatedness)
If they share something in common, then the variables are correlated (co-related) with one another
While correlation is about examining relationships between variables, experimental analysis is about making comparisons between variables
People use these comparisons to infer causality
…here’s how

Experimental Analysis:

Making comparisons means looking at the difference between two variables
It tells us whether 2 variables share nothing in common with each other
If they share nothing in common, then the variables are independent of one another

Performing an Experiment:

The goal is to systematically change (manipulate) the variable to observe its effect on another variable

Example: manipulating the independent variable (X) so see its effect on the dependent variable (Y)

A Simple Experimental Design Includes:

One dependent variable
One independent variable (manipulated 2 ways)

Treatment group
Control group

Experimental Analysis:

Between-Subjects Design: Manipulate one of the variables using different subjects

Assigning different subjects to different experimental conditions

Within-Subjects Design: Manipulate one of the variables using the same subjects

Give the same group of subjects to all the experimental conditions

[In Class Examples]

These examples show that the treatment causes the difference between the treatment and control groups
However, is this difference significant (meaningful)?
Use a t-Test to determine whether the difference between groups is significant

What are t-Tests:

They are tests of significance
They determine whether there is a significant difference between the mean values of two groups
Two Types (which one you use depends on whether you are testing the difference between related or unrelated groups:

T-Tests for Independent Means = Tests difference between the means of 2 unrelated groups
T-Tests for Dependent Means = Tests difference between the means of 2 related groups

T-Test for Independent Means

Asks the question: “Is there a [significant] difference between the means of two unrelated groups?

Unrelated Groups = When there are two experimental conditions and different subjects are assigned to each condition (Between-Subject design)

T-Test for Dependent Means

Asks the question: “Is there a difference between the means of two related groups

Related Groups= when there are two experimental conditions and the same subjects took part in both conditions of the experiment (Within-Subject design)

The logic behind t-Tests

Two samples of data are collected (in this case, samples for both groups)

You calculate sample means for each group

These means might differ a little or a lot

If the samples come from the same population, then we expect their means to be roughly equal (or at least very similar)

If this is true, then any differences in the means are due to the treatment

When testing hypotheses, the null would be that the manipulation had no effect on the subjects
This means that any differences between means were due to chance and not the treatment
To conduct this hypothesis test, we would compare the difference between the sample means we collected and the difference between the sample means that we would expect to obtain by chance
If the difference between the sample means we collect is larger than the difference in sample means that we would expect by chance, then:

Either our sample is somehow not representative of the population we were trying to tap
Or the two samples we collected come from different populations

The larger the difference between the obtained sample means and the expected sample means, the more confident we can be in rejecting the null hypothesis
If we can reject the null, then we are essentially saying that the two sample means differ because because of the manipulation

References (FYI):

Levin, Jack and James Alan Fox. 2003. Elementary Statistics in Social Research, 9th Edition. Boston, MA: Pearson Education Group, Inc.
Salkind, Neil J. 2003. Exploring Research, 5th Edition. Upper Saddle River, NJ: Prentice Hall.
Kranzler, John H. 2003. Statistics for the Terrified, 3rd Edition. Upper Saddle River, NJ: Prentice Hall.

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