Comparing Two Groups: Means and Proportions

Manual: General · Subject: Statistics

Apply inferential methods to detect differences between groups in real-world settings.

Two-Sample Inference

Comparing groups

Many practical questions ask whether two populations differ in mean outcome or proportion of success. The inferential strategy depends on whether the response is numerical or categorical.

Two-group methods

Question	Typical parameter	Typical method
Difference in average test scores	$\mu_1-\mu_2$	Two-sample $t$ procedures
Difference in success rates	$p_1-p_2$	Two-proportion z procedures
Before/after change in same units	Mean of paired differences	Paired $t$ procedures

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Paired data

Use a paired method when measurements come from matched units or the same subject measured twice, because observations are not independent across groups.

Independent vs paired designs

Independent samples

Different units in each group
Compare two separate populations
Use two-sample methods

Paired samples

Same units measured twice or matched pairs
Analyze differences within pairs
Usually more powerful when pairing is meaningful

Choosing the right two-group test

1

Step 1: Identify whether the response is quantitative or categorical.
2

Step 2: Decide whether the data are independent or paired.
3

Step 3: Form the parameter of interest, such as $\mu_1-\mu_2$ or $p_1-p_2$ .
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Step 4: Use the appropriate confidence interval or hypothesis test.

Which situation calls for a paired test?

Comparing average heights of two unrelated classes Comparing the same patients before and after treatment Comparing two different cities' populations Comparing two unrelated survey proportions

What is the main advantage of pairing when it is appropriate?

Neuroanatomy Explorer

Memory Deck

Match the Pairs

Concept Constellation

Comparing Two Groups: Means and Proportions

Two-Sample Inference

Comparing groups

Independent vs paired designs

Choosing the right two-group test