Comparative Statistics Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Comparative Statistics.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Comparative statistics involves using statistical measures to compare two or more groups, data sets, or distributions.

Is A bigger/better/different than B? By how much? Is the difference real?

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Comparison requires both size of difference and variability context.

Common stuck point: Statistical significance \neq practical importance. A 'significant' difference can be tiny.

Worked Examples

Example 1

easy
Compare two groups on a test: Group A (n=30): mean=75, SD=10. Group B (n=30): mean=82, SD=8. Calculate the difference in means and comment on whether group B performs better.

Solution

  1. 1
    Difference in means: \bar{x}_B - \bar{x}_A = 82 - 75 = 7 points
  2. 2
    Group B's mean is 7 points higher
  3. 3
    But consider variability: SD_A=10 and SD_B=8; the groups overlap substantially
  4. 4
    Cohen's d (effect size): d = \frac{7}{\sqrt{(10^2+8^2)/2}} = \frac{7}{\sqrt{82}} \approx \frac{7}{9.06} \approx 0.77 — medium-large effect

Answer

Group B scores 7 points higher on average with Cohen's d ≈ 0.77 (medium-large effect).
Comparing groups requires reporting both the difference in centers and the effect size (like Cohen's d). A 7-point difference might be meaningful or trivial depending on the variability. Effect size standardizes the difference for meaningful interpretation.

Example 2

medium
Three products have customer satisfaction scores. Product A: mean=4.2, SD=0.5 (n=50). Product B: mean=3.8, SD=1.2 (n=50). Explain which product is preferable and why SD matters.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Box plots for two classes show: Class A median=70, IQR=20. Class B median=75, IQR=5. Which class has better performance? Which is more consistent?

Example 2

hard
Men: mean height = 70", SD = 3". Women: mean height = 64", SD = 2.5". A person is 67" tall. Calculate their z-score in each distribution and determine which group they are more extreme in.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

meanstandard deviation