Experimental Design Examples in Math

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

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

Concept Recap

The deliberate planning of a study in which the researcher imposes treatments on subjects and measures responses, using control groups, randomization, replication, and (where possible) blinding to establish cause-and-effect relationships.

You want to know if a fertilizer helps plants grow. You can't just give it to some plants and hope for the bestā€”you need a plan: a group that gets the fertilizer, a group that doesn't (control), random assignment so the groups are fair, enough plants so one weird result doesn't fool you (replication), and ideally the person measuring growth doesn't know which group is which (blinding).

Read the full concept explanation ā†’

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: The four pillars of good experimental design are: (1) controlā€”compare treatment to a baseline, (2) randomizationā€”eliminate lurking variables, (3) replicationā€”use enough subjects to reduce chance variation, and (4) blindingā€”prevent bias from expectations.

Common stuck point: Students confuse the purpose of randomization (to create comparable groups) with the purpose of blinding (to prevent bias in measurement and response).

Worked Examples

Example 1

medium
Design an experiment to test whether caffeine improves test performance. Identify: explanatory variable, response variable, control group, treatment group, and how to control for confounders.

Solution

  1. 1
    Explanatory variable (factor): caffeine consumption (with/without)
  2. 2
    Response variable (outcome): test score
  3. 3
    Control group: no caffeine (placebo, e.g., decaf coffee that looks identical)
  4. 4
    Treatment group: caffeine dose
  5. 5
    Controlling confounders: random assignment to groups (balances sleep, stress, ability); double-blind design (neither participant nor tester knows which group); same test, same time of day, same environment

Answer

Explanatory: caffeine. Response: test score. Control for confounders via random assignment and double-blinding.
Good experimental design uses (1) comparison (control vs treatment), (2) randomization (eliminates confounders), (3) replication (multiple subjects per group), and often (4) blinding (prevents bias). The double-blind design is the gold standard.

Example 2

hard
A study wants to test three fertilizer types (A, B, C) on crop yield across 12 fields. Design a completely randomized design (CRD) and a randomized block design (RBD) if the 12 fields have 4 different soil quality levels.

Practice Problems

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

Example 1

easy
A teacher tests two teaching methods. She teaches Group A (morning) with Method 1 and Group B (afternoon) with Method 2. Identify the confounding variable and explain how to fix the design.

Example 2

hard
What is a placebo, and why is blinding important? Describe the placebo effect quantitatively: if 30% of placebo patients improve, what must a drug achieve to demonstrate its own effect?
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Background Knowledge

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

causationsampling biasvariability