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 best—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).

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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: An experiment deliberately assigns treatments using control, randomization, replication, and blinding so a difference can be pinned on the treatment.

Common stuck point: The procedure for experimental design is the easy part; the trap is skipping random assignment but still claiming causation. Asking "Does the researcher actively assign subjects to treatments (rather than just observe what they already do)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the researcher actively assign subjects to treatments (rather than just observe what they already do)?

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.

Answer

Explanatory: caffeine. Response: test score. Control for confounders via random assignment and double-blinding.

First step

1
Explanatory variable (factor): caffeine consumption (with/without)

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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.

Example 3

medium
A teacher tests a new study app on 60 students. She randomly assigns 30 to use the app and 30 to study without it; both groups take the same exam. Identify the explanatory variable, response variable, and one strength of this design.

Example 4

medium
A study has 8 fields with two soil types (4 sandy + 4 clay). The researcher tests fertilizers A and B. Explain how to use blocking by soil type.

Example 5

medium
A trial compares 4 medications across 200 patients, with a balanced CRD. How many per medication? How can the researcher randomly assign them using only a coin?

Example 6

hard
A researcher wants to test 2 fertilizers (A, B) and 3 watering levels (low, medium, high) on plant yield. How many treatment combinations? If each gets 5 plants, total plants?

Example 7

hard
In a matched-pairs design, two subjects with similar characteristics are paired, and within each pair one is randomly assigned to treatment, the other to control. How does this differ from a CRD?

Example 8

hard
A study randomly assigns 80 cars to two engine oils (40 each) and measures wear after 10,000 miles. What is one extra design step that would let the researcher block by car model?

Example 9

challenge
A school wants to test if a new lunch menu improves attendance. Two schools are picked: School A switches menus; School B does not. Why is this NOT a true experiment, and how would you fix it?

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?

Example 3

easy
In an experiment, what is the group that receives no treatment (or a placebo) called?

Example 4

easy
What design feature uses chance to assign subjects to treatment groups?

Example 5

easy
Repeating the experiment on many subjects (not just one) is which principle?

Example 6

easy
A study where neither the subjects nor the researchers know who got the treatment is called what?

Example 7

easy
True or false: an experiment without a control group can still clearly establish that the treatment caused the effect.

Example 8

easy
A drug trial gives one group the real pill and another an identical sugar pill. What is the sugar pill called?

Example 9

easy
What do we call a variable, like blocking by age group, used to reduce variability before assigning treatments?

Example 10

easy
The variable the researcher deliberately manipulates is called the what?

Example 11

medium
A fertilizer study gives half the plots fertilizer and half none, assigned by coin flip, with 5050 plots each. Name the three core design principles present.

Example 12

medium
In a clinical trial, the doctors recording outcomes do not know who received the drug, but patients do. Is this single- or double-blind?

Example 13

medium
A teacher gives a new method to her morning class and the old method to her afternoon class, then compares scores. Why might this design be confounded?

Example 14

medium
Subjects are split into age blocks (under-40, over-40), then randomly assigned to treatment or control within each block. What design is this?

Example 15

medium
A study claims its new app improves sleep but had no control group and no blinding. Name two design flaws.

Example 16

medium
A researcher selects participants randomly from a city (random sampling) but gives everyone the same treatment with no comparison group. Is this a well-designed experiment?

Example 17

medium
In a placebo-controlled trial, 30%30\% of the placebo group also reports improvement. What does this illustrate?

Example 18

challenge
Explain why random assignment, not random sampling, is what allows an experiment to establish causation.

Example 19

challenge
A trial finds the drug group improved more than placebo, but the drug group was younger on average. What likely went wrong, and what design fix prevents it?

Example 20

challenge
Why is a double-blind, placebo-controlled, randomized design considered the 'gold standard' for testing a new drug? Name what each feature controls.

Example 21

medium
A trial gives one group a drug and another a placebo, assigned randomly, with neither patients nor doctors knowing the assignment. Name the design type fully.

Example 22

medium
A study compares a treatment given to volunteers against records of people who declined. Why is this not a true experiment despite having two groups?

Example 23

easy
A scientist tests if music improves plant growth. What is the response variable?

Example 24

easy
A clinical trial gives one group a pill and another an identical sugar pill. What is the sugar pill called?

Example 25

easy
A researcher does NOT know which subjects got the real drug; subjects don't know either. What is this called?

Example 26

medium
In an experiment with 120 subjects and 3 treatments (A, B, C), how many should go to each group in a balanced completely randomized design?

Example 27

medium
Why is a control group typically necessary for an experiment?

Example 28

medium
Replication helps reduce what kind of variability in an experiment?

Example 29

medium
A factor in an experiment has 3 levels (low, medium, high dose). Including a placebo as a 4th level, what is the new total number of groups?

Example 30

hard
A medical trial has a strong placebo effect: 25% of placebo patients improve. The drug shows 30% improvement. Should the FDA conclude the drug is effective?

Example 31

hard
A trial randomly assigns 50 patients to drug, 50 to placebo. The drug group improves 60%, placebo 40%. Why is this NOT proof the drug caused improvement without further analysis?

Example 32

medium
Why is using volunteers (e.g., self-selected subjects) generally a weak experimental design?
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Background Knowledge

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

causationsampling biasvariability