You are reading the 2026 curriculum (exams up to November 2026). Sitting in February 2027 or later? Go to the 2027 Quantitative Methods guide.
Parametric vs nonparametric tests
Parametric tests are about a parameter (a mean, a variance) and assume a distribution, usually normal. Nonparametric tests make few assumptions and often work on ranks or signs. Use them when the data break the assumptions, when outliers dominate, when the data are ranks, or when the question is not about a parameter.
Before this: How a hypothesis test works: null, critical value, p-value
On the exam
- Compare and contrast parametric and nonparametric tests, and describe situations where each is the more appropriate type of test
Compare and contrast parametric and nonparametric tests, and describe when each is more appropriate. Expect the four situations, the matching alternative for a given t-test, and the cost of switching when assumptions hold.
How deeply to learn each part
- Know the idea
- Understand it in general. It frames other topics, but you will not calculate with it.
- Use it
- Understand it well enough to solve a question with it. Recognize when it applies.
- Memorize
- Learn it by heart. The exam will not give it to you.
You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.
You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.
What makes a test parametric?
A parametric test has two features: it is about a parameter (a mean, a variance), and its validity rests on assumptions about the population, such as normality. Every test in this module so far is parametric.
A nonparametric test either does not concern a parameter or makes minimal assumptions about the population. Many of them work on ranks (the order of the observations) or on signs (above or below a value), not on the raw numbers.
When should you switch?
Memorize
There are four situations.
- Assumptions fail Memorize small sample, clearly non-normal data
- Outliers Memorize extremes would swamp a mean-based test
- Ranked or ordinal data Memorize fund rankings, credit grades
- Question is not about a parameter Memorize is this sequence random? does it follow this distribution?
- Assumptions fail. A small sample from a clearly non-normal population leaves no valid t-test.
- Outliers. One extreme month can drag a mean, so a test based on ranks or the median is steadier.
- Ranks or ordinal data. Fund rankings or credit grades carry order but not size, so means are meaningless.
- The question is not about a parameter. “Is this sequence of up and down days random?” is answered by a runs test; “does this sample come from a given distribution?” is also nonparametric.
The cost: when the parametric assumptions do hold, the parametric test is more powerful. Switching without reason throws information away. Reports often show both, so you can see how sensitive the conclusion is to the assumptions.
Sort each scenario into its bucket, then name the specific test. Notice how often the trigger is one word: “ranks”, “outlier”, “random”.
Which test replaces which?
Memorize
- Single mean Memorize
- Parametric: t-test (or z-test) Memorize
- Nonparametric: Wilcoxon signed-rank test Memorize
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- Difference between two independent means Memorize
- Parametric: t-test (pooled) Memorize
- Nonparametric: Mann–Whitney U test (Wilcoxon rank-sum test) Memorize
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- Mean of paired differences Memorize
- Parametric: paired comparisons t-test Memorize
- Nonparametric: Wilcoxon signed-rank test or sign test Memorize
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The Wilcoxon signed-rank test ranks the sizes of the differences from the hypothesized value (or the paired differences), keeps their signs, and compares the sums of positive and negative ranks. The Mann–Whitney U test (also called the Wilcoxon rank-sum test) ranks all observations from two independent samples together and compares the rank sums. The sign test uses only whether each paired difference is positive or negative.
| Question | Parametric | Nonparametric |
|---|---|---|
| Single mean | t-test (or z) | Wilcoxon signed-rank test |
| Two independent means | t-test (pooled) | Mann–Whitney U test |
| Paired differences | paired t-test | Wilcoxon signed-rank test; sign test |
| Correlation | t-test of r | Spearman rank correlation (next module) |
Exam tip Paired or single sample: signed-rank. Two independent samples: Mann–Whitney. Only signs available: sign test.
Go deeper How a sign test works
Take 14 paired months and count how many show fund A ahead of fund B. If there is no difference, each month is like a fair coin toss, so the count of positive signs follows a binomial distribution with p = 0.5. Twelve or more heads out of 14 happen less than 1% of the time with a fair coin, so a count that high is strong evidence that fund A really does better. The size of each gap is ignored, which makes the test robust to outliers but weaker than the signed-rank test.
Go deeper Why ranking loses information
Replace 0.1%, 0.2% and 9.0% with ranks 1, 2 and 3 and the huge size of the last observation vanishes. That protects you from an outlier, and it also hides genuine large effects. So a nonparametric test is a safeguard, not an upgrade.
Memorize this
MemorizeParametric and nonparametric pairs
| Question | Parametric | Nonparametric |
|---|---|---|
| Single mean | t-test (or z) | Wilcoxon signed-rank |
| Two independent means | t-test (pooled) | Mann–Whitney U (Wilcoxon rank-sum) |
| Paired differences | paired t-test | Wilcoxon signed-rank; sign test |
Memory hook One signed-rank test serves two jobs (a single sample and paired differences). Two independent samples are the only ones that need the rank-sum / Mann–Whitney test.
Know it cold Nonparametric = few assumptions, but less power when the parametric assumptions do hold.
MemorizeFour situations for a nonparametric test
| # | Situation |
|---|---|
| 1 | Distributional assumptions fail |
| 2 | Outliers |
| 3 | Ranks or ordinal data |
| 4 | Hypothesis is not about a parameter |
Memory hook Break, Bump, Rank, Beyond: assumptions that break, outliers that bump the mean, data that are ranks, questions beyond parameters.
Know it cold If the assumptions are met, prefer the parametric test (more powerful).
- Assumptions fail Memorize small sample, clearly non-normal data
- Outliers Memorize extremes would swamp a mean-based test
- Ranked or ordinal data Memorize fund rankings, credit grades
- Question is not about a parameter Memorize is this sequence random? does it follow this distribution?
- Single mean Memorize
- Parametric: t-test (or z-test) Memorize
- Nonparametric: Wilcoxon signed-rank test Memorize
-
- Difference between two independent means Memorize
- Parametric: t-test (pooled) Memorize
- Nonparametric: Mann–Whitney U test (Wilcoxon rank-sum test) Memorize
-
- Mean of paired differences Memorize
- Parametric: paired comparisons t-test Memorize
- Nonparametric: Wilcoxon signed-rank test or sign test Memorize
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How it is asked
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Ranks only
The analyst has only rankings (1 to 15) of managers from two consultants. Test of association?
Nonparametric, rank-based. Trap: a parametric t-test of the correlation.
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Match the alternative
The nonparametric alternative to a t-test of the difference between two independent means?
Mann–Whitney U test. Traps: sign test (paired), chi-square (variance).
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Cost of switching
The parametric assumptions fully hold, but a nonparametric test is used. Effect?
Lower power. Trap: "more power because fewer assumptions".
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Not a parameter
Are daily up and down moves random?
Nonparametric (a runs test): the question is about a pattern, not a mean or variance.
Exam traps
- Thinking nonparametric tests are always better because they assume less. When the assumptions hold, they have less power.
- Using a parametric test on rank data. Ranks do not support means and variances.
- Pairing the wrong nonparametric test with the wrong parametric one: paired data use signed-rank or sign; independent samples use Mann–Whitney U.
Checkpoint
4 exam-style questions. About 90 seconds each.
1An analyst has only the rankings (1 to 15) of fund managers by two consultants. A test of their association should most appropriately be:
2The nonparametric alternative to a t-test of the difference between two independent means is the:
3When the assumptions of a parametric test are fully met, using a nonparametric test instead will most likely:
4To check whether a sequence of daily price changes is random, a nonparametric test is most appropriate because:
Key takeaways
- Parametric: about parameters, with a distribution assumed.
- Nonparametric: few assumptions; works on ranks and signs.
- Four triggers: broken assumptions, outliers, rank data, questions that are not about a parameter.
- If the parametric assumptions hold, the parametric test is more powerful.
Questions about this lesson?
Where this comes back
- Parametric and Non-Parametric Tests of Independence Quantitative Methods
- Market Efficiency Equity Investments
Questions candidates ask
What is the difference between parametric and nonparametric tests?
A parametric test is about a parameter, such as a mean or variance, and assumes a distribution, usually normal. A nonparametric test is either not about a parameter or makes few assumptions about the population, and often uses ranks or signs instead of the raw values.
When should a nonparametric test be used?
Use one when the data break the assumptions of the parametric test, when outliers would dominate, when the data are ranks or ordinal, or when the question is not about a parameter. When the assumptions hold, the parametric test is preferred because it is more powerful.
What is the Wilcoxon signed-rank test used for?
It is the nonparametric alternative for a single mean and for the mean of paired differences. It ranks the sizes of the differences from the hypothesized value, keeps their signs, and compares the sums of positive and negative ranks.
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