Parametric vs. Non-parametric#

The distinction between parametric and non-parametric tests is one of the most fundamental decisions when choosing the right statistical method.

What Does "Parametric" Mean?#

Parametric tests make assumptions about the distribution of data in the population. They estimate parameters such as means and variances and typically require:

Data are measured on a metric scale
Data are (approximately) normally distributed
Homogeneity of variance across groups

Non-parametric tests make no or less strict assumptions about the distribution. They often work with ranks rather than raw values.

Comparison#

Property	Parametric	Non-parametric
Distribution assumption	Normal distribution	None/minimal
Level of measurement	Metric (interval/ratio)	Ordinal or metric
Analyzes	Means, variances	Ranks, medians
Power	Higher (when assumptions met)	Lower
Sample size	Can work with small n	Also suitable for small n
Robustness	Sensitive to violations	Robust against outliers

Parametric Tests and Their Non-parametric Alternatives#

Research question	Parametric	Non-parametric
2 independent groups	t-test	Mann-Whitney U test
2 related groups	Paired t-test	Wilcoxon signed-rank test
3+ independent groups	One-way ANOVA	Kruskal-Wallis test
3+ related groups	Repeated-measures ANOVA	Friedman test
Association (2 variables)	Pearson correlation	Spearman rank correlation

When to Use Parametric Tests#

Use parametric tests when:

The dependent variable is measured on a metric scale
The data are approximately normally distributed (or n is large enough)
The variances across groups are similar
You need maximum power

Example: Parametric appropriate

A pharmaceutical company tests a new drug. The dependent variable is blood pressure (metric, in mmHg). Both groups (n = 50) show normally distributed data and similar variances.

→ Independent samples t-test is the right choice.

When to Use Non-parametric Tests#

Use non-parametric tests when:

The data are measured on an ordinal scale (e.g., individual Likert items)
Normality is clearly violated and the sample is small
There are strong outliers that cannot be removed
The distribution is heavily skewed
The sample size is very small (n < 15 per group)

Example: Non-parametric appropriate

A researcher examines service satisfaction on a 5-point scale (1 = very dissatisfied to 5 = very satisfied). The data are heavily right-skewed, and the sample includes only n = 12 per group.

→ Mann-Whitney U test is a better choice than the t-test.

How Do Rank Tests Work?#

Non-parametric tests often convert raw data into ranks:

Ranking principle

Original values: 12, 5, 28, 3, 19

Sorted: 3, 5, 12, 19, 28

Ranks: 1, 2, 3, 4, 5

The test then works with ranks instead of original values. This diminishes the impact of outliers and makes the distribution shape irrelevant.

The Power Trade-off#

Non-parametric tests have less power than their parametric counterparts when data are normally distributed. The asymptotic relative efficiency (ARE) indicates what percentage of power is retained:

Non-parametric test	ARE under normality
Mann-Whitney vs. t-test	~95.5%
Wilcoxon vs. paired t-test	~95.5%
Kruskal-Wallis vs. ANOVA	~95.5%
Spearman vs. Pearson	~91.2%

The power loss is often surprisingly small. With non-normal data, non-parametric tests can even have more power.

The Gray Area#

In practice, many situations are ambiguous:

Sum scores from Likert scales: Often treated as quasi-metric. With ≥ 5 items and approximate normality, parametric tests are generally justifiable.

Large samples with mild non-normality: The central limit theorem ensures that parametric tests are robust. With n ≥ 30 per group, mild deviations are usually unproblematic.

Small samples with normality: Parametric tests are appropriate here, even though the Shapiro-Wilk test has low power with small n.

Decision Guide#

Check level of measurement → Ordinal? → Non-parametric
Check normality → Clearly violated with small n? → Non-parametric
Check for outliers → Extreme outliers present? → Non-parametric
When in doubt: Run both tests. If the results agree, the choice is less critical.

Common Misconceptions#

"Non-parametric tests are always better because they have fewer assumptions." No. When the assumptions for parametric tests are met, parametric tests have more power. "Safer" does not mean "better."

"Non-parametric tests have no assumptions at all." Wrong. They also require independence. For comparing medians, the distribution shapes must be similar.

"When normality is violated, I must always use a non-parametric test." Not with large samples. The t-test is surprisingly robust against non-normality when n > 30 per group.

Parametric vs. Non-parametric