Difference between parametric and nonparametric test
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parametric vs nonparametric tests statistics differences
Here is a thorough comparison:
Parametric vs. Nonparametric Tests
Core Definitions
| Feature | Parametric | Nonparametric |
|---|---|---|
| Distributional assumption | Requires data to follow a known distribution (usually normal) | No distribution assumed ("distribution-free") |
| Data type | Interval or ratio (continuous) | Ordinal, nominal, or non-normal continuous |
| Central measure | Mean | Median or ranks |
| Statistical power | Higher (when assumptions met) | Lower for same sample size |
| Sample size | Works well with moderate to large samples | Better suited for small samples |
| Sensitivity to outliers | High - outliers can distort results | Low - rank-based methods are resistant |
Key Assumptions for Parametric Tests
- Normality - the data (or residuals) are normally distributed
- Homogeneity of variance - groups have roughly equal variances (homoscedasticity)
- Independence - observations are independent of each other
- Scale - data must be at least interval-level
If any of these assumptions are violated, parametric results can be misleading.
Common Test Pairs (Parametric → Nonparametric Equivalent)
| Parametric Test | Nonparametric Equivalent | Purpose |
|---|---|---|
| One-sample t-test | Wilcoxon signed-rank / Sign test | Test if a sample mean/median = a value |
| Independent samples t-test | Mann-Whitney U test | Compare two independent groups |
| Paired t-test | Wilcoxon signed-rank test | Compare two related/paired groups |
| One-way ANOVA | Kruskal-Wallis test | Compare 3+ independent groups |
| Repeated measures ANOVA | Friedman test | Compare 3+ related groups |
| Pearson correlation | Spearman / Kendall correlation | Measure association between variables |
When to Use Each
Use parametric tests when:
- Data is normally distributed (or sample size is large enough for the Central Limit Theorem to apply, typically n > 30)
- You have interval or ratio data
- Variances are roughly equal across groups
- You want maximum statistical power
Use nonparametric tests when:
- Data is clearly non-normal and sample size is small
- Data is ordinal (e.g., Likert scales, rankings)
- There are outliers you cannot justify removing
- The median is more meaningful than the mean for your domain (e.g., income data, survival times)
Power Trade-off
Parametric tests are generally more powerful - they detect a true effect with a smaller sample size. The cost of using a nonparametric test when parametric assumptions ARE met is roughly a 5-15% reduction in efficiency. However, if you use a parametric test when assumptions are violated, you risk Type I errors (false positives) or completely invalid results.
A useful rule of thumb from Einstein Medical: if your assumptions are met, use parametric - the greater power is worth it. When in doubt with small samples and non-normal data, default to nonparametric.
Summary in One Line
Parametric tests are more powerful but demanding; nonparametric tests are more flexible but require larger samples to achieve the same confidence.
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