Difference between parametric and nonparametric test
Parametric tests assume the data follows a specific distribution (usually normal), require interval/ratio data, and use parameters like mean and variance. Nonparametric tests make no such distributional assumptions, work on ordinal or nominal data, and are based on ranks or counts. Parametric tests are generally more statistically powerful when assumptions are met; nonparametric tests are more flexible but less powerful.
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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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