Nonparametric, Robust, and Resampling Methods

Introduction and Context

Many statistical procedures taught in introductory courses rely on assumptions about a population distribution. Common examples include normality, equal variances, independence, and measurement on an interval or ratio scale. These assumptions can be reasonable, but they are not guaranteed in practical data. Financial returns may contain extreme observations, biomedical measurements may be skewed, survey responses may be ordinal, and machine-learning errors may be dependent or heavy-tailed.

Nonparametric, robust, and resampling methods provide alternatives. Nonparametric methods make relatively weak assumptions about the form of a population distribution. Robust methods reduce the influence of unusual observations or remain reliable under moderate departures from model assumptions. Resampling methods use the observed sample, repeatedly reorganized or sampled, to approximate a sampling distribution.

These approaches are not assumption-free. They replace strong parametric assumptions with other requirements, such as independence, exchangeability, symmetry, or an appropriate sampling design. Good statistical practice therefore requires understanding both what a method assumes and what question it answers.

Learning Objectives

After studying this chapter, you should be able to:

1. Explain why nonparametric, robust, permutation, and bootstrap methods are useful.

2. Apply and interpret rank-based tests for one-sample, paired, and independent-sample problems.

3. Conduct goodness-of-fit and independence analyses using chi-square procedures.

4. Distinguish permutation tests from bootstrap methods.

5. Construct and interpret bootstrap confidence intervals.

6. Calculate and compare robust measures of location and spread.

7. Select an appropriate distribution-free method for an applied problem.

8. Evaluate limitations, computational issues, and ethical implications in data analysis.

1. The Logic of Distribution-Free Inference

A parametric method specifies a family of distributions using a small number of parameters. For example, a one-sample t procedure is based on a population mean and variance and uses a reference distribution derived under normality. A nonparametric method generally avoids specifying a particular population family. It may use ranks, signs, counts, or the observed data as the basis of inference.

The phrase distribution-free is more precise than nonparametric. A distribution-free procedure has a sampling distribution that does not depend on an unknown population distribution under its stated assumptions. Some nonparametric procedures are distribution-free only under particular conditions, such as continuity or symmetry.

Three ideas are central:

• Ranks: Replace numerical values by their order positions. This reduces sensitivity to scale and extreme values.
• Randomization: Generate a reference distribution by rearranging labels or signs in ways justified by the null hypothesis.
• Resampling: Repeatedly draw from the observed sample to approximate variation in a statistic.

The reduction of information caused by ranking can improve reliability but may reduce power when a parametric model is correct. Thus, “nonparametric” does not mean “always better.” It means that the method is suitable when its assumptions and target question fit the data.

2. Rank-Based Methods

2.1 Ranks and Their Interpretation

Suppose observations are ordered from smallest to largest. The smallest receives rank 1, the next rank 2, and so on. If tied observations occur, each tied value is assigned the average of the ranks it would occupy. For example, values 2, 4, 4, 9 have ranks 1, 2.5, 2.5, and 4.

Rank procedures test statements about distributions, locations, or paired differences rather than necessarily testing means. Interpretation must therefore match the procedure. A significant Mann–Whitney result, for example, does not automatically prove that two population means differ.

2.2 Sign Test

The sign test is used for a one-sample median or for paired data. For paired observations, calculate differences d_i = X_i − Y_i and discard zero differences. Under the null hypothesis that the median difference is zero, positive and negative differences are equally likely, so the number of positive differences follows a binomial distribution:

S ~ Binomial(n, 0.5),

where n is the number of nonzero differences. An exact two-sided p-value can be obtained from the binomial distribution. The sign test requires independent pairs and is valid for ordinal data, but it ignores the magnitude of differences and may have lower power than alternatives.

2.3 Wilcoxon Signed-Rank Test

The Wilcoxon signed-rank test uses both the signs and the absolute magnitudes of nonzero paired differences. Its usual assumptions are independent pairs, a continuous difference distribution, and approximate symmetry of that distribution when the test is interpreted as a test of a median difference.

Procedure:

1. Compute each difference d_i.

2. Remove zero differences.

3. Rank |d_i|, using average ranks for ties.

4. Restore the signs.

5. Sum positive ranks, T+, and negative ranks, T−.

6. Use the smaller rank sum, or an equivalent statistic, for inference.

For larger samples, a normal approximation may be used. If W is the sum of positive ranks and there are n nonzero differences without ties, then

E(W) = n(n + 1)/4,

Var(W) = n(n + 1)(2n + 1)/24.

A standardized statistic is approximately

Z = [W − E(W)] / sqrt[Var(W)],

with a continuity correction often applied. Ties and zero differences require adjusted variance calculations or software that uses exact or corrected methods.

2.4 Mann–Whitney U Test

The Mann–Whitney U test, also called the Wilcoxon rank-sum test, compares two independent groups. Combine all observations, rank them, and calculate the rank sum for one group. If group 1 has n1 observations and rank sum R1, then

U1 = R1 − n1(n1 + 1)/2.

Similarly,

U2 = n1n2 − U1,

where n2 is the size of group 2. The smaller U is often used for a two-sided test.

The test assumes independent observations, comparable measurement processes, and (for a simple location interpretation) similarly shaped distributions. If the two distributions have similar shape and spread, a significant result can be interpreted as evidence of a location difference. Without that condition, it is more accurate to say that the distributions differ in stochastic ordering or overall placement.

For larger samples, the mean and variance of U, without ties, are

E(U) = n1n2/2,

Var(U) = n1n2(n1 + n2 + 1)/12.

Ties require a variance correction. Exact methods are preferable for small samples when computationally feasible.

2.5 Kruskal–Wallis Test

The Kruskal–Wallis test extends rank-sum reasoning to k independent groups. Rank all N observations together. If R_j is the rank sum for group j and n_j is its sample size, the statistic is

H = [12/(N(N + 1))] Σ(R_j²/n_j) − 3(N + 1).

Under the null hypothesis that all groups come from the same distribution, H is approximately chi-square with k − 1 degrees of freedom for sufficiently large samples. A tie correction is needed when tied ranks are present.

A significant omnibus result indicates that at least one group differs, but it does not identify which groups. Post-hoc pairwise comparisons should use an appropriate multiplicity adjustment, such as Holm’s procedure. If group distributions differ in shape, interpretation should concern distributional differences rather than medians alone.

2.6 Friedman Test

The Friedman test is a rank-based alternative for k related treatments or repeated measurements. Rank the observations within each subject or block. If R_j is the sum of ranks for treatment j, with n blocks and k treatments, the statistic is

Q = [12/(nk(k + 1))] ΣR_j² − 3n(k + 1).

For moderate or large n, Q is approximately chi-square with k − 1 degrees of freedom. The method assumes independent blocks and a meaningful within-block ranking. It is useful when repeated measurements are not plausibly normal or when the outcome is ordinal.

3. Goodness-of-Fit and Categorical Data

3.1 Chi-Square Goodness-of-Fit Test

A goodness-of-fit test compares observed category counts O_i with expected counts E_i under a specified probability model. The Pearson statistic is

χ² = Σ[(O_i − E_i)²/E_i].

If probabilities are fully specified in advance and there are k categories, the approximate reference distribution has k − 1 degrees of freedom. If parameters are estimated from the data, the degrees of freedom are reduced by the number of independently estimated parameters.

Expected counts should generally be sufficiently large for the chi-square approximation to be reliable. A common practical guideline is that expected counts should not be very small, although exact thresholds depend on context and software. Sparse tables may require combining substantively similar categories or using an exact or simulation-based procedure.

3.2 Test of Independence

For a two-way table, the expected count in cell (i, j) under independence is

E_ij = (row total_i × column total_j) / grand total.

The same Pearson statistic, summed over all cells, tests independence. With r rows and c columns, degrees of freedom are (r − 1)(c − 1), provided the usual approximation is appropriate.

A significant result indicates association, not causation. Effect-size measures such as Cramér’s V help describe practical importance. For a table with r rows, c columns, and sample size N,

V = sqrt[χ² / (N × min(r − 1, c − 1))].

The statistic should be interpreted alongside the table, conditional percentages, study design, and substantive context.

4. Permutation Tests

A permutation test constructs a null distribution by rearranging observations or labels in all or many ways that are equally plausible under the null hypothesis. The observed statistic is compared with this reference distribution.

For two independent groups under a null hypothesis of no group effect, the group labels are exchangeable. A difference in means, medians, or another statistic is calculated for the observed labels. Labels are then permuted repeatedly, and the statistic is recalculated. For a two-sided test, a simulation p-value may be estimated by

p̂ = (1 + number of permuted statistics at least as extreme as the observed statistic) / (B + 1),

where B is the number of random permutations. The “+1” correction avoids reporting a zero p-value and aligns the simulation procedure with a conservative finite-randomization calculation.

Permutation tests are exact when all admissible permutations are enumerated and the exchangeability assumptions hold. Randomly sampling permutations produces a Monte Carlo approximation. A large B reduces simulation error but does not repair a flawed null model.

Exchangeability is crucial. Paired data require permutations within pairs, often by randomly switching the two values or randomly assigning a plus or minus sign. Clustered or time-series observations generally cannot be freely permuted because dependence destroys exchangeability. Restricted permutations, block permutations, or model-based methods may be required.

5. Bootstrap Inference

The bootstrap estimates the sampling distribution of a statistic by repeatedly sampling with replacement from the observed data. If the sample is x1, …, xn, one bootstrap sample consists of n draws from these observations, with replacement. For each bootstrap sample, calculate the statistic T*. Repeating this B times gives an empirical bootstrap distribution.

The bootstrap standard error is

SE_boot(T) = sqrt[Σ(T_b − T̄)²/(B − 1)],

where T̄* is the mean of the B bootstrap statistics. A simple percentile confidence interval uses the α/2 and 1 − α/2 quantiles of the bootstrap distribution:

[Q_(α/2), Q_(1−α/2)].

For a 95% percentile interval, α = 0.05, so the 2.5th and 97.5th percentiles are used.

Other intervals include the basic interval, bias-corrected and accelerated (BCa) interval, and bootstrap-t interval. BCa intervals can account for bias and skewness but require additional calculations. Percentile intervals are easy to explain, yet they may perform poorly when the statistic is strongly biased, the sample is very small, or the bootstrap distribution is highly irregular.

Worked Example: Bootstrap Median Interval

Suppose a sample of response times, in milliseconds, is [118, 121, 123, 125, 130, 132, 160]. The sample median is 125 ms. Because the sample is small and right-skewed, a normal-theory confidence interval for the mean may be unsuitable, and the median is a sensible robust summary.

A bootstrap analysis would repeatedly draw seven values with replacement from this list and calculate the median of each resample. For example, one resample might be [121, 121, 125, 130, 132, 160, 160], with median 130. After many repetitions, the empirical distribution of medians can be used to calculate a percentile or BCa interval.

This example illustrates the method rather than reporting a numerical interval: an actual interval depends on the chosen number of resamples and the resulting bootstrap quantiles. The bootstrap does not create new information beyond the sample; it assesses sampling variation under the assumption that the observed sample represents the population reasonably well.

Bootstrap Designs and Dependence

For independent observations, ordinary case resampling is appropriate. For paired data, resample complete pairs rather than individual values. For clustered data, resample clusters if clusters are the independent sampling units. For time series, ordinary bootstrap resampling destroys serial dependence; block bootstrap methods resample consecutive blocks. Stratified or weighted bootstrap methods may be needed in complex surveys.

6. Robust Estimation

6.1 Robust Location Measures

The mean uses every value and is efficient under normality, but it can be strongly influenced by outliers. The median is the middle ordered observation and has a 50% breakdown point: up to nearly half the observations can be contaminated before the median becomes arbitrarily unreliable. The trimmed mean removes a specified proportion from each tail and averages the remainder. A 10% trimmed mean removes the lowest 10% and highest 10%, subject to an appropriate integer rule.

The winsorized mean replaces extreme observations with the nearest retained values rather than deleting them. It can be useful when tail observations are considered informative but excessively influential.

6.2 Robust Spread Measures

The range is easy to calculate but depends only on two observations. The interquartile range is

IQR = Q3 − Q1,

where Q1 and Q3 are the first and third quartiles. The median absolute deviation is

MAD = median(|x_i − median(x)|).

For normally distributed data, a consistency-adjusted estimate of standard deviation is approximately 1.4826 × MAD. The adjustment should not be used automatically when normal consistency is not the intended interpretation.

6.3 Influence and Breakdown

The influence of an observation describes how much a small contamination can affect an estimator. The mean has unbounded influence; one sufficiently extreme value can change it arbitrarily. The median has bounded influence. The breakdown point is the smallest contamination fraction that can make an estimator arbitrarily bad. High breakdown is desirable for contamination resistance, but often involves a trade-off with efficiency under an ideal model.

6.4 Huber M-Estimator

An M-estimator can reduce the influence of large residuals while retaining more information than a median. Huber loss is defined by

ρ_c(u) = u²/2, if |u| ≤ c;

ρ_c(u) = c|u| − c²/2, if |u| > c.

Here u is a standardized residual and c is a tuning constant. Small residuals receive squared-error treatment; large residuals receive approximately linear treatment. The choice of c affects the balance between efficiency and robustness. Robust regression similarly uses weighted residuals to reduce the influence of unusual observations.

7. Choosing a Method

Method selection should begin with the estimand and design, not with a mechanical normality test.

• One sample or paired ordinal data: sign test or Wilcoxon signed-rank test.
• Two independent groups with ordinal, skewed, or outlier-prone outcomes: Mann–Whitney test or a permutation test.
• More than two independent groups: Kruskal–Wallis test, followed by adjusted comparisons if needed.
• More than two related conditions: Friedman test.
• Categorical counts: chi-square goodness-of-fit or independence test; use exact or simulation-based alternatives for sparse data.
• A statistic with a complicated sampling distribution: bootstrap confidence interval.
• Heavy contamination or influential observations: median, trimmed mean, MAD, or robust regression.

The analysis should report the estimand, sampling design, missing-data treatment, test statistic, p-value or interval, effect size, and limitations. A statistically significant rank test may have little practical importance in a very large data set, while a non-significant result in a small sample may reflect limited precision rather than evidence of no effect.

8. Worked Comparative Example

An AI service team compares prediction latency for two deployment systems. System A has 12 observations and System B has 12 observations. Latency is positive, right-skewed, and includes occasional spikes. The team wants to know whether typical latency differs.

A two-sample t test would compare means but could be highly affected by spikes. A Mann–Whitney test compares rank ordering and is less sensitive to extreme magnitudes. A permutation test using the difference in medians directly addresses a null hypothesis of exchangeable labels, provided observations are independent. A bootstrap confidence interval for the median difference can quantify uncertainty.

A sound workflow is:

1. Plot the observations and inspect the deployment process.

2. Confirm that observations are independent or identify clustering by server, user, or time.

3. Define the estimand, such as median latency difference.

4. Use a permutation test if exchangeability is justified.

5. Use a bootstrap interval, resampling complete independent units.

6. Report the estimated difference, interval, p-value, and operational effect.

7. Investigate extreme values rather than automatically deleting them.

If the spikes are caused by a known infrastructure fault, they may be analyzed as a separate operational regime. If they are genuine future events, deleting them would produce an optimistic and potentially misleading performance assessment.

9. Case Study: Robust Evaluation of a Machine-Learning Model

Background

An illustrative case concerns a hospital developing a model to predict emergency-department length of stay. The data contain 8,000 encounters, but many patients belong to the same hospital departments, and measurements are recorded over several months. Prediction errors are strongly right-skewed, and a small number of extreme errors occur during unusual events.

Problem

The data science team initially reports the mean absolute error and a conventional confidence interval. Clinical reviewers question whether the mean is dominated by a small number of extreme cases and whether random resampling of individual encounters ignores department-level dependence.

Analysis

The team defines several complementary estimands: median absolute error for typical performance, the 90th percentile of absolute error for high-risk performance, and the mean absolute error for comparison with existing benchmarks. They inspect error distributions by department and month. They find clustering within departments and changes during a staffing disruption.

Approach and Solution

The team reports the median, IQR, and MAD, alongside the mean. For uncertainty, they use a cluster bootstrap: hospital departments, rather than individual encounters, are resampled, and all encounters within selected departments are retained. They also conduct a sensitivity analysis excluding the disruption period, but report it separately rather than silently removing it.

For model comparison, paired prediction errors are calculated for each encounter. A paired permutation procedure is used by randomly switching the two model labels within encounter, preserving pairing. Because the clinical cost of large underestimation is high, the team additionally reports the proportion of cases exceeding a clinically defined error threshold.

Outcome

The robust summaries show that the proposed model improves typical error but does not substantially improve the upper tail during the staffing disruption. The mean alone had obscured this distinction. Department-level resampling produces wider intervals than individual-level resampling, reflecting the dependence structure.

Learning Points

1. The estimand should represent the practical decision being supported.

2. Robust summaries complement, rather than automatically replace, conventional summaries.

3. Resampling must respect the independent unit and dependence structure.

4. Extreme observations should be investigated and transparently reported.

5. Model evaluation should include subgroup and tail-performance analysis when harms are asymmetric.

10. Limitations, Ethics, and Governance

Nonparametric methods can lose power when a valid parametric model would provide greater efficiency. Rank tests may discard meaningful magnitude information. Their interpretation can also be misunderstood, especially when distribution shapes differ. Bootstrap methods can fail when the sample is unrepresentative, the sample is very small, observations are dependent but resampled independently, or the statistic is unstable at the boundary.

Statistical software can generate precise-looking output from an inappropriate design. Analysts should document the method, random seed where simulation is used, number of resamples or permutations, treatment of ties and missing data, and any stopping or selection rules. Repeatedly trying many methods and reporting only the favorable result creates researcher degrees of freedom and can inflate false-positive findings.

In AI and health applications, robust methods do not eliminate bias in data collection. A bootstrap confidence interval around a biased sample estimates uncertainty about the observed sampling process, not necessarily uncertainty about the target population. Fairness assessments should report subgroup-specific uncertainty, account for unequal sample sizes, and avoid suppressing poor performance in small or vulnerable groups. Privacy-preserving reporting is also important because resampling outputs, extreme cases, or detailed subgroup tables can disclose sensitive information.

11. Contemporary Developments and Best Practices

Modern statistical computing makes permutation and bootstrap methods practical for complex statistics, but computation should support—not replace—reasoning. Reproducible analysis pipelines can automate resampling, preserve random seeds, and record software versions. Parallel computing permits millions of resamples when needed, although the number of repetitions should be justified by the desired Monte Carlo precision.

Best practices include:

• Define the scientific question and estimand before selecting a test.
• Examine plots and data provenance before applying a procedure.
• Respect independence, pairing, clustering, stratification, and time order.
• Use exact methods for small samples when available.
• Report effect sizes and confidence intervals, not p-values alone.
• Correct for multiple comparisons in post-hoc analyses.
• Use robust and classical analyses as sensitivity analyses when appropriate.
• Treat outliers as observations requiring explanation, not automatic deletion.
• Validate resampling schemes through simulation when the design is complex.
• Document all analytical choices for reproducibility.

Chapter Summary

Nonparametric methods provide inference with weaker distributional assumptions by using ranks, signs, and category counts. The sign test and Wilcoxon signed-rank test address one-sample and paired questions, while the Mann–Whitney, Kruskal–Wallis, and Friedman tests extend rank-based reasoning to independent and repeated-group designs. Chi-square procedures analyze categorical goodness-of-fit and independence, provided expected counts and sampling assumptions support the approximation.

Permutation tests construct null distributions by rearranging exchangeable labels or signs. Bootstrap methods approximate sampling distributions by sampling with replacement and can produce standard errors and confidence intervals for complex statistics. Both methods depend on the sampling design and are not assumption-free.

Robust estimators, including the median, trimmed mean, IQR, MAD, and Huber-based procedures, reduce sensitivity to outliers and heavy tails. Method selection should be guided by the estimand, measurement scale, dependence structure, sample size, and practical consequences. In AI and professional applications, transparent reporting, subgroup analysis, reproducible computation, and ethical interpretation are essential.

Key Terms

Nonparametric method: A method that does not require a fully specified parametric population distribution.

Distribution-free: Having a null sampling distribution that does not depend on an unknown population distribution under stated assumptions.

Rank: The order position assigned to an observation.

Sign test: A binomial procedure based only on positive and negative deviations.

Wilcoxon signed-rank test: A paired or one-sample rank test using signed absolute differences.

Mann–Whitney U test: A rank-sum test for two independent groups.

Kruskal–Wallis test: A rank-based omnibus test for several independent groups.

Friedman test: A rank-based test for several related treatments or repeated measurements.

Goodness of fit: Agreement between observed categorical counts and a specified distribution.

Chi-square statistic: A sum of squared observed–expected differences scaled by expected counts.

Exchangeability: The property that rearrangements under the null hypothesis have the same distribution.

Permutation test: A test based on rearranging labels, signs, or observations under a null model.

Bootstrap: Repeated sampling with replacement from observed data to approximate sampling variation.

Robust estimator: An estimator relatively insensitive to contamination or assumption violations.

Trimmed mean: The mean after removing specified proportions from both tails.

Median absolute deviation: The median of absolute deviations from the sample median.

Influence: The sensitivity of an estimator to an individual observation.

Breakdown point: The smallest contamination proportion capable of making an estimator arbitrarily unreliable.

Cluster bootstrap: A bootstrap that resamples independent clusters rather than individual observations.

Effect size: A quantitative measure of the magnitude or practical importance of an association or difference.

Review Questions

Short Answer Questions

1. Why are nonparametric methods not completely assumption-free?

2. What information is lost when numerical observations are replaced by ranks?

3. State the main difference between a permutation test and a bootstrap procedure.

4. Define the breakdown point of an estimator.

5. Why must expected counts be considered in a chi-square analysis?

Long Answer Questions

1. Explain the assumptions, procedure, and interpretation of the Wilcoxon signed-rank test.

2. Compare the Mann–Whitney, Kruskal–Wallis, and Friedman tests, including their data structures and interpretations.

3. Explain how a bootstrap confidence interval is constructed and discuss situations in which it may be unreliable.

4. Compare the mean, median, trimmed mean, IQR, and MAD as summaries of skewed data.

5. Discuss why exchangeability is central to permutation tests and how dependence changes the resampling strategy.

Analytical / Application Questions

1. A study compares customer waiting times from two independent branches. The data are strongly right-skewed and contain several extreme delays. Select an appropriate primary analysis, define the estimand, and justify an uncertainty procedure.

2. A machine-learning model is evaluated on patients nested within hospitals. Explain why an ordinary individual-level bootstrap may be inappropriate and design a more suitable resampling scheme.

3. A chi-square test finds a significant association between model approval and demographic group. Describe the additional analyses needed before making a fairness or policy conclusion.

Multiple Choice Questions with Answers

Question 1

Which feature most directly distinguishes a permutation test from a bootstrap procedure?

A. A permutation test rearranges data under a null hypothesis, whereas a bootstrap samples with replacement to estimate sampling variation.

B. A permutation test always uses normal data, whereas a bootstrap requires ordinal data.

C. A permutation test estimates only medians, whereas a bootstrap estimates only means.

D. A permutation test requires larger samples than every bootstrap procedure.

Correct Answer: A. A permutation test rearranges data under a null hypothesis, whereas a bootstrap samples with replacement to estimate sampling variation.

Question 2

For paired data, which procedure is most appropriate when differences are ordinal and only their directions are considered?

A. Two-sample t test

B. Sign test

C. Chi-square goodness-of-fit test

D. Kruskal–Wallis test

Correct Answer: B. Sign test

Question 3

If group 1 has n1 observations and rank sum R1, which expression defines its Mann–Whitney statistic U1?

A. R1 − n1(n1 + 1)/2

B. R1 − n1n2/2

C. n1(n1 − 1)/2 − R1

D. R1/[n1(n1 + 1)]

Correct Answer: A. R1 − n1(n1 + 1)/2

Question 4

What does a significant Kruskal–Wallis test establish by itself?

A. Every group has a different median.

B. At least one group differs in distribution from another.

C. The group means are equal.

D. The data are normally distributed.

Correct Answer: B. At least one group differs in distribution from another.

Question 5

Under independence in a contingency table, the expected count for cell (i, j) is

A. the cell total divided by the grand total.

B. the row total multiplied by the column total, divided by the grand total.

C. the grand total divided by the number of cells.

D. the column total minus the row total.

Correct Answer: B. the row total multiplied by the column total, divided by the grand total.

Question 6

Which measure of spread is defined as Q3 − Q1?

A. Range

B. Variance

C. Interquartile range

D. Median absolute deviation

Correct Answer: C. Interquartile range

Question 7

Why might a cluster bootstrap be used in a study of patients nested within hospitals?

A. It makes all observations normally distributed.

B. It preserves the dependence among patients within hospitals.

C. It eliminates the need for an estimand.

D. It guarantees a narrower confidence interval.

Correct Answer: B. It preserves the dependence among patients within hospitals.

Question 8

For normally distributed data, which approximate multiplier makes MAD consistent with the standard deviation?

A. 0.6745

B. 1.4826

C. 2.0000

D. 3.1416

Correct Answer: B. 1.4826

Question 9

What is the principal effect of ranking observations in a rank-based test?

A. It preserves all numerical magnitudes exactly.

B. It generally reduces sensitivity to extreme magnitudes.

C. It guarantees greater power than a parametric test.

D. It removes the need for independent observations.

Correct Answer: B. It generally reduces sensitivity to extreme magnitudes.

Question 10

In a Monte Carlo permutation test with B random permutations, which p-value estimate avoids reporting exactly zero?

A. Extreme permutations divided by B − 1

B. Extreme permutations divided by B

C. (1 + extreme permutations)/(B + 1)

D. B/(1 + extreme permutations)

Correct Answer: C. (1 + extreme permutations)/(B + 1)