Exploring Parametric Test: Grades, Properties, and Practical Industrial Uses

Types of Parametric Tests

Parametric tests are statistical methods used to make inferences about population parameters based on sample data. These tests assume that the data follows a known distribution—typically the normal distribution—and require certain conditions such as normality, homogeneity of variances, and interval or ratio-level measurement. When these assumptions are met, parametric tests offer greater statistical power and precision compared to non-parametric alternatives.

This guide explores the most commonly used parametric tests in research and data analysis, detailing their purpose, underlying assumptions, appropriate use cases, and key differences.

Student's T-test

Purpose: Determines whether the means of two independent groups are significantly different from each other.

Key Assumptions

• Data is normally distributed within each group
• Homogeneity of variances (equal variances between groups)
• Observations are independent
• Data measured on an interval or ratio scale

Common Applications

• Comparing test scores between two classes
• Assessing effectiveness of two medical treatments
• Testing differences in customer satisfaction between two regions

Note: Also known as the independent samples t-test; uses the t-distribution to calculate significance.

Paired T-test

Purpose: Evaluates whether the mean difference between two related (paired) observations is significantly different from zero.

Key Assumptions

• Differences between pairs are normally distributed
• Data consists of paired or matched samples
• Dependent (related) measurements (e.g., before and after)
• Interval or ratio scale of measurement

Common Applications

• Pre-test and post-test comparisons in educational studies
• Measuring blood pressure before and after treatment
• Comparing performance of the same individuals under two conditions

Key Insight: More powerful than independent t-tests for repeated measures due to reduced variability from subject-to-subject differences.

ANOVA (Analysis of Variance)

Purpose: Compares the means of three or more independent groups to determine if at least one group mean is statistically different from the others.

Key Assumptions

• Normality of data within each group
• Homogeneity of variances across groups
• Independence of observations
• Continuous dependent variable

Common Applications

• Comparing crop yields across multiple fertilizer types
• Testing customer response to three or more advertising campaigns
• Evaluating performance of students from different schools

Important: ANOVA only tells you if a difference exists—not which groups differ. Post-hoc tests (e.g., Tukey’s HSD) are needed for pairwise comparisons.

Repeated Measures ANOVA

Purpose: Analyzes data where the same subjects are measured multiple times under different conditions or over time.

Key Assumptions

• Normality of difference scores
• Sphericity (equal variances of the differences between all pairs of repeated measures)
• Dependent (within-subjects) design
• Interval or ratio level data

Common Applications

• Tracking patient recovery over several weeks
• Measuring employee productivity across quarterly assessments
• Studying cognitive performance at different times of day

Note: When sphericity is violated, corrections like Greenhouse-Geisser or Huynh-Feldt are applied to adjust degrees of freedom.

Z-test

Purpose: Determines whether there is a significant difference between a sample mean and a population mean, or between the means of two large samples.

Key Assumptions

• Sample size is large (typically n > 30)
• Population standard deviation is known
• Data is normally distributed (or approximately normal due to Central Limit Theorem)
• Random sampling and independence of observations

Common Applications

• Quality control in manufacturing with known population parameters
• Standardized testing (e.g., comparing school average to national average)
• Market research with large survey samples

Distinction: Unlike the t-test, the z-test uses the standard normal distribution (z-distribution) and is preferred when population variance is known or sample sizes are very large.

Test	Number of Groups	Data Relationship	Sample Size Consideration	Primary Use Case
Student's T-test	2 (independent)	Independent samples	Small to moderate (n < 30 typical)	Compare two group means
Paired T-test	2 (related)	Dependent/matched pairs	Small to moderate	Before-and-after or matched designs
ANOVA	3 or more (independent)	Independent groups	Varies, balanced design preferred	Multiple group comparison
Repeated Measures ANOVA	3 or more (related)	Within-subjects / repeated	Moderate to large	Longitudinal or time-series data
Z-test	1 or 2	Independent	Large (n > 30), known σ	Large sample inference with known variance

Expert Tip: Always verify the assumptions of parametric tests before application. Use diagnostic plots (Q-Q plots for normality, Levene’s test for homogeneity) and consider non-parametric alternatives (e.g., Mann-Whitney U, Wilcoxon signed-rank, Kruskal-Wallis) if assumptions are severely violated.

Important Features of Parametric Tests

Parametric tests are a cornerstone of inferential statistics, widely used in research and data analysis due to their robustness and efficiency when underlying assumptions are met. These tests rely on specific distributional and measurement criteria, allowing researchers to draw precise conclusions about population parameters based on sample data. Below is a detailed breakdown of the key features that define parametric tests and make them preferable in many analytical contexts.

Assumptions

Parametric tests are built upon several critical statistical assumptions: normality, homogeneity of variance, and appropriate scale of measurement. The assumption of normality means that the data (or sampling distribution of the mean) should follow a normal distribution, especially important in small samples. For larger samples, the Central Limit Theorem allows some flexibility.

Homogeneity of variance (or homoscedasticity) assumes that variances across groups are approximately equal, which is particularly vital in tests like ANOVA and t-tests. Violations can lead to increased Type I error rates. Lastly, these tests require data measured on an interval or ratio scale, where differences between values are meaningful and consistent.

When these assumptions are satisfied, parametric tests provide more accurate and efficient estimates than their nonparametric counterparts, making them the preferred choice in many scientific studies.

Sensitivity

One of the major advantages of parametric tests is their high sensitivity in detecting true effects. Because they utilize information about the data’s distribution and variance, they are more likely to identify statistically significant differences or relationships even with subtle effect sizes.

This sensitivity stems from the fact that parametric methods make full use of the numerical properties of the data, such as mean and standard deviation, rather than relying on ranks or signs (as in nonparametric tests). As a result, they often yield more powerful results with smaller sample sizes, reducing the need for extensive data collection while maintaining statistical rigor.

For example, a t-test can detect a small but meaningful difference in crop yields between two fertilizer treatments, whereas a nonparametric alternative like the Mann-Whitney U test might fail to reach significance due to lower sensitivity.

Common Types of Parametric Tests

Several well-known statistical tests fall under the parametric category, each designed for specific research scenarios:

• Student’s t-test: Used to compare the means of two groups. The independent samples t-test assesses differences between unrelated groups (e.g., control vs. treatment), while the paired t-test evaluates changes within the same subjects over time or conditions.
• ANOVA (Analysis of Variance): Extends the t-test to three or more groups. One-way ANOVA compares means across multiple independent groups, while repeated-measures ANOVA is used for within-subject designs.
• Linear Regression: Assesses the relationship between one or more independent variables and a continuous dependent variable. It helps predict outcomes and quantify the strength and direction of associations.
• Multiple Regression & ANCOVA: More advanced forms that control for covariates or examine multiple predictors simultaneously.

These tests are foundational in fields such as psychology, agriculture, medicine, and economics, where precise measurement and hypothesis testing are essential.

Scale of Measurement Requirements

Parametric tests require data measured on interval or ratio scales, where numerical values have consistent intervals and a meaningful zero point (in the case of ratio data). Examples include temperature in degrees Celsius (interval), height in centimeters, weight in kilograms, income in dollars, or test scores with equal intervals.

These scales allow for meaningful computation of means, standard deviations, and correlations—key components in parametric analysis. In contrast, nominal (e.g., gender, color) or ordinal (e.g., satisfaction ratings on a 1–5 scale) data lack these properties and are better analyzed using nonparametric methods like chi-square or Kruskal-Wallis tests.

Ensuring proper measurement scaling during study design is crucial to avoid misapplication of statistical techniques and to maintain the validity of research findings.

Higher Statistical Power

Statistical power refers to the probability that a test will correctly reject a false null hypothesis (i.e., detect an effect when one truly exists). Parametric tests generally have higher power than nonparametric tests when their assumptions are met.

This increased power means that researchers can achieve statistically significant results with smaller sample sizes, making studies more efficient and cost-effective. For instance, a parametric t-test may detect a significant difference in plant growth between two soil types with 30 samples per group, while a nonparametric alternative might require 40 or more to achieve the same level of confidence.

The higher power arises from the efficient use of data through distributional assumptions and precise estimation of parameters. However, this advantage diminishes or reverses if assumptions are severely violated, emphasizing the importance of diagnostic checks before analysis.

Comparison Summary

Feature	Parametric Tests	Nonparametric Alternatives
Assumptions	Normality, homogeneity of variance, interval/ratio data	Fewer assumptions; suitable for non-normal or ranked data
Sensitivity	High – detects smaller effects	Moderate to low – less sensitive to subtle differences
Statistical Power	Higher when assumptions met	Lower, requires larger samples for same power
Data Scale	Interval or ratio	Ordinal, nominal, or non-normal interval/ratio
Common Examples	t-test, ANOVA, Linear Regression	Mann-Whitney, Kruskal-Wallis, Chi-square

Important: While parametric tests offer numerous advantages, their validity depends heavily on meeting underlying assumptions. Always conduct preliminary data analysis—including tests for normality (e.g., Shapiro-Wilk), homogeneity of variance (e.g., Levene’s test), and outlier detection—before applying parametric methods. If assumptions are violated, consider data transformation or switching to nonparametric alternatives to ensure reliable and interpretable results.

Scenarios of Using Parametric Tests

Parametric tests are powerful statistical tools used when data meets certain assumptions—primarily normality, homogeneity of variance, and interval or ratio measurement scales. These tests offer high statistical power and precision, making them ideal for analyzing continuous data across a wide range of real-world applications. Below are key scenarios where parametric tests are commonly and effectively applied.

1. Comparing Means Between Groups

Common Tests: Independent Samples t-test, Paired t-test, One-way ANOVA, Two-way ANOVA

Parametric tests are most frequently used to determine whether there are statistically significant differences in the means of two or more groups. This is essential in research settings where average outcomes need to be compared across distinct populations or experimental conditions.

• t-tests: Used to compare the means of two independent or related groups (e.g., pre-test vs. post-test scores, treatment vs. control group).
• ANOVA (Analysis of Variance): Extends t-testing to three or more groups, allowing researchers to assess whether at least one group mean differs significantly from the others.
• Widely applied in clinical studies, educational research, and social sciences to analyze differences in variables such as income, age, test performance, or psychological well-being.

Example: A researcher compares average blood pressure levels between patients receiving a new drug versus those on a placebo using an independent samples t-test.

2. Quality Control in Manufacturing

Common Tests: One-way ANOVA, Repeated Measures ANOVA

In industrial and manufacturing environments, maintaining consistent product quality is critical. Parametric tests help monitor production processes by detecting meaningful deviations in product measurements across batches or time periods.

• Manufacturers use ANOVA to compare the mean dimensions (e.g., diameter, thickness) of products from different production runs.
• Helps identify whether variations are due to random chance or systematic issues in machinery, materials, or human factors.
• Enables early detection of process drift, reducing waste and ensuring compliance with industry standards.

Example: An automotive parts factory uses ANOVA to verify that brake disc thickness remains consistent across five production shifts.

3. Clinical Trials and Medical Research

Common Tests: t-tests, ANOVA, Linear Regression

Parametric methods are foundational in evaluating the efficacy and safety of medical treatments. They allow researchers to quantify treatment effects and model relationships between variables with precision.

• t-tests and ANOVA: Compare average outcomes (e.g., symptom reduction, recovery time) between treatment and control groups.
• Regression analysis: Models how patient characteristics (e.g., age, BMI, dosage) influence health outcomes, enabling prediction and personalization of care.
• Clinical trial data often follows a normal distribution and has equal variances across groups, satisfying key assumptions for parametric testing.

Example: A pharmaceutical company uses ANOVA to assess the effectiveness of three different dosages of a new antidepressant on depression scale scores.

4. Market Research and Consumer Behavior Analysis

Common Tests: t-tests, ANOVA, Multiple Regression

Businesses leverage parametric tests to understand consumer preferences, segment audiences, and forecast market trends based on quantitative survey or sales data.

• ANOVA: Determines if different demographic groups (e.g., age, income level) have significantly different satisfaction ratings for a product.
• Regression models: Predict sales volume based on advertising spend, pricing, seasonality, and regional factors.
• Provides actionable insights for product development, pricing strategies, and targeted marketing campaigns.

Example: A beverage company uses multiple regression to analyze how household income and age affect monthly soft drink consumption.

5. A/B Testing in Digital Marketing

Common Tests: Independent Samples t-test, Z-test for proportions (when sample sizes are large)

Digital marketers use A/B testing (or split testing) to compare two or more versions of a webpage, email, or ad to determine which performs better in terms of user engagement or conversion rates.

• Parametric tests like the t-test are used to compare continuous metrics such as average session duration, time on page, or revenue per user.
• Even for binary outcomes (e.g., click/no click), large sample sizes often justify the use of parametric approximations due to the Central Limit Theorem.
• Results guide data-driven decisions to optimize user experience and maximize ROI.

Example: An e-commerce site runs an A/B test comparing two checkout button colors and uses a t-test to determine if the difference in average purchase value is statistically significant.

Note on Assumptions: Parametric tests require data to be approximately normally distributed, have similar variances across groups (homoscedasticity), and be measured on an interval or ratio scale. When these assumptions are violated, non-parametric alternatives (e.g., Mann-Whitney U, Kruskal-Wallis) should be considered.

Best Practice: Always conduct assumption checks—using tools like Shapiro-Wilk (normality), Levene’s test (equal variances), and visual inspection (histograms, Q-Q plots)—before applying parametric tests. Transforming data (e.g., log transformation) or using robust statistical methods can help meet assumptions when possible.

Application Area	Typical Parametric Test	Key Variables Analyzed	Business/Research Impact
Clinical Research	t-test, ANOVA	Blood pressure, pain scores, recovery time	Determines treatment efficacy and regulatory approval
Manufacturing	ANOVA	Product dimensions, weight, durability	Ensures quality consistency and reduces defects
Market Research	Regression, ANOVA	Customer satisfaction, spending habits	Guides product design and marketing strategy
Digital Marketing	t-test, Z-test	Click-through rate, conversion rate, revenue	Optimizes campaigns and improves user engagement
Educational Studies	Paired t-test, ANOVA	Test scores, learning outcomes	Evaluates teaching methods and program effectiveness

Why Choose Parametric Tests?

• Higher Statistical Power: More likely to detect a true effect when one exists, especially with smaller sample sizes.
• Precision in Estimation: Provide confidence intervals and effect size measures that enhance interpretation.
• Wide Software Support: Easily implemented in statistical packages like SPSS, R, Python (SciPy), and Excel.
• Well-Established Interpretation: Results are widely understood across scientific and business communities.

How to Choose a Parametric Test: A Step-by-Step Guide

Selecting the appropriate parametric statistical test is essential for drawing valid conclusions from your data. Parametric tests are powerful tools that assume certain characteristics about your data distribution and measurement scale. This guide walks you through the five key steps to confidently choose the right parametric test for your research, ensuring accuracy, reliability, and meaningful results.

Important Note: Using an incorrect statistical test can lead to misleading or invalid conclusions. Always verify that your data meet the necessary assumptions before applying any parametric method. When in doubt, consult a statistician or use nonparametric alternatives.

Step 1: Clearly Define Your Research Question

The foundation of any statistical analysis begins with a well-formulated research question. Understanding what you're trying to investigate determines the type of test you'll need. Ask yourself:

• Are you comparing group means (e.g., treatment vs. control)?
• Are you assessing the relationship between two continuous variables?
• Are you predicting one variable based on another (regression)?
• Do you have two independent groups, paired/matched samples, or more than two groups?

For example:

• Comparing average test scores between two different teaching methods → suggests a t-test.
• Analyzing changes in blood pressure before and after treatment in the same individuals → indicates a paired t-test.
• Examining the linear relationship between income and education level → points to Pearson’s correlation.

A precise research question guides the entire analytical process and prevents misapplication of statistical methods.

Step 2: Identify Your Data Type and Measurement Scale

Parametric tests require data measured on interval or ratio scales, where numerical values have meaningful distances and, in the case of ratio data, a true zero point. Examples include temperature (interval), weight, height, time, and income (ratio).

Data that are nominal (e.g., gender, color) or ordinal (e.g., satisfaction ratings on a 1–5 scale) generally do not meet the requirements for parametric testing unless they approximate interval properties (e.g., validated survey scales).

Choosing the correct test depends heavily on your data type:

• Two-group comparison with interval/ratio data → Independent or Paired t-test
• Three or more groups → ANOVA (Analysis of Variance)
• Relationship between two continuous variables → Pearson correlation coefficient
• Predicting outcomes → Linear regression

Misclassifying data type can invalidate your results—always confirm the scale of measurement before proceeding.

Step 3: Evaluate Key Statistical Assumptions

Parametric tests rely on three core assumptions. Violating these can compromise the validity of your results:

1. Normality: The data (or residuals in regression) should be approximately normally distributed, especially in small samples. Use histograms, Q-Q plots, or tests like Shapiro-Wilk to assess this.
2. Homogeneity of Variance (Homoscedasticity): Group variances should be roughly equal. Levene’s test or Bartlett’s test can help evaluate this assumption, particularly in t-tests and ANOVA.
3. Independence of Observations: Data points must not influence one another. This is typically ensured through proper study design (e.g., random sampling, no repeated measures unless accounted for).

If these assumptions are severely violated, consider:

• Transforming the data (e.g., log transformation)
• Using robust statistical methods
• Switching to nonparametric alternatives (e.g., Mann-Whitney U, Kruskal-Wallis, Spearman correlation)

Expert Tip: With large sample sizes (typically n > 30 per group), the Central Limit Theorem allows some flexibility with normality due to the sampling distribution of the mean becoming approximately normal—even if the original data aren't perfectly normal.

Step 4: Consider Sample Size and Statistical Power

Sample size plays a crucial role in both the robustness and power of parametric tests:

• Large samples (>30–50 per group): More resilient to minor assumption violations. Z-tests may be appropriate for known population variances.
• Small samples (<30): Highly sensitive to deviations from normality and equal variances. Use t-tests (which account for uncertainty in variance estimation) rather than z-tests.
• Power considerations: Small samples may lack the power to detect true effects, increasing the risk of Type II errors (false negatives).

Whenever possible, conduct a power analysis during the study design phase to determine the minimum sample size needed to detect a meaningful effect.

Step 5: Select the Most Appropriate Parametric Test

After evaluating the research question, data type, assumptions, and sample size, you can confidently choose the correct test. The table below summarizes common scenarios and recommended parametric tests:

Research Objective	Data Structure	Recommended Parametric Test	Key Assumptions to Verify
Compare means of two independent groups	Two unrelated groups, continuous outcome	Independent Samples t-test	Normality, homogeneity of variance, independence
Compare means of two related/paired groups	Pre-post measurements, matched pairs	Paired Samples t-test	Normality of differences, independence of pairs
Compare means across three or more groups	One independent variable with ≥3 levels	One-Way ANOVA	Normality, homogeneity of variance, independence
Assess linear relationship between two variables	Two continuous variables	Pearson Correlation (r)	Linearity, normality, absence of outliers
Predict one variable from another	Continuous outcome and predictor(s)	Simple or Multiple Linear Regression	Linearity, normality of residuals, homoscedasticity, independence

Quick Decision Checklist: Before finalizing your test, ask: (1) Is my outcome variable continuous (interval/ratio)? (2) Are my data approximately normal? (3) Are group variances similar? (4) Are observations independent? (5) Do I have adequate sample size? If yes to all, a parametric test is likely appropriate.

Final Recommendations

• Always document your rationale for test selection in your methodology section.
• Use statistical software (e.g., SPSS, R, Python) to check assumptions and run tests accurately.
• Report effect sizes and confidence intervals alongside p-values for a more complete interpretation.
• When assumptions are questionable, perform both parametric and nonparametric analyses to compare results.
• Seek peer or statistical consultation when dealing with complex designs (e.g., repeated measures, covariates).

Choosing the right parametric test is not just a technical step—it's a critical component of sound scientific practice. By systematically evaluating your research question, data characteristics, and statistical assumptions, you ensure that your findings are both valid and impactful.

Frequently Asked Questions About Parametric Tests in Statistics

Q1: Which factors are considered in the selection of a parametric test?

Selecting the appropriate parametric test involves a systematic evaluation of several key elements related to your research design and data characteristics. The decision should not be made in isolation but as part of a broader statistical planning process.

• Research Question: Determine whether you're comparing group means (e.g., using t-tests or ANOVA), assessing relationships between variables (e.g., linear regression), or evaluating changes over time (e.g., repeated measures ANOVA).
• Data Type: Ensure your data is continuous (interval or ratio level). Parametric tests are designed for numerical data that can meaningfully support operations like averaging and standard deviation calculations.
• Study Design: Consider the number of groups (two vs. more than two), whether samples are independent or paired, and the presence of covariates that may require analysis of covariance (ANCOVA).
• Sample Size: Larger samples (typically n > 30 per group) help satisfy the Central Limit Theorem, making parametric tests more robust even if normality is slightly violated.

For example, use an independent samples t-test to compare average test scores between two classes, one-way ANOVA for three or more groups, and Pearson correlation or simple linear regression to explore the relationship between study hours and exam performance.

Q2: What are the most important assumptions of parametric tests?

Parametric tests rely on three core statistical assumptions. Violating these can lead to inaccurate p-values, reduced power, or misleading conclusions. It's essential to verify them before interpreting results.

Assumption	Description	How to Check	Consequences of Violation
Normality	Data within each group should follow a normal distribution, especially important for small sample sizes.	Use Shapiro-Wilk test, Q-Q plots, or histogram inspection.	Inflated Type I error rates; unreliable confidence intervals.
Homogeneity of Variances	Variances across groups should be approximately equal (homoscedasticity).	Levene’s test or Bartlett’s test; visual inspection via boxplots.	Reduced test accuracy, especially in unbalanced designs.
Independence	Observations must be independent of one another—no repeated measures or clustering unless modeled.	Assessed through study design; avoid data duplication or correlated samples.	Invalidates standard errors and p-values; increases false positives.

When assumptions are not met, consider data transformations (e.g., log transformation), increasing sample size, or switching to non-parametric alternatives such as the Mann-Whitney U test or Kruskal-Wallis test.

Q3: Is it mandatory to use parametric tests only when proper conditions are met?

Yes, parametric tests should only be used when their underlying assumptions are reasonably satisfied. While they offer greater statistical power and precision under ideal conditions, applying them to inappropriate data can produce misleading or invalid results.

• High Power Advantage: When assumptions are met, parametric tests are more likely to detect true effects due to efficient use of data information.
• Risk of Misuse: Using parametric tests with skewed data, unequal variances, or outliers can inflate Type I error rates (false positives) or reduce sensitivity (Type II errors).
• Alternatives Exist: Non-parametric methods (e.g., Wilcoxon signed-rank, Kruskal-Wallis) make fewer assumptions and are robust to violations of normality and homogeneity.
• Robustness Consideration: Some parametric tests (like t-tests) are relatively robust to mild violations, particularly with large, balanced samples.

Always conduct assumption checks before analysis. If assumptions are violated, either transform the data to meet them or choose a more appropriate non-parametric method to ensure valid inference.

Q4: When dealing with more than two groups, which test can be more effective, ANOVA or T-test?

ANOVA (Analysis of Variance) is the statistically appropriate and more effective choice when comparing means across three or more independent groups. Using multiple t-tests instead increases the risk of Type I errors (false positives) due to the problem of multiple comparisons.

• Problem with Multiple T-Tests: Conducting repeated pairwise t-tests inflates the overall alpha level. For example, with 3 groups, you'd perform 3 comparisons, increasing the chance of at least one false positive from 5% to approximately 14%.
• ANOVA Advantage: It tests the null hypothesis that all group means are equal in a single omnibus test, controlling the family-wise error rate at the desired alpha level (e.g., 0.05).
• Post-Hoc Testing: If ANOVA is significant, follow up with post-hoc tests (e.g., Tukey’s HSD, Bonferroni) to identify which specific groups differ, while maintaining error control.
• Flexibility: ANOVA can be extended to factorial designs (e.g., two-way ANOVA), repeated measures, and mixed models for more complex experimental setups.

In contrast, t-tests are best reserved for comparing exactly two groups. For paired or matched data with multiple time points, repeated measures ANOVA is preferred over repeated t-tests for the same reasons.

Q5: Are outliers important in affecting parametric tests?

Yes, outliers can have a substantial impact on parametric tests because these methods are sensitive to extreme values due to their reliance on means and variances, which are not resistant to outliers.

• Distorted Means: A single extreme value can pull the mean away from the central tendency, leading to incorrect interpretations of group differences.
• Inflated Variances: Outliers increase within-group variability, reducing the test's power to detect real differences (lower F-ratios in ANOVA).
• Violation of Normality: Extreme values often result in skewed distributions, violating the normality assumption required for validity.
• Leverage in Regression: In regression models, outliers can disproportionately influence the slope and intercept, leading to misleading relationships.

To address outliers, first verify they are not data entry errors. Then, use visualization tools like boxplots or scatterplots to identify them. Options include winsorizing, transformation, robust statistical methods, or conducting sensitivity analyses with and without outliers to assess their impact on conclusions.