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Can You Use a T-Test to Compare Three Groups?

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Can you use a t-test to compare three groups? No, a t-test is designed for comparing the means of only two groups. For comparing three or more groups, you should use ANOVA (Analysis of Variance) or other multiple comparison methods to avoid inflating the Type I error rate. COMPARE.EDU.VN provides comprehensive comparisons and analyses to help you choose the right statistical test. Alternatives to t-tests include ANOVA, Tukey-Kramer, and Dunnett’s test, offering robust solutions for multiple group comparisons and ensuring accurate statistical analysis.

1. Understanding the T-Test

A t-test, also known as Student’s t-test, is a statistical hypothesis test used to determine if there is a significant difference between the means of two groups. It is widely used in various fields, including medicine, psychology, and engineering, to compare two sets of data. The core principle of a t-test involves calculating a t-statistic, which is then compared to a critical value from the t-distribution to determine whether the null hypothesis can be rejected.

1.1. Types of T-Tests

There are three main types of t-tests, each suited for different scenarios:

  • One-Sample T-Test: This test is used to determine whether the mean of a single group is equal to a specific value. For example, you might use a one-sample t-test to determine if the average height of students in a school is significantly different from a known population average.
  • Independent Two-Sample T-Test: This test, also known as an unpaired t-test, is used to determine if there is a significant difference between the means of two independent groups. For instance, you might use an independent two-sample t-test to compare the test scores of students who were taught using two different methods.
  • Paired T-Test: This test is used to determine if there is a significant difference in paired measurements from the same group. For example, you might use a paired t-test to compare the blood pressure of patients before and after a treatment.

1.2. Assumptions of T-Tests

T-tests rely on several key assumptions to ensure the validity of their results:

  • Continuous Data: The data should be continuous, meaning that it can take on any value within a range.
  • Random Sampling: The sample data must be randomly sampled from the population to ensure that it is representative.
  • Homogeneity of Variance: The variability of the data in each group should be similar. This assumption is particularly important for independent two-sample t-tests.
  • Approximate Normality: The distribution of the data should be approximately normal. While t-tests are relatively robust to deviations from this assumption, it is important to check for extreme departures from normality.
  • Independence: For two-sample t-tests, the samples must be independent. If the samples are not independent, a paired t-test may be more appropriate.

1.3. How T-Tests Work

The process of conducting a t-test involves several steps:

  1. Formulate Hypotheses: Define the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis typically states that there is no difference between the means of the groups being compared, while the alternative hypothesis states that there is a difference.
  2. Select Significance Level: Choose a significance level (α), which represents the probability of rejecting the null hypothesis when it is actually true. Common values for α are 0.05 and 0.01.
  3. Calculate the T-Statistic: Compute the t-statistic using the appropriate formula for the type of t-test being performed.
  4. Determine Degrees of Freedom: Calculate the degrees of freedom (df), which depend on the sample size(s) and the type of t-test.
  5. Find the Critical Value: Look up the critical value from the t-distribution table using the chosen significance level and degrees of freedom.
  6. Compare T-Statistic to Critical Value: Compare the calculated t-statistic to the critical value. If the absolute value of the t-statistic is greater than the critical value, reject the null hypothesis.
  7. Draw Conclusion: Based on the comparison, draw a conclusion about whether there is a significant difference between the means of the groups being compared.

2. Why T-Tests Are Not Suitable for Comparing Three Groups

While t-tests are powerful tools for comparing the means of two groups, they are not appropriate for comparing three or more groups. The primary reason for this limitation is the increased risk of making a Type I error, also known as a false positive.

2.1. The Problem of Multiple Comparisons

When conducting multiple t-tests to compare all possible pairs of groups, the probability of making at least one Type I error increases dramatically. This is known as the multiple comparisons problem.

Suppose you want to compare the means of three groups (A, B, and C). You could perform three separate t-tests:

  • Compare Group A to Group B
  • Compare Group A to Group C
  • Compare Group B to Group C

If each t-test has a significance level of α = 0.05, the probability of not making a Type I error in each test is 1 – α = 0.95. However, the probability of not making a Type I error in all three tests is:

$0.95 0.95 0.95 = 0.857$

Therefore, the probability of making at least one Type I error in the three tests is:

$1 – 0.857 = 0.143$ or 14.3%

This means that there is a 14.3% chance of incorrectly concluding that there is a significant difference between at least one pair of groups when, in reality, there is no difference. As the number of groups increases, the probability of making a Type I error grows even larger.

2.2. Inflated Type I Error Rate

The inflated Type I error rate makes it difficult to draw accurate conclusions from the data. If you rely on multiple t-tests to compare three or more groups, you are more likely to find statistically significant differences that are actually due to chance. This can lead to incorrect interpretations and flawed decision-making.

2.3. Example Scenario

Consider a scenario where you want to compare the effectiveness of three different teaching methods (A, B, and C) on student test scores. You randomly assign students to one of the three methods and then administer a test to measure their performance.

If you use multiple t-tests to compare the means of the three groups, you might find a statistically significant difference between Method A and Method B. However, this difference could simply be due to chance, and in reality, there may be no true difference in the effectiveness of the three teaching methods.

3. Alternatives to T-Tests for Comparing Three Groups

To address the limitations of using t-tests for comparing three or more groups, several alternative statistical methods can be used. These methods are designed to control the Type I error rate and provide more accurate and reliable results.

3.1. Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA) is a statistical test that compares the means of two or more groups. It is a generalization of the t-test and can be used to determine whether there is a significant difference between the means of multiple groups.

ANOVA works by partitioning the total variance in the data into different sources of variation. It compares the variance between the groups to the variance within the groups. If the variance between the groups is significantly larger than the variance within the groups, it suggests that there is a significant difference between the means of the groups.

3.1.1. Types of ANOVA

There are several types of ANOVA, each suited for different experimental designs:

  • One-Way ANOVA: This is the simplest type of ANOVA and is used to compare the means of two or more independent groups on a single factor. For example, you might use a one-way ANOVA to compare the test scores of students who were taught using three different methods.
  • Two-Way ANOVA: This type of ANOVA is used to compare the means of two or more independent groups on two factors. For example, you might use a two-way ANOVA to compare the effects of different teaching methods and different classroom environments on student test scores.
  • Repeated Measures ANOVA: This type of ANOVA is used when the same subjects are measured multiple times. For example, you might use a repeated measures ANOVA to compare the blood pressure of patients at different time points after a treatment.

3.1.2. Assumptions of ANOVA

ANOVA relies on several key assumptions to ensure the validity of its results:

  • Independence: The observations must be independent of each other.
  • Normality: The data within each group should be approximately normally distributed.
  • Homogeneity of Variance: The variance of the data in each group should be approximately equal.

3.1.3. Post-Hoc Tests

If the ANOVA test indicates that there is a significant difference between the means of the groups, it is important to conduct post-hoc tests to determine which specific pairs of groups are significantly different from each other. Post-hoc tests are designed to control the Type I error rate and provide more accurate results.

Common post-hoc tests include:

  • Tukey’s Honestly Significant Difference (HSD)
  • Bonferroni correction
  • Scheffé’s method
  • Dunnett’s test

3.2. Tukey-Kramer Pairwise Comparison

Tukey’s Honestly Significant Difference (HSD) test, also known as the Tukey-Kramer method, is a post-hoc test used to compare all possible pairs of means after an ANOVA test has shown a significant difference between the groups. It is designed to control the familywise error rate, which is the probability of making at least one Type I error when comparing multiple pairs of means.

The Tukey-Kramer method is based on the studentized range distribution and provides a more conservative approach to multiple comparisons than some other post-hoc tests. It is widely used in various fields, including agriculture, psychology, and engineering, to compare the means of multiple groups.

3.3. Dunnett’s Comparison to a Control

Dunnett’s test is a post-hoc test used to compare the means of several treatment groups to the mean of a control group. It is designed to control the Type I error rate and provide more accurate results when the primary interest is in comparing each treatment group to a single control group.

Dunnett’s test is particularly useful in situations where there is a control group that represents a standard treatment or a baseline condition. For example, in a clinical trial, you might use Dunnett’s test to compare the effectiveness of several new drugs to the effectiveness of a placebo or a standard treatment.

3.4. Analysis of Means (ANOM)

Analysis of Means (ANOM) is a statistical method used to compare the means of several groups to an overall mean. It is a graphical procedure that provides a visual representation of the differences between the group means and the overall mean.

ANOM is particularly useful in situations where you want to identify which groups have means that are significantly different from the overall mean. It is based on the concept of control limits, which are calculated based on the sample size, the number of groups, and the chosen significance level.

Groups with means that fall outside the control limits are considered to be significantly different from the overall mean. ANOM can be used in a wide range of applications, including quality control, process improvement, and experimental design.

4. Choosing the Right Statistical Test

Selecting the appropriate statistical test is crucial for ensuring the validity and reliability of your research findings. When comparing the means of three or more groups, it is important to consider the characteristics of your data, the research question you are trying to answer, and the assumptions of the statistical tests.

4.1. Factors to Consider

Several factors should be considered when choosing between ANOVA, Tukey-Kramer, Dunnett’s test, and ANOM:

  • Number of Groups: If you are comparing the means of three or more groups, ANOVA is generally the most appropriate choice.
  • Research Question: If you want to compare all possible pairs of means, Tukey-Kramer is a good option. If you want to compare several treatment groups to a control group, Dunnett’s test is more suitable. If you want to identify which groups have means that are significantly different from the overall mean, ANOM can be used.
  • Assumptions: It is important to check the assumptions of each statistical test before using it. ANOVA requires that the data be independent, normally distributed, and have equal variances. Tukey-Kramer, Dunnett’s test, and ANOM also have their own specific assumptions that must be met.
  • Type of Data: The type of data you are working with can also influence your choice of statistical test. ANOVA, Tukey-Kramer, Dunnett’s test, and ANOM are all designed for use with continuous data.

4.2. Guidelines for Selecting a Test

Here are some general guidelines for selecting a statistical test when comparing the means of three or more groups:

  1. If you want to determine whether there is a significant difference between the means of the groups, start with ANOVA.
  2. If ANOVA indicates that there is a significant difference between the means of the groups, conduct post-hoc tests to determine which specific pairs of groups are significantly different from each other.
  3. If you want to compare all possible pairs of means, use Tukey-Kramer.
  4. If you want to compare several treatment groups to a control group, use Dunnett’s test.
  5. If you want to identify which groups have means that are significantly different from the overall mean, use ANOM.
  6. Always check the assumptions of the statistical tests and ensure that they are met.
  7. Consult with a statistician if you are unsure which statistical test to use.

4.3. Example Scenario

Suppose you are conducting a study to compare the effectiveness of four different fertilizers on plant growth. You randomly assign plants to one of the four fertilizers and then measure the height of the plants after a certain period of time.

In this scenario, you would start by using ANOVA to determine whether there is a significant difference between the means of the four groups. If ANOVA indicates that there is a significant difference, you would then conduct post-hoc tests to determine which specific pairs of fertilizers are significantly different from each other.

If you want to compare all possible pairs of fertilizers, you would use Tukey-Kramer. If you have a control group (e.g., plants that receive no fertilizer), you would use Dunnett’s test to compare the effectiveness of each fertilizer to the control group.

5. Real-World Applications and Examples

Understanding how to apply the correct statistical test in real-world scenarios is crucial for drawing meaningful conclusions from data. Here are some examples illustrating the use of ANOVA and other methods in various fields:

5.1. Education

Scenario: A school district wants to evaluate the effectiveness of three different teaching methods on student performance in mathematics.

Approach: Students are randomly assigned to one of the three teaching methods. At the end of the semester, all students take a standardized math test. An ANOVA test is used to compare the mean test scores of the three groups. If the ANOVA test shows a significant difference, Tukey’s HSD test can be used to determine which pairs of teaching methods differ significantly.

Why ANOVA is suitable: ANOVA allows for the comparison of multiple groups (three teaching methods) simultaneously, controlling for the overall Type I error rate.

5.2. Healthcare

Scenario: A pharmaceutical company is testing the efficacy of three different drugs for lowering blood pressure.

Approach: Patients with hypertension are randomly assigned to one of the three drug groups or a placebo group. Blood pressure measurements are taken after six weeks of treatment. ANOVA is used to compare the mean blood pressure reduction in the four groups. Dunnett’s test is then used to compare each drug group to the placebo group to determine which drugs are significantly more effective than the placebo.

Why Dunnett’s Test is suitable: Dunnett’s test is appropriate because the primary interest is in comparing each treatment group to a single control (placebo) group.

5.3. Agriculture

Scenario: An agricultural researcher wants to compare the yield of four different varieties of wheat.

Approach: The researcher plants each variety of wheat in multiple plots of land. At the end of the growing season, the yield (measured in bushels per acre) is recorded for each plot. ANOVA is used to compare the mean yield of the four wheat varieties. If the ANOVA test is significant, Tukey’s HSD test can be used to determine which varieties have significantly different yields.

Why Tukey’s HSD is suitable: Tukey’s HSD test is used because the researcher wants to compare all possible pairs of wheat varieties to identify which ones perform significantly differently.

5.4. Manufacturing

Scenario: A manufacturing company wants to compare the performance of three different machines used to produce widgets.

Approach: The number of widgets produced by each machine is recorded over a series of shifts. ANOVA is used to compare the mean production rate of the three machines. Analysis of Means (ANOM) is used to identify which machines have production rates that are significantly different from the overall mean production rate.

Why ANOM is suitable: ANOM is used because the company wants to identify which machines are performing significantly above or below the average performance.

5.5. Marketing

Scenario: A marketing team wants to compare the effectiveness of four different advertising campaigns on product sales.

Approach: Each advertising campaign is run in a different region. Sales data is collected for each region. ANOVA is used to compare the mean sales increase in the four regions. Post-hoc tests, such as Bonferroni correction, are used to determine which campaigns had a significantly different impact on sales.

Why Bonferroni Correction is suitable: Bonferroni correction is a conservative approach used to control the familywise error rate when conducting multiple comparisons.

6. Step-by-Step Guide to Performing ANOVA

Performing an ANOVA test involves several steps, from data preparation to interpretation of results. Here is a step-by-step guide to help you through the process:

6.1. Step 1: State the Hypotheses

  • Null Hypothesis (H₀): The means of all groups are equal. Mathematically, this can be represented as: $μ₁ = μ₂ = μ₃ = … = μₖ$, where $μ$ represents the mean of each group, and $k$ is the number of groups.
  • Alternative Hypothesis (Hₐ): At least one group mean is different from the others.

6.2. Step 2: Set the Significance Level (α)

The significance level (α) is the probability of rejecting the null hypothesis when it is true. Common values for α are 0.05 (5%) and 0.01 (1%).

6.3. Step 3: Collect and Prepare the Data

Collect data for each group you want to compare. Ensure that the data meets the assumptions of ANOVA:

  • Independence: Observations within each group and between groups are independent.
  • Normality: Data within each group is approximately normally distributed.
  • Homogeneity of Variance: The variance is approximately equal across all groups.

6.4. Step 4: Calculate the Test Statistic (F-statistic)

The F-statistic is the test statistic for ANOVA. It is calculated as follows:

  1. Calculate the Mean for Each Group:

    $μᵢ = frac{Σxᵢ}{nᵢ}$

    where $μᵢ$ is the mean of group $i$, $xᵢ$ are the individual data points in group $i$, and $nᵢ$ is the number of data points in group $i$.

  2. Calculate the Overall Mean:

    $μ = frac{Σμᵢnᵢ}{N}$

    where $N$ is the total number of data points across all groups.

  3. Calculate the Sum of Squares Between Groups (SSB):

    $SSB = Σnᵢ(μᵢ – μ)²$

  4. Calculate the Sum of Squares Within Groups (SSW):

    $SSW = ΣΣ(xᵢ – μᵢ)²$

    This is the sum of the squared differences between each data point and its group mean.

  5. Calculate the Degrees of Freedom:

    • Degrees of Freedom Between Groups ($df_B$): $k – 1$, where $k$ is the number of groups.
    • Degrees of Freedom Within Groups ($df_W$): $N – k$, where $N$ is the total number of data points.

  6. Calculate the Mean Squares:

    • Mean Square Between Groups (MSB): $MSB = frac{SSB}{df_B}$
    • Mean Square Within Groups (MSW): $MSW = frac{SSW}{df_W}$

  7. Calculate the F-statistic:

    $F = frac{MSB}{MSW}$

6.5. Step 5: Determine the Critical Value

The critical value is obtained from the F-distribution table using the chosen significance level (α) and the degrees of freedom ($df_B$ and $df_W$).

6.6. Step 6: Compare the F-statistic to the Critical Value

If the calculated F-statistic is greater than the critical value, reject the null hypothesis.

6.7. Step 7: Draw a Conclusion

  • Reject H₀: There is a statistically significant difference between the means of at least two groups.
  • Fail to Reject H₀: There is no statistically significant difference between the means of the groups.

6.8. Step 8: Perform Post-Hoc Tests (if necessary)

If the ANOVA test indicates a significant difference, perform post-hoc tests (e.g., Tukey’s HSD, Bonferroni, Dunnett’s) to determine which specific pairs of groups differ significantly.

7. Addressing Common Concerns and Misconceptions

When dealing with statistical tests like t-tests and ANOVA, several common concerns and misconceptions can arise. Addressing these can help ensure that you apply these tests correctly and interpret the results accurately.

7.1. Misconception: T-tests can be used for any number of groups if you adjust the significance level.

Reality: While it’s true that you can adjust the significance level (e.g., using Bonferroni correction) to account for multiple comparisons, using t-tests for more than two groups is generally not recommended. ANOVA is specifically designed to handle multiple groups, and it partitions the variance in a way that is more appropriate for this scenario.

7.2. Concern: ANOVA assumes that the data is perfectly normally distributed.

Reality: ANOVA is relatively robust to deviations from normality, especially with larger sample sizes. The central limit theorem suggests that the distribution of sample means will approach normality as the sample size increases. However, if the data is severely non-normal, consider using non-parametric alternatives like the Kruskal-Wallis test.

7.3. Misconception: If ANOVA shows a significant difference, all groups are significantly different from each other.

Reality: A significant result in ANOVA only indicates that at least one group mean is different from the others. To determine which specific pairs of groups differ significantly, you need to perform post-hoc tests like Tukey’s HSD, Bonferroni, or Dunnett’s test.

7.4. Concern: Post-hoc tests always agree with each other.

Reality: Different post-hoc tests have different assumptions and methods for controlling the Type I error rate. As a result, they may not always agree on which pairs of groups are significantly different. It’s important to choose a post-hoc test that is appropriate for your research question and to interpret the results cautiously.

7.5. Misconception: ANOVA can only be used for independent groups.

Reality: While one-way ANOVA is used for independent groups, there are other types of ANOVA (e.g., repeated measures ANOVA) that are designed for situations where the same subjects are measured multiple times.

7.6. Concern: ANOVA is too complicated to perform manually.

Reality: While the calculations involved in ANOVA can be complex, statistical software packages like SPSS, R, and Excel can automate the process. These tools make it easy to perform ANOVA and post-hoc tests with just a few clicks.

7.7. Misconception: ANOVA is only useful for experimental data.

Reality: ANOVA can be used for both experimental and observational data. However, when working with observational data, it’s important to be cautious about drawing causal conclusions, as there may be confounding variables that are not accounted for in the analysis.

8. Resources for Further Learning

To deepen your understanding of t-tests, ANOVA, and other statistical methods, several resources are available:

8.1. Online Courses and Tutorials

  • Coursera: Offers a variety of statistics courses, including those covering ANOVA and t-tests.
  • edX: Provides courses from top universities on statistical analysis and research methods.
  • Khan Academy: Offers free tutorials on basic statistics concepts.
  • Udacity: Provides nanodegree programs in data analysis and statistics.

8.2. Textbooks and Reference Materials

  • “Statistics” by David Freedman, Robert Pisani, and Roger Purves: A comprehensive introduction to statistics with clear explanations and examples.
  • “Statistical Methods for Psychology” by David Howell: A popular textbook for students in psychology and related fields.
  • “Design and Analysis of Experiments” by Douglas Montgomery: A classic textbook on experimental design and ANOVA.
  • “Discovering Statistics Using SPSS” by Andy Field: A user-friendly guide to statistics using SPSS software.

8.3. Statistical Software Documentation

  • SPSS Documentation: Provides detailed information on using SPSS for statistical analysis.
  • R Documentation: Offers comprehensive documentation for the R statistical programming language.
  • SAS Documentation: Provides information on using SAS for statistical analysis.
  • Excel Help: Offers tutorials and help articles on using Excel for basic statistical analysis.

8.4. Online Statistical Calculators

  • GraphPad QuickCalcs: Provides a variety of online statistical calculators, including t-tests and ANOVA.
  • Social Science Statistics: Offers calculators for various statistical tests and procedures.
  • VassarStats: Provides online calculators and tutorials for various statistical concepts.

8.5. Academic Journals and Articles

  • Journal of Applied Statistics: Publishes articles on statistical methods and their applications.
  • Biometrics: Focuses on statistical methods in the biological sciences.
  • Psychological Methods: Publishes articles on statistical methods in psychology.
  • Statistics in Medicine: Focuses on statistical methods in medical research.

9. The Role of COMPARE.EDU.VN

Navigating the complexities of statistical analysis can be challenging. COMPARE.EDU.VN offers a valuable resource by providing detailed comparisons and analyses to guide you in choosing the right statistical test for your specific needs.

9.1. Comprehensive Comparisons

COMPARE.EDU.VN offers comprehensive comparisons of various statistical methods, including t-tests, ANOVA, Tukey-Kramer, Dunnett’s test, and ANOM. These comparisons highlight the strengths and limitations of each method, helping you make an informed decision.

9.2. Real-World Examples

The website provides real-world examples illustrating how to apply different statistical tests in various fields, such as education, healthcare, agriculture, and manufacturing. These examples help you understand the practical applications of statistical methods and how to interpret the results.

9.3. User-Friendly Interface

COMPARE.EDU.VN features a user-friendly interface that makes it easy to find the information you need. The website is designed to be accessible to both novice and experienced researchers.

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9.5. Decision Support

COMPARE.EDU.VN serves as a valuable decision support tool, helping you choose the most appropriate statistical test for your research question and data. By providing clear and concise explanations of complex statistical concepts, the website empowers you to make informed decisions and draw meaningful conclusions from your data.

Don’t struggle with statistical comparisons alone. Visit COMPARE.EDU.VN today to find the detailed comparisons you need to make confident decisions.

10. Frequently Asked Questions (FAQ)

1. When is it appropriate to use a t-test?

A t-test is appropriate when comparing the means of two groups. There are different types of t-tests for different scenarios: one-sample t-test (comparing a sample mean to a known value), independent two-sample t-test (comparing means of two independent groups), and paired t-test (comparing paired measurements from the same group).

2. Why can’t I use a t-test to compare three or more groups?

Using multiple t-tests to compare three or more groups increases the risk of making a Type I error (false positive). ANOVA is a more appropriate method for comparing the means of multiple groups while controlling for the overall Type I error rate.

3. What is ANOVA, and when should I use it?

ANOVA (Analysis of Variance) is a statistical test that compares the means of two or more groups. It should be used when you want to determine whether there is a significant difference between the means of multiple groups.

4. What are post-hoc tests, and why are they necessary?

Post-hoc tests are used after ANOVA to determine which specific pairs of groups differ significantly from each other. They are necessary because ANOVA only tells you whether there is a significant difference between the means of the groups, but not which groups are different.

5. What is Tukey’s HSD test, and when should I use it?

Tukey’s Honestly Significant Difference (HSD) test is a post-hoc test used to compare all possible pairs of means after an ANOVA test has shown a significant difference between the groups. It is designed to control the familywise error rate.

6. What is Dunnett’s test, and when should I use it?

Dunnett’s test is a post-hoc test used to compare the means of several treatment groups to the mean of a control group. It is particularly useful in situations where there is a control group that represents a standard treatment or a baseline condition.

7. What is ANOM, and when should I use it?

Analysis of Means (ANOM) is a statistical method used to compare the means of several groups to an overall mean. It is particularly useful in situations where you want to identify which groups have means that are significantly different from the overall mean.

8. What are the assumptions of ANOVA?

ANOVA relies on several key assumptions to ensure the validity of its results: independence of observations, normality of data within each group, and homogeneity of variance across all groups.

9. How do I check the assumptions of ANOVA?

The assumptions of ANOVA can be checked using various methods, such as examining residual plots, performing normality tests (e.g., Shapiro-Wilk test), and performing homogeneity of variance tests (e.g., Levene’s test).

10. What if my data does not meet the assumptions of ANOVA?

If your data does not meet the assumptions of ANOVA, you may need to use a non-parametric alternative, such as the Kruskal-Wallis test. Alternatively, you may be able to transform your data to meet the assumptions of ANOVA.

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