What Statistical Test Is Used To Compare Two Groups? The right statistical test to compare two groups depends on the type of data you have and the nature of the comparison you want to make. This comprehensive guide, brought to you by compare.edu.vn, will walk you through the process of selecting the most appropriate test. Whether you’re dealing with numerical or categorical data, understanding your options is the first step toward accurate analysis, and using statistical tools effectively enhances data-driven decision-making capabilities, enabling you to draw meaningful insights from your research and comparisons.
1. Understanding Statistical Tests
Statistical tests are mathematical tools used to determine whether there is a significant difference between two or more sets of data. These tests use statistical measures such as the mean, standard deviation, and coefficient of variation to compare data against predetermined criteria. If the data meet these criteria, the test concludes that a significant difference exists.
2. Types of Statistical Tests
There are two main categories of statistical tests: parametric and non-parametric. Parametric tests have stricter requirements and make stronger inferences, while non-parametric tests are more flexible but less accurate.
2.1. Parametric Statistical Tests
Parametric tests require data to meet certain assumptions, such as normal distribution and homogeneity of variance. These tests are powerful and can provide detailed insights when used appropriately.
2.1.1. Regression Tests
Regression tests are used to determine cause-and-effect relationships between variables. They estimate the effect of one or more continuous variables on another variable.
- Simple Linear Regression: Describes the relationship between a dependent and an independent variable using a straight line. It’s used to determine the relationship between two quantitative variables.
- Multiple Linear Regression: Measures the relationship between a quantitative dependent variable and two or more independent variables, using a straight line.
- Logistic Regression: Predicts and classifies research problems, helping to identify data anomalies that could indicate fraud.
2.1.2. Comparison Tests
Comparison tests determine the differences among group means. They are used to test the effect of a categorical variable on the mean value of other characteristics.
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T-Test: Compares the means of two groups when population parameters are unknown. There are several types of t-tests:
- Paired T-Test: Tests the difference between two variables from the same population, such as pre- and post-test scores.
- Independent T-Test: Also known as a two-sample t-test, it determines whether there is a statistically significant difference between the means in two unrelated groups.
- One-Sample T-Test: Compares the mean of a single group with a given mean.
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ANOVA (Analysis of Variance): Analyzes the difference between the means of more than two groups.
- One-Way ANOVA: Determines how one factor impacts another.
- Two-Way ANOVA: Compares samples with different variables.
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MANOVA (Multivariate Analysis of Variance): Provides regression analysis and analysis of variance for multiple dependent variables by one or more factor variables or covariates. It examines the statistical difference between one continuous dependent variable and an independent grouping variable.
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Z-Test: Determines whether two population means are different, provided the variances are known and the sample size is large.
2.1.3. Correlation Tests
Correlation tests check if variables are related without hypothesizing a cause-and-effect relationship. These tests can be used to check if two variables are correlated before using them in a multiple regression test.
- Pearson Correlation Coefficient: Measures the linear correlation between two variables. The coefficient ranges from -1 to 1, indicating the strength and direction of the relationship.
2.2. Non-Parametric Statistical Tests
Non-parametric tests make fewer assumptions about the data and are useful when parametric assumptions are violated. However, the inferences from these tests are generally less accurate.
- Chi-Square Test: Compares two categorical variables to assess whether the observed frequency is significantly different from the expected frequency.
3. How to Choose the Right Statistical Test
Selecting the right statistical test involves several key considerations. Here are seven essential steps to guide your decision-making process.
3.1. Research Question
The choice of a statistical test depends on the research question you need to answer. The research question will help you formulate the data structure and research design.
3.2. Formulation of Null Hypothesis
Develop a null hypothesis after defining the research question. A null hypothesis suggests that no statistical significance exists in the expected observations.
3.3. Level of Significance in Study Protocol
Specify a level of significance before conducting the study. The level of significance determines the statistical importance, defining the acceptance or rejection of the null hypothesis.
3.4. One-Tailed vs. Two-Tailed Test
Decide whether your study should be a one-tailed or two-tailed test. Use a one-tailed test if you have clear evidence indicating the direction of the statistics. If there is no particular direction of the expected difference, use a two-tailed test.
3.5. Number of Variables
Statistical tests and procedures are divided according to the number of variables they are designed to analyze. Consider how many variables you want to analyze when choosing a test.
3.6. Type of Data
Determine whether your data is continuous, categorical, or binary. For continuous data, check if it is normally distributed or skewed to further refine your choice of statistical test.
3.7. Paired vs. Unpaired Study Designs
A paired design includes comparison studies where the two population means are compared when the two samples depend on each other. In an unpaired or independent study design, the results of the two samples are grouped and then compared.
4. Common Statistical Tests for Comparing Two Groups
Several statistical tests are commonly used to compare two groups, each suited to different types of data and research questions.
4.1. T-Tests
T-tests are among the most frequently used statistical tests for comparing the means of two groups. They are particularly useful when you want to determine if there is a significant difference between the average values of two sets of data.
4.1.1. Independent Samples T-Test
The independent samples t-test, also known as the two-sample t-test, is used to determine if there is a statistically significant difference between the means of two independent groups. “Independent” here means that the two groups being compared are not related in any way; the data from one group does not influence the data from the other group.
When to Use:
- When you have two separate and unrelated groups.
- When you want to compare the average values (means) of these two groups.
- When the data is approximately normally distributed.
- When you don’t know the population standard deviations.
Examples:
- Comparing the test scores of students from two different schools.
- Comparing the effectiveness of two different drugs by measuring patient outcomes in two separate treatment groups.
- Comparing the average income of men and women in two distinct professions.
Hypotheses:
- Null Hypothesis (H0): There is no significant difference between the means of the two groups (μ1 = μ2).
- Alternative Hypothesis (H1): There is a significant difference between the means of the two groups (μ1 ≠ μ2).
Assumptions:
- Independence: The observations within each group are independent of each other.
- Normality: The data within each group is approximately normally distributed.
- Homogeneity of Variance: The variance (spread) of data in each group is approximately equal. If this assumption is violated, a modified t-test (Welch’s t-test) can be used.
How it Works:
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Calculate the Means and Standard Deviations: Compute the mean (average) and standard deviation for each group.
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Calculate the T-Statistic: The t-statistic measures the difference between the means of the two groups relative to the variability within the groups. The formula for the t-statistic is:
t = (X1 – X2) / √((s1^2/n1) + (s2^2/n2))
Where:
- X1 is the mean of the first group.
- X2 is the mean of the second group.
- s1^2 is the variance of the first group.
- s2^2 is the variance of the second group.
- n1 is the sample size of the first group.
- n2 is the sample size of the second group.
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Determine the Degrees of Freedom: The degrees of freedom (df) is a value that depends on the sample sizes of the two groups. For an independent samples t-test, the degrees of freedom is typically calculated as:
df = n1 + n2 – 2
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Find the P-Value: The p-value is the probability of observing a t-statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true. The p-value is obtained from a t-distribution table or using statistical software.
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Make a Decision: Compare the p-value to a predetermined significance level (alpha), usually set at 0.05.
- If the p-value ≤ alpha: Reject the null hypothesis. This means there is a statistically significant difference between the means of the two groups.
- If the p-value > alpha: Fail to reject the null hypothesis. This means there is not enough evidence to conclude that there is a statistically significant difference between the means of the two groups.
Example Calculation:
Suppose we want to compare the test scores of students from two different schools. We have the following data:
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School A:
- Sample Size (n1) = 30
- Mean Score (X1) = 82
- Standard Deviation (s1) = 6
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School B:
- Sample Size (n2) = 40
- Mean Score (X2) = 78
- Standard Deviation (s2) = 8
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Calculate the T-Statistic:
t = (82 – 78) / √((6^2/30) + (8^2/40))
t = 4 / √((36/30) + (64/40))
t = 4 / √(1.2 + 1.6)
t = 4 / √2.8
t ≈ 4 / 1.673
t ≈ 2.39
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Determine the Degrees of Freedom:
df = 30 + 40 – 2
df = 68
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Find the P-Value:
Using a t-distribution table or statistical software with df = 68, we find that the p-value for t = 2.39 is approximately 0.02.
-
Make a Decision:
Since the p-value (0.02) is less than the significance level (0.05), we reject the null hypothesis.
Conclusion:
There is a statistically significant difference between the test scores of students from School A and School B.
Advantages:
- Simple and widely used.
- Effective for comparing means of two independent groups.
Disadvantages:
- Assumes data is approximately normally distributed.
- Less accurate if sample sizes are small and data is not normally distributed.
- Requires homogeneity of variance (or use Welch’s t-test).
4.1.2. Paired Samples T-Test
The paired samples t-test, also known as the dependent samples t-test, is used to determine if there is a statistically significant difference between the means of two related groups. This test is appropriate when you have two sets of observations from the same subjects or matched pairs, and you want to assess whether a significant change has occurred.
When to Use:
- When you have two sets of data from the same individuals or matched pairs.
- When you want to compare the means of these two related sets of data.
- When you are assessing the effect of an intervention, treatment, or change on the same subjects.
- When the data is approximately normally distributed.
Examples:
- Measuring the blood pressure of patients before and after taking a new medication.
- Comparing the performance scores of employees before and after completing a training program.
- Comparing the ratings of a product by the same group of users before and after a product update.
Hypotheses:
- Null Hypothesis (H0): There is no significant difference between the means of the two related groups (μ1 = μ2).
- Alternative Hypothesis (H1): There is a significant difference between the means of the two related groups (μ1 ≠ μ2).
Assumptions:
- Dependence: The observations in the two groups are dependent on each other (i.e., they come from the same subjects or matched pairs).
- Normality: The differences between the paired observations are approximately normally distributed.
How it Works:
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Calculate the Differences: For each subject or pair, calculate the difference between the two observations (d = X1 – X2).
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Calculate the Mean of the Differences: Compute the mean (average) of these differences (d̄).
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Calculate the Standard Deviation of the Differences: Compute the standard deviation of the differences (sd).
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Calculate the T-Statistic: The t-statistic measures the difference between the means of the paired observations relative to the variability of the differences. The formula for the t-statistic is:
t = d̄ / (sd / √n)
Where:
- d̄ is the mean of the differences.
- sd is the standard deviation of the differences.
- n is the number of pairs.
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Determine the Degrees of Freedom: The degrees of freedom (df) is a value that depends on the number of pairs. For a paired samples t-test, the degrees of freedom is calculated as:
df = n – 1
-
Find the P-Value: The p-value is the probability of observing a t-statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true. The p-value is obtained from a t-distribution table or using statistical software.
-
Make a Decision: Compare the p-value to a predetermined significance level (alpha), usually set at 0.05.
- If the p-value ≤ alpha: Reject the null hypothesis. This means there is a statistically significant difference between the means of the two related groups.
- If the p-value > alpha: Fail to reject the null hypothesis. This means there is not enough evidence to conclude that there is a statistically significant difference between the means of the two related groups.
Example Calculation:
Suppose we want to compare the performance scores of employees before and after completing a training program. We have the following data for 10 employees:
| Employee | Before Training (X1) | After Training (X2) | Difference (d = X2 – X1) |
|---|---|---|---|
| 1 | 65 | 75 | 10 |
| 2 | 70 | 80 | 10 |
| 3 | 68 | 72 | 4 |
| 4 | 72 | 78 | 6 |
| 5 | 75 | 85 | 10 |
| 6 | 60 | 65 | 5 |
| 7 | 63 | 70 | 7 |
| 8 | 67 | 75 | 8 |
| 9 | 72 | 82 | 10 |
| 10 | 78 | 85 | 7 |
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Calculate the Differences: The differences are already calculated in the table above.
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Calculate the Mean of the Differences:
d̄ = (10 + 10 + 4 + 6 + 10 + 5 + 7 + 8 + 10 + 7) / 10
d̄ = 77 / 10
d̄ = 7.7
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Calculate the Standard Deviation of the Differences:
First, calculate the squared differences:
Difference (d) Squared Difference (d^2) 10 100 10 100 4 16 6 36 10 100 5 25 7 49 8 64 10 100 7 49 Sum of squared differences = 639
Variance = (Sum of squared differences – n * d̄^2) / (n – 1)
Variance = (639 – 10 * 7.7^2) / 9
Variance = (639 – 10 * 59.29) / 9
Variance = (639 – 592.9) / 9
Variance = 46.1 / 9
Variance ≈ 5.122
Standard Deviation (sd) = √Variance
sd ≈ √5.122
sd ≈ 2.263
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Calculate the T-Statistic:
t = 7.7 / (2.263 / √10)
t = 7.7 / (2.263 / 3.162)
t = 7.7 / 0.7157
t ≈ 10.76
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Determine the Degrees of Freedom:
df = 10 – 1
df = 9
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Find the P-Value:
Using a t-distribution table or statistical software with df = 9, we find that the p-value for t = 10.76 is very close to 0.000.
-
Make a Decision:
Since the p-value (0.000) is less than the significance level (0.05), we reject the null hypothesis.
Conclusion:
There is a statistically significant difference between the performance scores of employees before and after completing the training program.
Advantages:
- Controls for individual differences by comparing each subject to themselves.
- More powerful than independent samples t-test when appropriate.
Disadvantages:
- Can only be used when data is paired or matched.
- Assumes that the differences are approximately normally distributed.
4.1.3. One-Sample T-Test
The one-sample t-test is used to determine whether the mean of a single group is significantly different from a known or hypothesized value. This test is useful when you want to compare a sample mean to a specific target or benchmark.
When to Use:
- When you have one set of data from a single group.
- When you want to compare the mean of this group to a known or hypothesized population mean.
- When you don’t know the population standard deviation.
- When the data is approximately normally distributed.
Examples:
- Comparing the average height of students in a class to the national average height.
- Comparing the average test score of students in a school to a target score set by the administration.
- Comparing the average processing time of a computer program to a benchmark time.
Hypotheses:
- Null Hypothesis (H0): The mean of the sample is equal to the known or hypothesized population mean (μ = μ0).
- Alternative Hypothesis (H1): The mean of the sample is significantly different from the known or hypothesized population mean (μ ≠ μ0).
Assumptions:
- Independence: The observations within the sample are independent of each other.
- Normality: The data in the sample is approximately normally distributed.
How it Works:
-
Calculate the Sample Mean: Compute the mean (average) of the sample data (X̄).
-
Calculate the Sample Standard Deviation: Compute the standard deviation of the sample data (s).
-
Calculate the T-Statistic: The t-statistic measures the difference between the sample mean and the known or hypothesized population mean relative to the variability within the sample. The formula for the t-statistic is:
t = (X̄ – μ0) / (s / √n)
Where:
- X̄ is the sample mean.
- μ0 is the known or hypothesized population mean.
- s is the sample standard deviation.
- n is the sample size.
-
Determine the Degrees of Freedom: The degrees of freedom (df) is a value that depends on the sample size. For a one-sample t-test, the degrees of freedom is calculated as:
df = n – 1
-
Find the P-Value: The p-value is the probability of observing a t-statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true. The p-value is obtained from a t-distribution table or using statistical software.
-
Make a Decision: Compare the p-value to a predetermined significance level (alpha), usually set at 0.05.
- If the p-value ≤ alpha: Reject the null hypothesis. This means there is a statistically significant difference between the mean of the sample and the known or hypothesized population mean.
- If the p-value > alpha: Fail to reject the null hypothesis. This means there is not enough evidence to conclude that there is a statistically significant difference between the mean of the sample and the known or hypothesized population mean.
Example Calculation:
Suppose we want to compare the average test score of students in a school to a target score of 75 set by the administration. We have the following data for a sample of 25 students:
- Sample Size (n) = 25
- Sample Mean (X̄) = 78
- Sample Standard Deviation (s) = 10
- Target Score (μ0) = 75
-
Calculate the T-Statistic:
t = (78 – 75) / (10 / √25)
t = 3 / (10 / 5)
t = 3 / 2
t = 1.5
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Determine the Degrees of Freedom:
df = 25 – 1
df = 24
-
Find the P-Value:
Using a t-distribution table or statistical software with df = 24, we find that the p-value for t = 1.5 is approximately 0.145.
-
Make a Decision:
Since the p-value (0.145) is greater than the significance level (0.05), we fail to reject the null hypothesis.
Conclusion:
There is not enough evidence to conclude that there is a statistically significant difference between the average test score of students in the school and the target score of 75.
Advantages:
- Simple and straightforward to use.
- Useful for comparing a sample mean to a known or hypothesized value.
Disadvantages:
- Only applicable to one sample.
- Assumes data is approximately normally distributed.
4.2. ANOVA (Analysis of Variance)
ANOVA is used to compare the means of three or more groups. It determines if there is a significant difference between the groups, but it doesn’t identify which specific groups are different.
4.2.1. When to Use ANOVA
ANOVA is a powerful statistical test used to compare the means of three or more independent groups. It is particularly useful when you want to determine if there is a significant difference among the groups, but you are not necessarily interested in comparing each pair of groups individually.
When to Use:
- When you have three or more independent groups.
- When you want to compare the means of these groups to determine if there is a significant difference overall.
- When you want to avoid inflating the Type I error rate that would occur if you performed multiple t-tests.
- When the data is approximately normally distributed.
- When the variances of the groups are approximately equal (homogeneity of variance).
Examples:
- Comparing the effectiveness of three different teaching methods on student test scores.
- Comparing the yields of four different varieties of wheat in an agricultural study.
- Comparing the customer satisfaction scores of five different product designs.
Hypotheses:
- Null Hypothesis (H0): The means of all groups are equal (μ1 = μ2 = μ3 = … = μk).
- Alternative Hypothesis (H1): At least one group mean is different from the others.
Assumptions:
- Independence: The observations within each group are independent of each other.
- Normality: The data within each group is approximately normally distributed.
- Homogeneity of Variance: The variances of the data in each group are approximately equal.
How it Works:
-
Calculate the Overall Mean: Compute the overall mean by combining all data points from all groups.
-
Calculate the Sum of Squares Between Groups (SSB): SSB measures the variability between the group means and the overall mean. It is calculated as:
SSB = Σ n_i * (X̄_i – X̄)^2
Where:
- n_i is the sample size of group i.
- X̄_i is the mean of group i.
- X̄ is the overall mean.
- Σ indicates the sum over all groups.
-
Calculate the Sum of Squares Within Groups (SSW): SSW measures the variability within each group. It is calculated as:
SSW = Σ (n_i – 1) * s_i^2
Where:
- s_i^2 is the variance of group i.
- Σ indicates the sum over all groups.
-
Calculate the Degrees of Freedom:
- Degrees of freedom between groups (dfb) = k – 1, where k is the number of groups.
- Degrees of freedom within groups (dfw) = N – k, where N is the total number of observations across all groups.
-
Calculate the Mean Squares:
- Mean square between groups (MSB) = SSB / dfb.
- Mean square within groups (MSW) = SSW / dfw.
-
Calculate the F-Statistic: The F-statistic is the ratio of the mean square between groups to the mean square within groups. It is calculated as:
F = MSB / MSW
-
Find the P-Value: The p-value is the probability of observing an F-statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true. The p-value is obtained from an F-distribution table or using statistical software.
-
Make a Decision: Compare the p-value to a predetermined significance level (alpha), usually set at 0.05.
- If the p-value ≤ alpha: Reject the null hypothesis. This means there is a statistically significant difference among the means of the groups.
- If the p-value > alpha: Fail to reject the null hypothesis. This means there is not enough evidence to conclude that there is a statistically significant difference among the means of the groups.
Example Calculation:
Suppose we want to compare the effectiveness of three different teaching methods on student test scores. We have the following data for each teaching method:
| Teaching Method | Sample Size (n) | Mean Score (X̄) | Variance (s^2) |
|---|---|---|---|
| Method A | 20 | 80 | 25 |
| Method B | 20 | 85 | 30 |
| Method C | 20 | 75 | 20 |
- Calculate the Overall Mean:
X̄ = (80 + 85 + 75) / 3 = 80 - Calculate the Sum of Squares Between Groups (SSB):
SSB = 20 (80 – 80)^2 + 20 (85 – 80)^2 + 20 (75 – 80)^2
SSB = 20 0 + 20 25 + 20 25
SSB = 0 + 500 + 500
SSB = 1000 - Calculate the Sum of Squares Within Groups (SSW):
SSW = (20 – 1) 25 + (20 – 1) 30 + (20 – 1) 20
SSW = 19 25 + 19 30 + 19 20
SSW = 475 + 570 + 380
SSW = 1425 - Calculate the Degrees of Freedom:
- dfb = 3 – 1 = 2
- dfw = (20 + 20 + 20) – 3 = 60 – 3 = 57
- Calculate the Mean Squares:
- MSB = 1000 / 2 = 500
- MSW = 1425 / 57 = 25
- Calculate the F-Statistic:
F = 500 / 25 = 20 - Find the P-Value:
Using an F-distribution table or statistical software with dfb = 2 and dfw = 57, we find that the p-value for F = 20 is very close to 0.000. - Make a Decision:
Since the p-value (0.000) is less than the significance level (0.05), we reject the null hypothesis.
Conclusion:
There is a statistically significant difference among the means of the teaching methods.
Advantages:
- Allows for comparison of means of three or more groups.
- Avoids inflating the Type I error rate.
Disadvantages:
- Only indicates that there is a difference, but not which specific groups are different.
- Assumes data is approximately normally distributed and variances are approximately equal.
4.2.2. Post-Hoc Tests
If ANOVA indicates a significant difference, post-hoc tests are used to determine which specific groups differ from each other. Common post-hoc tests include Tukey’s HSD (Honestly Significant Difference), Bonferroni, and Scheffé.
4.3. Non-Parametric Tests: Mann-Whitney U Test
When the data is not normally distributed, non-parametric tests like the Mann-Whitney U test are used to compare two independent groups. This test assesses whether two samples are likely to derive from the same population.
4.3.1. When to Use the Mann-Whitney U Test
The Mann-Whitney U test, also known as the Wilcoxon rank-sum test, is a non-parametric statistical test used to compare two independent groups. It is particularly useful when your data does not meet the assumptions of parametric tests such as t-tests, especially when the data is not normally distributed or when the sample sizes are small.
When to Use:
- When you have two independent groups.
- When your data is not normally distributed.
- When you have small sample sizes.
- When you want to compare the medians (rather than the means) of the two groups.
- When your data is ordinal (ranked) or interval/ratio but violates normality assumptions.
Examples:
- Comparing the satisfaction scores of customers who use two different product versions, where the scores are on a non-continuous scale (e.g., 1 to 5 stars).
- Comparing the time it takes for employees to complete a task using two different software programs, where the data is skewed or has outliers.
- Comparing the test scores of two groups of students where the scores are not normally distributed due to factors like a ceiling or floor effect.
Hypotheses:
- Null Hypothesis (H0): The two groups come from the same population, and there is no difference in their distributions.
- Alternative Hypothesis (H1): The two groups do not come from the same population, and there is a difference in their distributions.
Assumptions:
- Independence: The observations within each group are independent of each other, and the two groups are independent of each other.
- Ordinal Scale: The data should be measured on at least an ordinal scale, meaning that the values can be ranked.
- Similar Distribution Shape: The two groups should have a similar shape of distribution (although not necessarily normal).
How it Works:
-
Combine and Rank the Data: Combine all the data points from both groups into a single dataset and rank them from lowest to highest. Assign ranks starting from 1 to the smallest value, and continue assigning ranks in ascending order. In case of ties (identical values), assign the average rank to each tied value.
-
Calculate the Rank Sums: Calculate the sum of the ranks for each group. Let R1 be the sum of the ranks for Group 1, and R2 be the sum of the ranks for Group 2.
-
Calculate the U Statistics: Calculate the U statistic for each group using the following formulas:
U1 = n1 n2 + (n1 (n1 + 1)) / 2 – R1
U2 = n1 n2 + (n2 (n2 + 1)) / 2 – R2
Where:
- n1 is the sample size of Group 1.
- n2 is the sample size of Group 2.
- R1 is the sum of the ranks for Group 1.
- R2 is the sum of the ranks for Group 2.
-
Determine the Test Statistic: The test statistic U is the smaller of U1 and U2.
U = min(U1, U2)
-
Find the P-Value: The p-value is the probability of observing a U statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true. The p-value is obtained from Mann-Whitney U test tables or using statistical software.
-
Make a Decision: Compare the p-value to a predetermined significance level (alpha), usually set at 0.05.
- If the p-value ≤ alpha: Reject the null hypothesis. This means there is a statistically significant difference between the distributions of the two groups.
- If the p-value > alpha: Fail to reject the null hypothesis. This means there is not enough evidence to conclude that there is a statistically significant difference between the distributions of the two groups.
Example Calculation:
Suppose we want to compare the satisfaction scores of customers who use two different product versions. The scores are on a scale of 1 to 10 and are not normally distributed. We have the following data:
| Customer | Product Version A (Scores)
