Differences Between two-sample, t-test and paired t-test

Statistical tests are essential tools in data analysis, helping researchers make inferences about populations based on sample data. Two common tests used to compare the means of different groups are the two-sample t-test and the paired t-test. Both tests are based on the t-distribution, but they have distinct use cases and assumptions. In this article, we’ll explore the differences between these two tests in R, when to use each one, and how to conduct them in practice.

Two-Sample T-Test

The two-sample t-test, also known as an independent t-test, is used to determine whether there is a significant difference between the means of two independent (unrelated) groups. It is typically used when you have two separate groups and want to assess whether their means are statistically different from each other.

The formula for the Two-Sample t-test is given by:

where

• and are the sample means,

• n1 and n2 are the sample sizes,
• degree of freedom = n1 + n2 – 2
• and where sp is calculated as:

Here are some key characteristics of the two-sample t-test:

Assumptions:

• The data in each group should follow a normal distribution.
• The variances of the two groups should be approximately equal (homogeneity of variances).

Use Cases:

• Comparing the average test scores of students from two different schools.
• Assessing whether there is a significant difference in the average salaries of employees in two different departments.

Hypotheses:

Null Hypothesis (H0): There is no significant difference between the means of the two groups.

Alternative Hypothesis (Ha): There is a significant difference between the means of the two groups.

Paired T-Test

The paired t-test, also known as a dependent t-test or matched-pairs t-test, is used when you want to compare the means of two related groups or when each data point in one group is naturally paired with a data point in the other group. The formula for the paired t-test is given by:

Where,

• Σd is the sum of the differences
• degree of freedom = n – 1

Here are some key characteristics of the paired t-test:

Assumptions:

• The differences between the pairs should follow a normal distribution.
• The paired differences should be independent.

Use Cases:

• Comparing the performance of students before and after a tutoring program (where each student’s score is measured both before and after).
• Evaluating whether a new drug has a significant effect on blood pressure (with measurements taken before and after administering the drug).

Hypotheses:

Null Hypothesis (H0): There is no significant difference between the means of the paired groups (the mean of the differences is zero).

H0: u1 = u2 or H0: u1 –u2 = 0

Alternative Hypothesis (Ha): There is a significant difference between the means of the paired groups.

H1: u1 is not equal to u2 or H1: u1 – u2 is not equal to zero.

Key differences between the two-sample t-test and paired t-test

Function	Two-Sample T-Test	Paired T-Test
Data Relationship	Compares means of two independent groups with no natural pairing between the observations.	Compares means of two related groups where each data point in one group is paired with a data point in the other.
Assumptions	Assumes independence of samples and may assume equal variances.	Assumes that the paired differences follow a normal distribution and are independent.
Use Cases	Used when you want to compare two distinct groups or populations.	Used when you have before-and-after measurements or paired data points.

Code for Two-Sample t-Test: Comparing School Scores

R

# Generate example data
set.seed(123)
school1_scores <- rnorm(30, mean = 75, sd = 10)
school2_scores <- rnorm(30, mean = 80, sd = 12)

# Perform a two-sample t-test
t_test_result <- t.test(school1_scores, school2_scores)

# Print the result
print(t_test_result)

Output:

    Welch Two Sample t-test
data:  school1_scores and school2_scores
t = -2.9726, df = 57.974, p-value = 0.004295
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -12.736395  -2.485801
sample estimates:
mean of x mean of y 
 74.52896  82.14006 

• We generate two sets of example data, school1_scores and school2_scores, each representing the exam scores of students from two different schools.
• We perform a two-sample t-test using the t.test function, comparing the means of the two groups.
• The result includes the t-statistic, degrees of freedom, and p-value, which can be used to assess whether there is a significant difference between the means of the two groups.
• The p-value is exceptionally small, smaller than 0.05. This suggests strong evidence against the null hypothesis, indicating that there is a statistically significant difference in the mean scores between the two schools.

Code for Paired t-Test: Before and After Treatment Comparison

R

# Generate example data
set.seed(456)
before_treatment <- rnorm(20, mean = 140, sd = 10)
after_treatment <- before_treatment - rnorm(20, mean = 5, sd = 4)

# Perform a paired t-test
paired_t_test_result <- t.test(before_treatment, after_treatment, paired = TRUE)

# Print the result
print(paired_t_test_result)

Output:

    Paired t-test
data:  before_treatment and after_treatment
t = 4.6673, df = 19, p-value = 0.0001679
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
 2.227111 5.848578
sample estimates:
mean difference 
       4.037844 

• We generate example data for the blood pressure of patients before and after a drug treatment. The before_treatment and after_treatment vectors represent paired measurements.
• We perform a paired t-test using the t.test function with the paired = TRUE argument, indicating that the measurements are paired.
• The result includes the t-statistic, degrees of freedom, and p-value, allowing us to assess whether there is a significant difference in blood pressure before and after treatment.
• The p-value is exceptionally small, much smaller than the common significance level of 0.05. This suggests strong evidence against the null hypothesis, indicating that there is a statistically significant difference in the means before and after the treatment.

Comparing Product Sales with Two-Sample T-Test

R

# Generate example data
set.seed(456)
product_A_sales <- rnorm(40, mean = 500, sd = 50)
product_B_sales <- rnorm(45, mean = 480, sd = 60)

# Perform a two-sample t-test
t_test_result <- t.test(product_A_sales, product_B_sales)

# Print the result
print(t_test_result)

Output:

    Welch Two Sample t-test
data:  product_A_sales and product_B_sales
t = 1.6953, df = 82.328, p-value = 0.0938
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -3.609848 45.253099
sample estimates:
mean of x mean of y 
 506.0621  485.2404 

• The p-value is 0.09 which is greater than 0.05, therefore we accept our null hypothesis.

Paired T-Test for Exam Scores Before and After a Training Course

R

# Generate example data
set.seed(987)
before_scores <- rnorm(35, mean = 60, sd = 8)
after_scores <- before_scores + rnorm(35, mean = 10, sd = 5)

# Perform a paired t-test
paired_t_test_result <- t.test(before_scores, after_scores, paired = TRUE)

# Print the result
print(paired_t_test_result)

Output:

    Paired t-test
data:  before_scores and after_scores
t = -13.078, df = 34, p-value = 8.018e-15
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
 -12.877510  -9.413617
sample estimates:
mean difference 
      -11.14556 

• The p-value is exceedingly small (8.018e-15), much smaller than the common significance level of 0.05. This suggests a very strong evidence against the null hypothesis. Hence, we reject our null hypothesis indicating that there is a statistically significant difference in means between the “before_scores” and “after_scores.”.

Conclusion

In summary, understanding the differences between the two-sample t-test and paired t-test is crucial for selecting the appropriate statistical test for your research or data analysis. Each test has its own set of assumptions and use cases, and choosing the wrong test can lead to incorrect conclusions. By matching the test to your data and research question, you can make valid statistical inferences and draw meaningful conclusions from your analyses.