Paired Samples t-Test Calculator

Paired Samples T-Test Calculator

Enter paired sample data to calculate the t-score, p-value, and visualize the t-distribution.

Data Input Type:

Sample 1 (comma-separated values):

Sample 2 (comma-separated values):

Hypothesis Type:

Right-tailed Left-tailed Two-tailed

Significance Level (α):

0.01 0.05 0.10

T-Score:

Degrees of Freedom:

P-Value:

Understanding the Paired Samples T-Test

The paired samples t-test (also known as the dependent t-test) is used to determine whether there’s a significant difference between the means of two related groups. This test is commonly applied in situations where the same subjects are measured under two different conditions, or matched pairs are compared.

Key Components of the Paired Samples T-Test

Differences (\( d \)): The difference between each pair of observations from the two groups, calculated as \( d = X_1 - X_2 \), where \( X_1 \) and \( X_2 \) are values from the two related samples.
Mean Difference (\( \bar{d} \)): The average of all differences, representing the mean of the paired differences.
Standard Deviation of Differences (\( s_d \)): The standard deviation of the differences, which helps measure the variability of the paired differences.
T-Score: A standardized measure of the mean difference between the pairs, expressed in terms of standard deviations.

Formula for the Paired Samples T-Test

The t-score for a paired samples t-test is calculated as:

\[ t = \frac{\bar{d}}{s_d / \sqrt{n}} \]

where:

\( \bar{d} \): Mean of the differences
\( s_d \): Standard deviation of the differences
\( n \): Number of pairs (sample size)

Programmatically Calculating the Paired Samples T-Test

Below are examples for calculating the paired samples t-test in JavaScript, Python, and R.

1. Using JavaScript (with jStat)

In JavaScript, the jStat library can be used to calculate the paired samples t-test:

// Define inputs
const sample1 = [20, 21, 19, 23]; // Sample 1 values
const sample2 = [18, 20, 17, 22]; // Sample 2 values

// Calculate differences and statistics
const differences = sample1.map((val, index) => val - sample2[index]);
const meanDifference = jStat.mean(differences);
const stdDevDifference = jStat.stdev(differences, true);
const n = differences.length;

// Calculate t-score
const tScore = meanDifference / (stdDevDifference / Math.sqrt(n));

// Calculate p-values based on the t-score
const df = n - 1; // degrees of freedom
const pValueTwoTailed = 2 * (1 - jStat.studentt.cdf(Math.abs(tScore), df));
const pValueRightTailed = 1 - jStat.studentt.cdf(tScore, df);
const pValueLeftTailed = jStat.studentt.cdf(tScore, df);

console.log(`Two-tailed p-value: ${pValueTwoTailed.toFixed(5)}`);
console.log(`Right-tailed p-value: ${pValueRightTailed.toFixed(5)}`);
console.log(`Left-tailed p-value: ${pValueLeftTailed.toFixed(5)}`);

2. Using Python (with SciPy)

In Python, the SciPy library can be used to calculate the paired samples t-test:

from scipy.stats import t

# Define inputs
sample1 = [20, 21, 19, 23]
sample2 = [18, 20, 17, 22]

# Calculate differences and statistics
differences = [x1 - x2 for x1, x2 in zip(sample1, sample2)]
mean_difference = sum(differences) / len(differences)
std_dev_difference = (sum((d - mean_difference) ** 2 for d in differences) / (len(differences) - 1)) ** 0.5
n = len(differences)

# Calculate t-score
t_score = mean_difference / (std_dev_difference / (n ** 0.5))
df = n - 1  # degrees of freedom

# Calculate p-values
p_value_two_tailed = 2 * (1 - t.cdf(abs(t_score), df))
p_value_right_tailed = 1 - t.cdf(t_score, df)
p_value_left_tailed = t.cdf(t_score, df)

print(f"Two-tailed p-value: {p_value_two_tailed:.5f}")
print(f"Right-tailed p-value: {p_value_right_tailed:.5f}")
print(f"Left-tailed p-value: {p_value_left_tailed:.5f}")

3. Using R

In R, you can calculate the paired samples t-test as follows:

# Define inputs
sample1 <- c(20, 21, 19, 23)
sample2 <- c(18, 20, 17, 22)

# Calculate differences
differences <- sample1 - sample2
mean_difference <- mean(differences)
std_dev_difference <- sd(differences)
n <- length(differences)
df <- n - 1  # degrees of freedom

# Calculate t-score
t_score <- mean_difference / (std_dev_difference / sqrt(n))

# Calculate p-values
p_value_two_tailed <- 2 * (1 - pt(abs(t_score), df))
p_value_right_tailed <- 1 - pt(t_score, df)
p_value_left_tailed <- pt(t_score, df)

cat("Two-tailed p-value:", round(p_value_two_tailed, 5), "\n")
cat("Right-tailed p-value:", round(p_value_right_tailed, 5), "\n")
cat("Left-tailed p-value:", round(p_value_left_tailed, 5), "\n")