How to Calculate Paired T-Test?

Calculating a paired t-test involves determining if there's a statistically significant difference between two related sets of observations. This is typically used when you have data points that are naturally paired, like pre-test and post-test scores for the same individuals, or measurements taken on the same subject under different conditions. Here's a step-by-step guide:

1. Calculate the Differences

For each pair of data points, subtract one value from the other. Let's say you have data pairs (X₁, Y₁), (X₂, Y₂), ..., (X_n, Y_n). Calculate the difference d_i = X_i - Y_i for each pair.

2. Calculate the Mean of the Differences (d̄)

Calculate the average of all the differences you computed in step 1.

d̄ = (d₁ + d₂ + ... + d_n) / n

Where:

• d̄ is the mean of the differences
• d_i is the difference for each pair
• n is the number of pairs

3. Calculate the Standard Deviation of the Differences (s_d)

Calculate the standard deviation of the differences. This measures the spread of the differences around the mean difference.

s_d = √[ Σ (d_i - d̄)² / (n - 1) ]

Where:

• s_d is the standard deviation of the differences
• d_i is the difference for each pair
• d̄ is the mean of the differences
• n is the number of pairs

4. Calculate the t-statistic

The t-statistic measures how many standard errors the mean difference is away from zero (the null hypothesis value).

t = d̄ / (s_d / √n)

Where:

• t is the t-statistic
• d̄ is the mean of the differences
• s_d is the standard deviation of the differences
• n is the number of pairs

5. Determine the Degrees of Freedom

The degrees of freedom (df) for a paired t-test are calculated as:

df = n - 1

Where:

• n is the number of pairs

6. Determine the p-value

Using the calculated t-statistic and degrees of freedom, find the corresponding p-value from a t-distribution table or using statistical software. The p-value represents the probability of observing a t-statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.

7. Make a Decision

Compare the p-value to your chosen significance level (alpha), typically 0.05.

• If p-value ≤ alpha: Reject the null hypothesis. There is statistically significant evidence that there's a difference between the two related groups.
• If p-value > alpha: Fail to reject the null hypothesis. There is not enough statistically significant evidence to conclude that there is a difference between the two related groups.

Example:

Let's say you want to test if a training program improved employee performance. You measure the performance of 5 employees before and after the training.

Employee	Before (X)	After (Y)	Difference (d = X - Y)
1	70	75	-5
2	65	72	-7
3	80	85	-5
4	75	82	-7
5	60	68	-8

1. Calculate Differences: (See table above)
2. Mean of Differences (d̄): (-5 + -7 + -5 + -7 + -8) / 5 = -6.4
3. Standard Deviation of Differences (s_d): ≈ 1.34
4. t-statistic: -6.4 / (1.34 / √5) ≈ -10.66
5. Degrees of Freedom: 5 - 1 = 4
6. P-value: Using a t-distribution table or software, the p-value for t = -10.66 and df = 4 is very small (close to 0).
7. Decision: Since the p-value is less than 0.05, we reject the null hypothesis. There is statistically significant evidence that the training program improved employee performance.

In summary, calculating a paired t-test allows you to determine if a significant difference exists between two related sets of data by quantifying the mean difference, its variability, and comparing this to an expected null scenario.