Paired t-Test Formula

Paired t-test is a hypothesis test for the mean difference in a paired (matched) data design, where each subject provides two related measurements.

The Formula

t=dห‰โˆ’ฮผd0sd/nt = \frac{\bar{d} - \mu_{d_0}}{s_d / \sqrt{n}}

When to use: You want to know if a tutoring program improves math scores. Instead of comparing two separate groups, you test the SAME students before and after tutoring. Each student is their own control. By looking at the difference (after โˆ’- before) for each student, you eliminate individual variation and focus purely on the change.

Quick Example

5 students' scores before and after tutoring: differences d=+8,+3,+12,+5,+7d = +8, +3, +12, +5, +7. dห‰=7,sd=3.39,t=7โˆ’03.39/5=71.52โ‰ˆ4.61(df=4)\bar{d} = 7, \quad s_d = 3.39, \quad t = \frac{7 - 0}{3.39 / \sqrt{5}} = \frac{7}{1.52} \approx 4.61 \quad (df = 4) Strong evidence that tutoring improved scores (p<0.01p < 0.01).

Notation

dห‰\bar{d} = mean of differences, sds_d = standard deviation of differences, nn = number of pairs, df=nโˆ’1df = n - 1. Usually H0:ฮผd=0H_0: \mu_d = 0.

What This Formula Means

A hypothesis test for the mean difference in a paired (matched) data design, where each subject provides two related measurements. The test analyzes the differences di=x1iโˆ’x2id_i = x_{1i} - x_{2i} as a single sample.

You want to know if a tutoring program improves math scores. Instead of comparing two separate groups, you test the SAME students before and after tutoring. Each student is their own control. By looking at the difference (after โˆ’- before) for each student, you eliminate individual variation and focus purely on the change.

Formal View

t=dห‰โˆ’ฮผd0sd/nt = \frac{\bar{d} - \mu_{d_0}}{s_d / \sqrt{n}} with df=nโˆ’1df = n - 1 where di=x1iโˆ’x2id_i = x_{1i} - x_{2i} and dห‰=1nโˆ‘di\bar{d} = \frac{1}{n}\sum d_i

Worked Examples

Example 1

medium
Students' scores before and after tutoring: Before: {70,65,80,75,60}\{70, 65, 80, 75, 60\}, After: {75,70,85,80,70}\{75, 70, 85, 80, 70\}. Conduct a paired t-test at ฮฑ=0.05\alpha=0.05 to test if tutoring improved scores.

Answer

tโ‰ˆ6.51>2.132t \approx 6.51 > 2.132. Reject H0H_0. Tutoring significantly improved scores.

First step

1
Differences di=Afterโˆ’Befored_i = \text{After} - \text{Before}: 5,5,5,5,105, 5, 5, 5, 10

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Example 2

hard
Explain when to use a paired t-test vs. a two-sample t-test. If shoe comfort is measured on the same subjects wearing Brand A and Brand B, which test is appropriate and why?

Example 3

easy
Five pairs give differences {0,2,0,4,4}\{0, 2, 0, 4, 4\}. Compute dห‰\bar{d} and state the null hypothesis for a paired t-test.

Common Mistakes

  • Treating paired data as two independent samples - form the per-pair differences and run a one-sample test on them.
  • Using df=n1+n2โˆ’2df=n_1+n_2-2 - the paired test uses df=nโˆ’1df=n-1 where nn is the number of PAIRS.
  • Using ss of one group instead of sds_d - the test uses the standard deviation of the differences, not of either original column.

Why This Formula Matters

Pairing cancels out person-to-person variation so the test sees only the change, giving far more power than treating the groups as independent. Recognizing a paired design and reducing to differences is the move that makes a real effect detectable; treating paired data as two independent samples throws away the pairing and weakens the test. Recognizing it by "Is every value in one group paired with exactly one specific value in the other (so I can form a difference per pair)?" โ€” rather than by familiar numbers โ€” is what lets a student tell it apart from two-sample t-test and one-sample t-test and two-proportion z-test in a mixed problem set.

Frequently Asked Questions

What is the Paired t-Test formula?

A hypothesis test for the mean difference in a paired (matched) data design, where each subject provides two related measurements. The test analyzes the differences di=x1iโˆ’x2id_i = x_{1i} - x_{2i} as a single sample.

How do you use the Paired t-Test formula?

You want to know if a tutoring program improves math scores. Instead of comparing two separate groups, you test the SAME students before and after tutoring. Each student is their own control. By looking at the difference (after โˆ’- before) for each student, you eliminate individual variation and focus purely on the change.

What do the symbols mean in the Paired t-Test formula?

dห‰\bar{d} = mean of differences, sds_d = standard deviation of differences, nn = number of pairs, df=nโˆ’1df = n - 1. Usually H0:ฮผd=0H_0: \mu_d = 0.

Why is the Paired t-Test formula important in Math?

Pairing cancels out person-to-person variation so the test sees only the change, giving far more power than treating the groups as independent. Recognizing a paired design and reducing to differences is the move that makes a real effect detectable; treating paired data as two independent samples throws away the pairing and weakens the test. Recognizing it by "Is every value in one group paired with exactly one specific value in the other (so I can form a difference per pair)?" โ€” rather than by familiar numbers โ€” is what lets a student tell it apart from two-sample t-test and one-sample t-test and two-proportion z-test in a mixed problem set.

What do students get wrong about Paired t-Test?

The procedure for paired t-test is the easy part; the trap is treating paired data as two independent samples. Asking "Is every value in one group paired with exactly one specific value in the other (so I can form a difference per pair)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Paired t-Test formula?

Before studying the Paired t-Test formula, you should understand: hypothesis testing, confidence interval, sampling distribution, mean, standard deviation.

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