Paired t-test dependent sample

In this tutorial we will discuss paired $t$-test, assumptions for paired $t$-test and step by step procedure for paired $t$-test. Paired $t$ test is also known as dependent sample t-test.

Paired t-test dependent sample

Let $X_1, X_2, \cdots, X_n$ and $Y_1, Y_2, \cdots, Y_n$ be two dependent samples each of size $n$ with respective means $\mu_1$ and $\mu_2$.

Define $d_i = X_i - Y_i$, $i=1,2,\cdots, n$. Then $\mu_d= \mu_1 - \mu_2$.

Let $\overline{d}=\frac{1}{n} \sum d_i$ be the mean of the difference and $s_d=\sqrt{\frac{1}{n-1}\sum (d_i - \overline{d})^2}$ be the sample standard deviation of the difference.


a. The samples are dependent (matched pairs).

b. Both the samples are simple random sample.

c. The two samples are both large ($n_1 > 30$ and $n_2 >30$) or both the samples comes from population having normal distribution.

Step by Step Procedure

We wish to test the hypothesis $H_0: \mu_1 = \mu_2$, i.e., $H_0:\mu_d=0$.

The standard error of $\overline{d}$ is

$$ \begin{aligned} SE(\overline{d}) = \frac{s_d}{\sqrt{n}} \end{aligned} $$

where $s_d$ is the sample standard deviation of the difference $d_i$.

Step 1 State the hypothesis testing problem

The hypothesis testing problem can be structured in any one of the three situations as follows:

Situation Hypothesis Testing Problem
Situation A $H_0: \mu_d=0$ against $H_a : \mu_d < 0$ (Left-tailed)
Situation B $H_0: \mu_d=0$ against $H_a : \mu_d > 0$ (Right-tailed)
Situation C $H_0: \mu_d=0$ against $H_a : \mu_d \neq 0$ (Two-tailed)

Step 2 Define the test statistic

The test statistic for testing above hypothesis is

$$ \begin{eqnarray*} t & =& \frac{\overline{d}-\mu_d}{SE(\overline{d})}\\ & =& \frac{\overline{d}-\mu_d}{s_d/\sqrt{n}} \end{eqnarray*} $$

where $\overline{d} =\frac{1}{n}\sum d_i$ and $s_d=\sqrt{\frac{\sum (d_i -\overline{d})^2}{n-1}}$.

The test statistic defined above follows Students $t$ distribution with $n-1$ degrees of freedom.

Step 3 Specify the level of significance $\alpha$.

Step 4 Determine the critical values

For the specified value of $\alpha$ determine the critical region depending upon the alternative hypothesis.

  • For left-tailed alternative hypothesis: Find the $t$-critical value using

$$ \begin{aligned} P(t<-t_\alpha) = \alpha. \end{aligned} $$

  • For right-tailed alternative hypothesis: $t_\alpha$.

$$ \begin{aligned} P(t>t_\alpha) = \alpha. \end{aligned} $$

  • For two-tailed alternative hypothesis: $t_{\alpha/2}$.

$$ P(t<-t_{\alpha/2} \text{ or } t> t_{\alpha/2}) = \alpha. $$

Step 5 Computation

Compute the test statistic under the null hypothesis $H_0$ using equation

$$ \begin{aligned} t_{obs} & = \frac{\overline{d}-0}{s_d/\sqrt{n}} \end{aligned} $$

Step 6 Decision (Traditional Approach)

It is based on the critical values.

  • For left-tailed alternative hypothesis: Reject $H_0$ if $t_{obs}\leq -t_\alpha$.

  • For right-tailed alternative hypothesis: Reject $H_0$ if $t_{obs}\geq t_\alpha$.

  • For two-tailed alternative hypothesis: Reject $H_0$ if $|t_{obs}|\geq t_{\alpha/2}$.


Step 6 Decision ($p$-value Approach)

It is based on the $p$-value.

Alternative Hypothesis Type of Hypothesis $p$-value
$H_a: \mu_1<\mu_2$ Left-tailed $p$-value $= P(t\leq t_{obs})$
$H_a: \mu_1>\mu_2$ Right-tailed $p$-value $= P(t\geq t_{obs})$
$H_a: \mu_1\neq \mu_2$ Two-tailed $p$-value $= 2P(t\geq abs(t_{obs}))$

If $p$-value is less than $\alpha$, then reject the null hypothesis $H_0$ at $\alpha$ level of significance, otherwise fail to reject $H_0$ at $\alpha$ level of significance.


In this tutorial, you learned the paired $t$-test and the assumptions for paired $t$-test. You also learned about the step by step procedure to apply paired $t$-test.

To learn more about other hypothesis testing problems, hypothesis testing calculators and step by step procedure, please refer to the following tutorials:

Let me know in the comments if you have any questions on paired $t$-test and your thought on this article.

VRCBuzz co-founder and passionate about making every day the greatest day of life. Raju is nerd at heart with a background in Statistics. Raju looks after overseeing day to day operations as well as focusing on strategic planning and growth of VRCBuzz products and services. Raju has more than 25 years of experience in Teaching fields. He gain energy by helping people to reach their goal and motivate to align to their passion. Raju holds a Ph.D. degree in Statistics. Raju loves to spend his leisure time on reading and implementing AI and machine learning concepts using statistical models.

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