# Paired sample t test - overview

This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table

Paired sample $t$ test
One sample $z$ test for the mean
Independent variableIndependent variable
2 paired groupsNone
Dependent variableDependent variable
One quantitative of interval or ratio levelOne quantitative of interval or ratio level
Null hypothesisNull hypothesis
H0: $\mu = \mu_0$

Here $\mu$ is the population mean of the difference scores, and $\mu_0$ is the population mean of the difference scores according to the null hypothesis, which is usually 0. A difference score is the difference between the first score of a pair and the second score of a pair.
H0: $\mu = \mu_0$

Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis.
Alternative hypothesisAlternative hypothesis
H1 two sided: $\mu \neq \mu_0$
H1 right sided: $\mu > \mu_0$
H1 left sided: $\mu < \mu_0$
H1 two sided: $\mu \neq \mu_0$
H1 right sided: $\mu > \mu_0$
H1 left sided: $\mu < \mu_0$
AssumptionsAssumptions
• Difference scores are normally distributed in the population
• Sample of difference scores is a simple random sample from the population of difference scores. That is, difference scores are independent of one another
• Scores are normally distributed in the population
• Population standard deviation $\sigma$ is known
• Sample is a simple random sample from the population. That is, observations are independent of one another
Test statisticTest statistic
$t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
Here $\bar{y}$ is the sample mean of the difference scores, $\mu_0$ is the population mean of the difference scores according to the null hypothesis, $s$ is the sample standard deviation of the difference scores, and $N$ is the sample size (number of difference scores).

The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$.
$z = \dfrac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $\sigma$ is the population standard deviation, and $N$ is the sample size.

The denominator $\sigma / \sqrt{N}$ is the standard deviation of the sampling distribution of $\bar{y}$. The $z$ value indicates how many of these standard deviations $\bar{y}$ is removed from $\mu_0$.
Sampling distribution of $t$ if H0 were trueSampling distribution of $z$ if H0 were true
$t$ distribution with $N - 1$ degrees of freedomStandard normal distribution
Significant?Significant?
Two sided:
Right sided:
Left sided:
Two sided:
Right sided:
Left sided:
$C\%$ confidence interval for $\mu$$C\% confidence interval for \mu \bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}} where the critical value t^* is the value under the t_{N-1} distribution with the area C / 100 between -t^* and t^* (e.g. t^* = 2.086 for a 95% confidence interval when df = 20). The confidence interval for \mu can also be used as significance test. \bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}} where the critical value z^* is the value under the normal curve with the area C / 100 between -z^* and z^* (e.g. z^* = 1.96 for a 95% confidence interval). The confidence interval for \mu can also be used as significance test. Effect sizeEffect size Cohen's d: Standardized difference between the sample mean of the difference scores and \mu_0:$$d = \frac{\bar{y} - \mu_0}{s}$$Cohen's d indicates how many standard deviations s the sample mean of the difference scores \bar{y} is removed from \mu_0. Cohen's d: Standardized difference between the sample mean and \mu_0:$$d = \frac{\bar{y} - \mu_0}{\sigma}$$Cohen's$d$indicates how many standard deviations$\sigma$the sample mean$\bar{y}$is removed from$\mu_0.$Visual representationVisual representation Equivalent ton.a. • One sample$t$test on the difference scores. • Repeated measures ANOVA with one dichotomous within subjects factor. - Example contextExample context Is the average difference between the mental health scores before and after an intervention different from$\mu_0 = 0$?Is the average mental health score of office workers different from$\mu_0 = 50$? Assume that the standard deviation of the mental health scores in the population is$\sigma = 3.\$
SPSSn.a.
Analyze > Compare Means > Paired-Samples T Test...
• Put the two paired variables in the boxes below Variable 1 and Variable 2
-
Jamovin.a.
T-Tests > Paired Samples T-Test
• Put the two paired variables in the box below Paired Variables, one on the left side of the vertical line and one on the right side of the vertical line
• Under Hypothesis, select your alternative hypothesis
-
Practice questionsPractice questions