Paired sample t test  overview
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons (max. of 3) by clicking on the dropdown button in the righthand column. To practice with a specific method click the button at the bottom row of the table
Paired sample $t$ test 


Independent variable  
2 paired groups  
Dependent variable  
One quantitative of interval or ratio level  
Null hypothesis  
H_{0}: $\mu = \mu_0$
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.  
Alternative hypothesis  
H_{1} two sided: $\mu \neq \mu_0$ H_{1} right sided: $\mu > \mu_0$ H_{1} left sided: $\mu < \mu_0$  
Assumptions  
 
Test 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 $\mu_0$.  
Sampling distribution of $t$ if H_{0} were true  
$t$ distribution with $N  1$ degrees of freedom  
Significant?  
Two sided:
 
$C\%$ confidence interval for $\mu$  
$\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
where the critical value $t^*$ is the value under the $t_{N1}$ 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.  
Effect 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 $\mu_0.$  
Visual representation  
Equivalent to  
 
Example context  
Is the average difference between the mental health scores before and after an intervention different from $\mu_0 = 0$?  
SPSS  
Analyze > Compare Means > PairedSamples T Test...
 
Jamovi  
TTests > Paired Samples TTest
 
Practice questions  