One sample t test for the mean  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 righthand column. To practice with a specific method click the button at the bottom row of the table
One sample $t$ test for the mean  Cochran's Q test 


Independent variable  Independent variable  
None  One within subject factor ($\geq 2$ related groups)  
Dependent variable  Dependent variable  
One quantitative of interval or ratio level  One categorical with 2 independent groups  
Null hypothesis  Null hypothesis  
$\mu = \mu_0$
$\mu$ is the unknown population mean; $\mu_0$ is the population mean according to the null hypothesis  $\pi_1 = \pi_2 = \ldots = \pi_I$
$\pi_1$ is the population proportion of 'successes' in group 1; $\pi_2$ is the population proportion of 'successes' in group 2; $\pi_I$ is the population proportion of 'successes' in group $I$  
Alternative hypothesis  Alternative hypothesis  
Two sided: $\mu \neq \mu_0$ Right sided: $\mu > \mu_0$ Left sided: $\mu < \mu_0$  Not all population proportions are equal  
Assumptions  Assumptions  
 Sample of 'blocks' (usually the subjects) is a simple random sample from the population. That is, blocks are independent of one another  
Test statistic  Test statistic  
$t = \dfrac{\bar{y}  \mu_0}{s / \sqrt{N}}$
$\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to H0, $s$ is the sample standard deviation, $N$ is the sample size. 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 $\mu_0$  If a failure is scored as 0 and a success is scored as 1:
$Q = k(k  1) \dfrac{\sum_{groups} \Big (\mbox{group total}  \frac{\mbox{grand total}}{k} \Big)^2}{\sum_{blocks} \mbox{block total} \times (k  \mbox{block total})}$ Here $k$ is the number of related groups (usually the number of repeated measurements), a group total is the sum of the scores in a group, a block total is the sum of the scores in a block (usually a subject), and the grand total is the sum of all the scores. Before computing $Q$, first exclude blocks with equal scores in all $k$ groups  
Sampling distribution of $t$ if H0 were true  Sampling distribution of $Q$ if H0 were true  
$t$ distribution with $N  1$ degrees of freedom  If the number of blocks (usually the number of subjects) is large, approximately the chisquared distribution with $k  1$ degrees of freedom  
Significant?  Significant?  
Two sided:
 If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
 
$C\%$ confidence interval for $\mu$  n.a.  
$\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
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  n.a.  
Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{s}$$ Indicates how many standard deviations $s$ the sample mean $\bar{y}$ is removed from $\mu_0$    
Visual representation  n.a.  
  
n.a.  Equivalent to  
  Friedman test, with a categorical dependent variable consisting of two independent groups  
Example context  Example context  
Is the average mental health score of office workers different from $\mu_0$ = 50?  Subjects perform three different tasks, which they can either perform correctly or incorrectly. Is there a difference in task performance between the three different tasks?  
SPSS  SPSS  
Analyze > Compare Means > OneSample T Test...
 Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
 
Jamovi  Jamovi  
TTests > One Sample TTest
 Jamovi does not have a specific option for the Cochran's Q test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the $p$ value that would have resulted from the Cochran's Q test. Go to:
ANOVA > Repeated Measures ANOVA  Friedman
 
Practice questions  Practice questions  