One sample t test for the mean  overview
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One sample $t$ test for the mean  MannWhitneyWilcoxon test 


Independent variable  Independent variable  
None  One categorical with 2 independent groups  
Dependent variable  Dependent variable  
One quantitative of interval or ratio level  One of ordinal level  
Null hypothesis  Null hypothesis  
$\mu = \mu_0$
$\mu$ is the unknown population mean; $\mu_0$ is the population mean according to the null hypothesis  If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Formulation 1:
 
Alternative hypothesis  Alternative hypothesis  
Two sided: $\mu \neq \mu_0$ Right sided: $\mu > \mu_0$ Left sided: $\mu < \mu_0$  If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Formulation 1:
 
Assumptions  Assumptions  
 Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations 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 $\mu_0$  Two different types of test statistics can be used; both will result in the same test outcome. The first is the Wilcoxon rank sum statistic $W$:
Note: we could just as well base W and U on group 2. This would only 'flip' the right and left sided alternative hypotheses. Also, tables with critical values for $U$ are often based on the smaller of $U$ for group 1 and for group 2.  
Sampling distribution of $t$ if H0 were true  Sampling distribution of $W$ and of $U$ if H0 were true  
$t$ distribution with $N  1$ degrees of freedom  Sampling distribution of $W$:
Sampling distribution of $U$: For small samples, the exact distribution of $W$ or $U$ should be used. Note: the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated if ties are present in the data.  
Significant?  Significant?  
Two sided:
 For large samples, the table for standard normal probabilities can be used: Two sided:
 
$C\%$ 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  
  If no ties in the data: two sided MannWhitneyWilcoxon test is equivalent to KruskalWallis test with an independent variable with 2 levels ($I = 2$)  
Example context  Example context  
Is the average mental health score of office workers different from $\mu_0$ = 50?  Do men tend to score higher on social economic status than women?  
SPSS  SPSS  
Analyze > Compare Means > OneSample T Test...
 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples...
 
Jamovi  Jamovi  
TTests > One Sample TTest
 TTests > Independent Samples TTest
 
Practice questions  Practice questions  