One sample z 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 $z$ test for the mean  Spearman's rho 


Independent variable  Variable 1  
None  One of ordinal level  
Dependent variable  Variable 2  
One quantitative of interval or ratio level  One of ordinal level  
Null hypothesis  Null hypothesis  
H_{0}: $\mu = \mu_0$
$\mu$ is the population mean; $\mu_0$ is the population mean according to the null hypothesis  H_{0}: $\rho_s = 0$
$\rho_s$ is the unknown Spearman correlation in the population. The Spearman correlation is a measure for the strength and direction of the monotonic relationship between two variables of at least ordinal measurement level. In words, the null hypothesis would be: H_{0}: there is no monotonic relationship between the two variables in the population  
Alternative hypothesis  Alternative hypothesis  
H_{1} two sided: $\mu \neq \mu_0$ H_{1} right sided: $\mu > \mu_0$ H_{1} left sided: $\mu < \mu_0$  H_{1} two sided: $\rho_s \neq 0$ H_{1} right sided: $\rho_s > 0$ H_{1} left sided: $\rho_s < 0$  
Assumptions  Assumptions  

 
Test statistic  Test statistic  
$z = \dfrac{\bar{y}  \mu_0}{\sigma / \sqrt{N}}$
$\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $\sigma$ is the population standard deviation, $N$ 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 $\mu_0$.  $t = \dfrac{r_s \times \sqrt{N  2}}{\sqrt{1  r_s^2}} $ where $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores.  
Sampling distribution of $z$ if H_{0} were true  Sampling distribution of $t$ if H_{0} were true  
Standard normal distribution  Approximately the $t$ distribution with $N  2$ degrees of freedom  
Significant?  Significant?  
Two sided:
 Two sided:
 
$C\%$ confidence interval for $\mu$  n.a.  
$\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
where $z^*$ 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 size  n.a.  
Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{\sigma}$$ Indicates how many standard deviations $\sigma$ the sample mean $\bar{y}$ is removed from $\mu_0$    
Visual representation  n.a.  
  
Example context  Example context  
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.  Is there a monotonic relationship between physical health and mental health?  
n.a.  SPSS  
  Analyze > Correlate > Bivariate...
 
n.a.  Jamovi  
  Regression > Correlation Matrix
 
Practice questions  Practice questions  