Pearson correlation  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
Pearson correlation  One sample $z$ test for the mean 


Variable 1  Independent variable  
One quantitative of interval or ratio level  None  
Variable 2  Dependent variable  
One quantitative of interval or ratio level  One quantitative of interval or ratio level  
Null hypothesis  Null hypothesis  
H_{0}: $\rho = \rho_0$
Here $\rho$ is the Pearson correlation in the population, and $\rho_0$ is the Pearson correlation in the population according to the null hypothesis (usually 0). The Pearson correlation is a measure for the strength and direction of the linear relationship between two variables of at least interval measurement level.  H_{0}: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis.  
Alternative hypothesis  Alternative hypothesis  
H_{1} two sided: $\rho \neq \rho_0$ H_{1} right sided: $\rho > \rho_0$ H_{1} left sided: $\rho < \rho_0$  H_{1} two sided: $\mu \neq \mu_0$ H_{1} right sided: $\mu > \mu_0$ H_{1} left sided: $\mu < \mu_0$  
Assumptions of test for correlation  Assumptions  

 
Test statistic  Test statistic  
Test statistic for testing H0: $\rho = 0$:
 $z = \dfrac{\bar{y}  \mu_0}{\sigma / \sqrt{N}}$
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 $t$ and of $z$ if H_{0} were true  Sampling distribution of $z$ if H_{0} were true  
Sampling distribution of $t$:
 Standard normal distribution  
Significant?  Significant?  
$t$ Test two sided:
 Two sided:
 
Approximate $C$% confidence interval for $\rho$  $C\%$ confidence interval for $\mu$  
First compute the approximate $C$% confidence interval for $\rho_{Fisher}$:
Then transform back to get the approximate $C$% confidence interval for $\rho$:
 $\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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.  
Properties of the Pearson correlation coefficient  Effect size  
 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 $\mu_0.$  
n.a.  Visual representation  
  
Equivalent to  n.a.  
OLS regression with one independent variable:
   
Example context  Example context  
Is there a linear relationship between physical health and mental health?  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 $\sigma = 3.$  
SPSS  n.a.  
Analyze > Correlate > Bivariate...
   
Jamovi  n.a.  
Regression > Correlation Matrix
   
Practice questions  Practice questions  