Pearson correlation  overview
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Pearson correlation  Two sample $t$ test  equal variances not assumed 


Variable 1  Independent/grouping variable  
One quantitative of interval or ratio level  One categorical with 2 independent groups  
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}: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2.  
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: $\mu_1 \neq \mu_2$ H_{1} right sided: $\mu_1 > \mu_2$ H_{1} left sided: $\mu_1 < \mu_2$  
Assumptions of test for correlation  Assumptions  

 
Test statistic  Test statistic  
Test statistic for testing H0: $\rho = 0$:
 $t = \dfrac{(\bar{y}_1  \bar{y}_2)  0}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}} = \dfrac{\bar{y}_1  \bar{y}_2}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s^2_1$ is the sample variance in group 1, $s^2_2$ is the sample variance in group 2, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis. The denominator $\sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}$ is the standard error of the sampling distribution of $\bar{y}_1  \bar{y}_2$. The $t$ value indicates how many standard errors $\bar{y}_1  \bar{y}_2$ is removed from 0. Note: we could just as well compute $\bar{y}_2  \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$.  
Sampling distribution of $t$ and of $z$ if H_{0} were true  Sampling distribution of $t$ if H_{0} were true  
Sampling distribution of $t$:
 Approximately the $t$ distribution with $k$ degrees of freedom, with $k$ equal to $k = \dfrac{\Bigg(\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}\Bigg)^2}{\dfrac{1}{n_1  1} \Bigg(\dfrac{s^2_1}{n_1}\Bigg)^2 + \dfrac{1}{n_2  1} \Bigg(\dfrac{s^2_2}{n_2}\Bigg)^2}$ or $k$ = the smaller of $n_1$  1 and $n_2$  1 First definition of $k$ is used by computer programs, second definition is often used for hand calculations.  
Significant?  Significant?  
$t$ Test two sided:
 Two sided:
 
Approximate $C$% confidence interval for $\rho$  Approximate $C\%$ confidence interval for $\mu_1  \mu_2$  
First compute the approximate $C$% confidence interval for $\rho_{Fisher}$:
Then transform back to get the approximate $C$% confidence interval for $\rho$:
 $(\bar{y}_1  \bar{y}_2) \pm t^* \times \sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}$
where the critical value $t^*$ is the value under the $t_{k}$ 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 $\mu_1  \mu_2$ can also be used as significance test.  
Properties of the Pearson correlation coefficient  n.a.  
   
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 different between men and women?  
SPSS  SPSS  
Analyze > Correlate > Bivariate...
 Analyze > Compare Means > IndependentSamples T Test...
 
Jamovi  Jamovi  
Regression > Correlation Matrix
 TTests > Independent Samples TTest
 
Practice questions  Practice questions  