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 or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates)
One quantitative of interval or ratio level
Dependent variable
Variable 2
One quantitative of interval or ratio level
One quantitative of interval or ratio level
THIS TABLE IS YET TO BE COMPLETED
Null hypothesis

$\rho = \rho_0$
$\rho$ is the unknown Pearson correlation in the population, $\rho_0$ is the correlation in the population according to the null hypothesis (usually 0)
n.a.
Alternative hypothesis

Two sided: $\rho \neq \rho_0$
Right sided: $\rho > \rho_0$
Left sided: $\rho < \rho_0$
n.a.
Assumptions of tests for correlation

In the population, the two variables are jointly normally distributed (this covers the normality, homoscedasticity, and linearity assumptions)
Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Note: these assumptions are only important for the significance test and confidence interval, not for the correlation coefficient itself. The correlation coefficient just measures the strength of the linear relationship between two variables.
n.a.
Test statistic

Test statistic for testing H0: $\rho = 0$:
$t = \dfrac{r \times \sqrt{N  2}}{\sqrt{1  r^2}} $
where $r$ is the sample correlation $r = \frac{1}{N  1} \sum_{j}\Big(\frac{x_{j}  \bar{x}}{s_x} \Big) \Big(\frac{y_{j}  \bar{y}}{s_y} \Big)$ and $N$ is the sample size
Test statistic for testing values for $\rho$ other than $\rho = 0$:
$r_{Fisher} = \dfrac{1}{2} \times \log\Bigg(\dfrac{1 + r}{1  r} \Bigg )$, where $r$ is the sample correlation
$\rho_{0_{Fisher}} = \dfrac{1}{2} \times \log\Bigg( \dfrac{1 + \rho_0}{1  \rho_0} \Bigg )$, where $\rho_0$ is the population correlation according to H0
where $r_{Fisher} = \frac{1}{2} \times \log\Bigg(\dfrac{1 + r}{1  r} \Bigg )$ and $z^*$ 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).
Then transform back to get approximate $C$% confidence interval for $\rho$:
The Pearson correlation coefficient is a measure for the linear relationship between two quantitative variables.
The Pearson correlation coefficient squared reflects the proportion of variance explained in one variable by the other variable.
The Pearson correlation coefficient can take on values between 1 (perfect negative relationship) and 1 (perfect positive relationship). A value of 0 means no linear relationship.
The absolute size of the Pearson correlation coefficient is not affected by any linear transformation of the variables. However, the sign of the Pearson correlation will flip when the scores on one of the two variables are multiplied by a negative number (reversing the direction of measurement of that variable). For example:
the correlation between $x$ and $y$ is equivalent to the correlation between $3x + 5$ and $2y  6$.
the absolute value of the correlation between $x$ and $y$ is equivalent to the absolute value of the correlation between $3x + 5$ and $2y  6$. However, the signs of the two correlation coefficients will be in opposite directions, due to the multiplication of $x$ by $3$.
The Pearson correlation coefficient does not say anything about causality.
The Pearson correlation coefficient is sensitive to outliers.