This calculator will compute the confidence interval for a correlation coefficient, given the raw data or calculated correlation coefficient and sample size.
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What is a Confidence Interval for Correlation Coefficient?
A confidence interval for a correlation coefficient provides a range of plausible values for the true population correlation, given the sample data. It helps quantify the uncertainty associated with the estimated correlation coefficient.
Formula
There are two common methods for calculating the standard error of a correlation coefficient:
Direct Method (typically used for hypothesis testing):
Where r is the correlation coefficient and n is the sample size.
Fisher's Z-transformation Method (used for confidence intervals):
First, transform r to z:
Then calculate the standard error of z:
Constructing the Confidence Interval (using Fisher's method):
The confidence interval is constructed using Fisher's z-transformation because it provides better statistical properties.
- Calculate the confidence interval for z:
- Transform back to correlation scale:
Where is the critical value from the standard normal distribution
Where tanh is the hyperbolic tangent function
Note:
- Fisher's z-transformation method is preferred for confidence intervals
- The direct method is typically used for testing if a correlation differs from zero
Interpretation
A 95% confidence interval for the correlation coefficient means that if we repeated the sampling process many times and calculated the confidence interval each time, about 95% of these intervals would contain the true population correlation coefficient.
If the confidence interval does not include zero, we can conclude that there is a statistically significant correlation between the two variables at the chosen confidence level.