# Correlation Confidence Interval Calculator

The correlation confidence interval calculator computes the confidence interval of the Pearson correlation coefficient.

Enter comma , space or Enter after each data value.
The tool doesn't count empty cells or non-numeric cells.

### Correlation confidence interval calculator

The correlation confidence interval calculator computes the confidence interval of Pearson's correlation coefficient. The distribution of the correlation coefficient is not normal.
After performing Fisher's transformation on the sample correlation the result is approximately normal.
When using sample data, we know the sample's correlation, but we don't know the true value of the population correlation. Instead, we may treat the population correlation as a random variable and calculate the confidence interval.
First, we need to define the confidence level, the required certainty level that the parameter's true value will be in the confidence interval. Researchers commonly use a confidence level of 0.95.
The default is a 95 confidence interval calculator, but you may change the confidence level.
This confidence interval calculator reports the results in APA style.
The online confidence interval calculator shows the formulas and step-by-step calculations.

### What is a correlation confidence interval?

The correlation confidence interval is the range in which the population correlation is most likely to be found.
The degree of certainty for which it is likely to be within that range is called the confidence level.
When you collect sample data, you can not know the exact value of the correlation.

### What is a confidence level?

The confidence level is the required degree of certainty that the population parameter will be in the confidence interval. This is the probability that the calculated confidence interval contains the population parameter.
Note: researchers commonly use a confidence level of 0.95.

#### What is the correlation confidence interval formula?

1. Transform the correlation with the Fisher's transformation.
 r' = 1 ln( 1 + r ) = arctanh(r) 2 1 - r
2. Calculate the standard deviation of the transformed correlation.
 S'= 1 √(n-3)

3. Calculate the confidence interval using the Z statistic.

CI' = r' ± Z1-α/2 * S'
4. Transform back the lower and upper values to the correlation scale.
 Lower = exp(2*Lower') - 1 = tanh(Lower') exp(2*Lower') + 1 Upper = exp(2*Upper') - 1 = tanh(Upper') exp(2*Upper') + 1

Where:

r - sample Pearson correlation coefficient.
r'- transformed correlation (Fisher, 1921).
S' - the approximate standard deviation of the transformed correlation.
n - the sample size (the number of observations).
C -confidence level.
α = 1 - C.
Lower' - lower limit of the transformed correlation (r').
Upper' - Upper limit of the transformed correlation (r').
Lower - lower limit of the correlation (r).
Upper - Upper limit of the correlation (r).