Calculations For Confidence Interval

The equations behind the MitchellScience Confidence Interval Calculators are listed below. If you are interested in the theory behind these equations, and other background information, a good statistics textbook may be helpful.

Definition of Terms

We define the following terms:

confidence level (%)
finite population correction
margin of error (converted from a percentage by dividing by 100)
population percentage (converted from a percentage by dividing by 100)
population size
sample size
z score (1.96 if C is 95%, or 2.576 if C is 99%)

These terms are defined more completely on the definitions page.

Margin Of Error

In the MitchellScience calculators, the margin of error is computed from the other parameters using the following unrestricted equation:

(for large and small populations)

The unrestricted margin of error above includes the "finite population correction". The finite population correction is needed when the population size is small, but has negligible effect when the population size is large. The finite population correction term is:

As the population size increases, the finite population correction tends towards one, and the margin of error reduces to:

(for large populations only)

Comparing the large population margin of error to the unrestricted margin of error shows that the two are related by the simple equation:

Confidence Interval

The confidence interval is computed from the unrestricted margin of error in either "whole number" or "percentage" formats.

In the "whole number" format, the confidence interval is specified as a range of population values between a lower limit and an upper limit. Those limits are computed as:

(lower limit)
(upper limit)

In the "percentage" format, the confidence interval is specified as a range of percentages between a lower limit and an upper limit. Those limits are computed as:

(lower limit as a percent)
(upper limit as a percent)