Confidence Interval Calculator

What is a Confidence Interval?

A confidence interval is a statistical measure that estimates the range in which an unknown parameter, such as the population mean, is likely to fall. It is based on sample data and is calculated at a specific confidence level, like 95%. The confidence level represents the reliability of the estimation, but it doesn’t guarantee that the computed interval contains the true value. Instead, it indicates how often confidence intervals will capture the true value over multiple trials.

For example, if 100 confidence intervals are calculated at a 95% confidence level, you would expect about 95 of them to contain the true value of the parameter. However, it doesn’t mean that any given interval has a 95% chance of containing the true value—it will either contain the true value or it won’t.

Confidence intervals are typically expressed as (value) ± (range). The range can be shown as a specific value or as a percentage. Here are a few equivalent examples:

• 20.6 ± 0.887

• 20.6 ± 4.3%

• [19.713 – 21.487]

How to Calculate Confidence Intervals

This calculator calculates confidence intervals for data that follows a normal distribution with an unknown population mean and a known standard deviation. It does not compute confidence intervals for data where both the mean and standard deviation are unknown.

To calculate a confidence interval, you need the sample mean (X̄) and the population standard deviation (σ), if known. If the population standard deviation isn’t available, the sample standard deviation (s) can be used when the sample size is greater than 30, as the two values become similar with larger samples. This calculator assumes the population standard deviation is known or approximated by the sample standard deviation for large samples.

The formula for calculating a confidence interval is:

                          X̄ ± Z * (σ / √n)

Where:

• X̄ is the sample mean

• Z is the Z-value corresponding to the chosen confidence level

• σ is the standard deviation

• n is the sample size

For example, at a 95% confidence level, if:

• Sample mean (X̄) = 22.8

• Z-value (Z) = 1.960

• Standard deviation (σ) = 2.7

• Sample size (n) = 100

The confidence interval would be:

Where:

• X̄ is the sample mean

• Z is the Z-value corresponding to the chosen confidence level

• σ is the standard deviation

• n is the sample size

For example, at a 95% confidence level, if:

• Sample mean (X̄) = 22.8

• Z-value (Z) = 1.960

• Standard deviation (σ) = 2.7

• Sample size (n) = 100

The confidence interval would be:
                      22.8 ± 1.960 * (2.7 / √100) = 22.8 ± 0.5292

Z-Values for Confidence Intervals

Here are the Z-values for different confidence levels: