# Confidence Interval Calculator

Use this calculator to compute the confidence interval or margin of error, assuming the sample mean most likely follows a normal distribution. Use the Standard Deviation Calculator if you have raw data only.

## Result

Confidence Interval: 19.76 ±1.656 (±8.4%) [18.104 – 21.416]

Error Bar:    Steps:

CI =
 X̄ ± Z× s √n
=
 19.76 ± 1.6954× 6.48 √44
= 19.76 ± 1.656
 Sample size (amount), n Sample Mean (average), X̄ Standard Deviation, σ or s Confidence Level ### What is the confidence interval?

A confidence interval is a statistical measure used to indicate the range of estimates within which an unknown statistical parameter is likely to fall. If the parameter is the population mean, the confidence interval is an estimate of possible values of the population mean.

A confidence interval is determined through use of observed (sample) data and is calculated at a selected confidence level (chosen prior to the computation of the confidence interval). This confidence level, such as a 95% confidence level, indicates the reliability of the estimation procedure; it is not the degree of certainty that the computed confidence interval contains the true value of the parameter being studied. Specifically, the confidence level indicates the proportion of confidence intervals, that when constructed given the chosen confidence level over an infinite number of independent trials, will contain the true value of the parameter.

For example, if 100 confidence intervals are computed at a 95% confidence level, it is expected that 95 of these 100 confidence intervals will contain the true value of the given parameter; it does not say anything about individual confidence intervals. If 1 of these 100 confidence intervals is selected, we cannot say that there is a 95% chance it contains the true value of the parameter – this is a common misconception. The selected confidence interval will either contain or will not contain the true value, but we cannot say anything about the probability of a specific confidence interval containing the true value of the parameter.

Confidence intervals are typically written as (some value) ± (a range). The range can be written as an actual value or a percentage. It can also be written as simply the range of values. For example, the following are all equivalent confidence intervals:

20.6 ±0.887

or

20.6 ±4.3%

or

[19.713 – 21.487]

Calculating confidence intervals:

This calculator computes confidence intervals for normally distributed data with an unknown mean, but known standard deviation. It does not calculate confidence intervals for data with an unknown mean and unknown standard deviation.

Calculating a confidence interval involves determining the sample mean, X̄, and the population standard deviation, σ, if possible. If the population standard deviation cannot be used, then the sample standard deviation, s, can be used when the sample size is greater than 30. For a sample size greater than 30, the population standard deviation and the sample standard deviation will be similar. Depending on which standard deviation is known, the equation used to calculate the confidence interval differs. For the purposes of this calculator, it is assumed that the population standard deviation is known or the sample size is larger enough therefore the population standard deviation and sample standard deviation is similar. Only the equation for a known standard deviation is shown.

 X̄ ± Z× σ √n

where Z is the Z-value for the chosen confidence level, X̄ is the sample mean, σ is the standard deviation, and n is the sample size. Assuming the following with a confidence level of 95%:

X = 22.8

Z = 1.960

σ = 2.7

n = 100

The confidence interval is:
 22.8 ±1.960× 2.7 √100

22.8 ±0.5292

Z-values for Confidence Intervals

 Confidence Level Z Value 70% 1.036 75% 1.150 80% 1.282 85% 1.440 90% 1.645 95% 1.960 98% 2.326 99% 2.576 99.5% 2.807 99.9% 3.291 99.99% 3.891 99.999% 4.417