Does 95 percent confidence mean a 95 percent chance the value is in this interval?

No. The true value is fixed and the interval varies between samples. It means intervals built this way capture the true value 95 percent of the time, which is a statement about the method.

How do I halve the interval?

Quadruple the sample. The width shrinks with the square root of the sample size, so there is no cheap route to much greater precision.

Math & Statisticsstatistics

Confidence Interval Calculator

A confidence interval for a mean or a proportion. A 95 percent interval does not mean there is a 95 percent chance the true value sits inside this one: it means intervals built this way contain the true value 95 percent of the time.

Also called: ci calculator, margin of error interval.

Confidence interval

9.3234 to 18.3432

9.3234 to 18.3432 at 95% confidence, a margin of error of 4.51. Intervals built this way contain the true value 95 times in 100. It is a statement about the procedure, not about this particular interval. Quadrupling the sample to 24 would halve the margin, since width shrinks with the square root of the sample.

Lower bound

9.32

Upper bound

18.34

Margin of error

4.51

Point estimate

13.83

Standard error

2.3

Sample size used

What the interval means

Intervals built this way contain the true value 95 times in 100. It is a statement about the procedure, not about this particular interval.

On the width

Quadrupling the sample to 24 would halve the margin, since width shrinks with the square root of the sample.

How this is calculated

Method and background

How this is calculated

The interval is the estimate plus and minus the margin of error, which is the standard error times a critical value. The interpretation is the part worth getting right: the true value is fixed and the interval is what varies between samples, so the probability statement is about the procedure rather than about this particular interval. Width shrinks with the square root of the sample, which means quadrupling the data halves the interval and there is no cheap way to be much more precise.

the interval narrows with the square root of the sample, so quadrupling the data halves the width

s: Standard deviation
n: Sample size

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

an interval for a mean

Interval for: A mean
Values, for a mean: 12, 18, 7, 22, 15, 9
Or, the mean directly: 0
Standard deviation: 0
Sample size: 0
Proportion, for a proportion interval: 50%
Confidence level: 95%

Confidence interval9.3234 to 18.3432

the sample mean of six values, with the interval from the sample deviation

Open this example

an interval for a proportion

Interval for: A proportion
Values, for a mean
Or, the mean directly: 0
Standard deviation: 0
Sample size: 1,000
Proportion, for a proportion interval: 50%
Confidence level: 95%

Confidence interval46.901 to 53.099%

boundary: the roughly three percent margin a thousand respondents buys

Open this example

Method and limits

What it assumes

Roughly normal sampling distribution, which holds for reasonable sample sizes.

What it deliberately does not model

Uses the normal critical value rather than the t distribution, which understates the width below about thirty observations.
A confidence interval says nothing about bias: a badly drawn sample gives a precise wrong answer.

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Frequently asked questions

Does 95 percent confidence mean a 95 percent chance the value is in this interval?: No. The true value is fixed and the interval varies between samples. It means intervals built this way capture the true value 95 percent of the time, which is a statement about the method.
How do I halve the interval?: Quadruple the sample. The width shrinks with the square root of the sample size, so there is no cheap route to much greater precision.