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Confidence Interval Calculator
Calculate a confidence interval for a population mean. Enter the sample mean, the standard deviation, the sample size and a confidence level to get the interval, the margin of error, the standard error and the critical z value used.
- Lower bound
- 48.04
- Upper bound
- 51.96
- Margin of error (±)
- 1.96
- Standard error
- 1
- Critical value (z*)
- 1.96
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Formula
How to use it
- Enter the sample mean.
- Enter the standard deviation and the sample size.
- Choose the confidence level.
- Read the interval: the range that plausibly contains the true population mean.
Worked examples
Mean 50, standard deviation 10, 100 observations, 95% confidence
- Confidence interval
- 48.04 to 51.96
- Lower bound
- 48.04
- Upper bound
- 51.96
- Margin of error (±)
- 1.96
- Standard error
- 1
- Critical value (z*)
- 1.96
The same sample at 99% confidence
- Confidence interval
- 47.4242 to 52.5758
- Margin of error (±)
- 2.5758
- Critical value (z*)
- 2.5758
Mean height 68.2 in, SD 2.9 in, 40 people, 95% confidence
- Lower bound
- 67.3013
- Upper bound
- 69.0987
- Margin of error (±)
- 0.8987
What the interval means
A 95% confidence level describes the method, not one particular interval: if you repeated the sampling many times, about 95% of the intervals built this way would contain the true mean.
Higher confidence needs a wider interval. Quadrupling the sample size halves the width, because the standard error shrinks with the square root of n.
This is a z interval
The calculator uses critical values from the normal distribution. That is appropriate when the population standard deviation is known or the sample is large (roughly 30 or more).
For a small sample with the standard deviation estimated from the same data, statisticians use the t distribution, which gives a wider interval. With 10 observations at 95%, the t critical value is 2.262 rather than 1.960, so the z interval here would be about 13% too narrow.
Questions people ask
What is the 95% confidence interval for a mean of 50, standard deviation 10 and n = 100?
48.04 to 51.96. The standard error is 1, and 1.96 × 1 = 1.96 either side of the mean.
What is the z value for a 95% confidence interval?
1.96 (more precisely 1.959964). For 90% it is 1.645 and for 99% it is 2.576.
Does a 99% interval come out wider than a 95% one?
Yes. For the same data the 99% interval is 47.42 to 52.58 — being more confident requires covering more ground.