Confidence Interval Calculator
Enter the sample mean, the sample standard deviation, the sample size and the confidence level. You get the margin of error and the lower and upper ends of the interval.
Your numbers
How we calculate it
The standard error is the standard deviation divided by the square root of the sample size. The margin of error is the critical value z times the standard error, and the interval is the mean plus or minus the margin. The critical value z is 1.645 for 90%, 1.960 for 95% and 2.576 for 99%.
With a mean of 50, a standard deviation of 10 and 100 values, the standard error is 1, so at 95% the margin is 1.96 and the interval runs from 48.04 to 51.96.
Assumptions and limits
- This uses the normal distribution (z). With fewer than 30 values, an interval based on the t distribution is more accurate, and the calculator tells you when that applies.
- It assumes a random sample and that the sample mean is roughly normally distributed.
- A 95% interval does not mean there is a 95% chance that the true mean is inside your particular interval. It means the method captures the true mean 95% of the time.
Frequently asked questions
What is a confidence interval?
It is a range of values that is likely to contain the true mean of the whole group, based on a sample. A wider interval means more uncertainty.
Why is a 99% interval wider than a 90% interval?
To be more confident you must allow a wider range. The critical value grows from 1.645 at 90% to 2.576 at 99%, so the margin grows with it.
How can I make the interval narrower?
Take a larger sample, because the margin shrinks with the square root of the sample size, or accept a lower confidence level. Less variation in the data also helps.
Related calculators
- Sample Size CalculatorChoose a confidence level, a margin of error and the proportion you expect.
- Z-Score CalculatorChoose whether you know a value with its mean and standard deviation, or you already have a z-score.
- Standard Deviation CalculatorPaste or type a list of numbers and choose whether it is a whole population or a sample.
- Mean, Median and Mode CalculatorType or paste a list of numbers.
Worked example (a fixed example using the starting numbers, it does not change as you type)
Inputs: Sample mean = 50; Sample standard deviation = 10; Sample size = 100; Confidence level = 95%.
- Margin of error 1.96
- Lower bound 48.04
- Upper bound 51.96
- Standard error 1
- Critical value (z) 1.960
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