Statistics & Probability

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95% Confidence Interval Calculator.

Calculate a normal-approximation confidence interval using z = 1.96.

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Set your values

Results update as you type.

lower: 46.864

lower

46.864000
upper: 53.136

upper

53.136000
marginOfError: 3.136

marginOfError

3.136000

Results update automatically as you type.

Use Cases

Estimate population mean from sample data

When you have a sample from a larger population, use this calculator to estimate the range where the true population mean likely falls with 95% confidence.

Example: Sample mean = 50, standard deviation = 10, sample size = 30 gives an interval of 46.42 to 53.58.

Check if a hypothesized value is plausible

If a hypothesized population mean falls outside the 95% confidence interval, it is unlikely to be the true mean. This helps in hypothesis testing.

Example: If your interval is 100 to 110, a hypothesized mean of 95 is not plausible.

Frequently Asked Questions

What is a 95% confidence interval?
A 95% confidence interval is a range of values that you can be 95% confident contains the true population mean. It is calculated from your sample mean, standard deviation, and sample size using the normal distribution.
How do I calculate a 95% confidence interval?
Enter your sample mean, standard deviation, and sample size. The calculator uses the formula: mean ± (1.96 * (standard deviation / sqrt(sample size))). The result gives you the lower and upper bounds of the interval.
What does the margin of error mean?
The margin of error is the amount added and subtracted from the sample mean to create the interval. It reflects the uncertainty in your estimate due to sampling variability. A larger sample size or smaller standard deviation reduces the margin of error.

Tips & Common Mistakes

Tips

  • Ensure your sample size is large enough (typically n > 30) for the normal approximation to be valid.
  • Use the sample standard deviation, not the population standard deviation, unless you know the population value.
  • The confidence interval is only as good as your sample. Use a random sample to avoid bias.
  • Remember that 95% confidence means that if you repeated the sampling many times, 95% of the intervals would contain the true mean.

Common Mistakes to Avoid

  • Using the population standard deviation when you only have the sample standard deviation.
  • Forgetting to take the square root of the sample size in the denominator.
  • Interpreting the interval as a range that contains 95% of the data, rather than the range for the mean.

Last updated: August 12, 2026