Confidence Interval Calculator

How to use this calculator

Enter the sample mean, sample standard deviation, sample size and confidence level (95% is pre-filled). The calculator shows the critical value, standard error, margin of error and the lower and upper bounds. Change the confidence level to see how the interval widens at 99% or narrows at 90%.

For samples smaller than 30 the critical value comes from Student’s t distribution with n − 1 degrees of freedom, which widens the interval to account for the uncertainty in the estimated standard deviation; from n = 30 upwards the normal z value is used. A note under the sample size tells you which one applies. With n = 10 at 95% confidence, t = 2.262 instead of z = 1.960.

The reverse direction is useful when reading papers: enter a reported interval's lower and upper bounds with the sample size and confidence level to recover the mean, standard error and standard deviation — handy for meta-analysis or checking a result. You can also enter a desired margin of error with the SD to see the sample size required.

The formula

SE = s / √n

E = t₍₁₋α/₂, n−1₎ × SE for n < 30, otherwise E = z₍₁₋α/₂₎ × SE

CI = x̄ ± E

x̄: sample mean
s: standard deviation
n: sample size
CL: confidence level
z or t: critical value (t if n < 30, else z)
SE: standard error
E: margin of error
L: lower bound
U: upper bound

Worked example

A sample of 100 measurements has mean 50 and standard deviation 10. What is the 95% confidence interval for the population mean?

Critical value z = Φ⁻¹(0.975) = 1.96.
Standard error SE = 10 / √100 = 1.
Margin E = 1.96 × 1 = 1.96, so the interval is 48.04 to 51.96.

We are 95% confident the population mean lies between 48.04 and 51.96.

Assumptions and limits

Below n = 30 the Student’s t critical value with n − 1 degrees of freedom is used; at n ≥ 30 the normal z value (the difference is under 4% at n = 30). If the population SD is known exactly, a z interval is appropriate at any n.
Observations are independent and the sampling distribution of the mean is approximately normal.

Frequently asked questions

What does a 95% confidence interval mean?

If the study were repeated many times, about 95% of the intervals computed this way would contain the true population mean. It is not a 95% probability for this particular interval.

Why does a larger sample give a narrower interval?

The standard error shrinks with √n. Quadrupling the sample size halves the margin of error.

What z-value goes with 99% confidence?

2.576. For 90% it is 1.645, and for 95% it is 1.960.

When should I use t instead of z?

When the standard deviation is estimated from a small sample. The t distribution has heavier tails, so the critical value is larger: 2.262 for n = 10 and 2.045 for n = 30 at 95% confidence, versus 1.960 for z.