Ninety-five percent is the default confidence level in most of science, medicine and polling, and it pairs naturally with the 5% significance level used in hypothesis tests: a value outside the 95% interval would be rejected by a two-sided test at α = 0.05. The interval leaves 2.5% in each tail of the normal curve, which gives the famous critical value 1.96 – often rounded to 2, which leads to the handy rule “estimate plus or minus two standard errors”.
For a mean, the interval is x̄ ± 1.96 σ/√n when the population standard deviation is known. In practice σ is almost always estimated from the same sample, so the calculator switches to the t distribution, whose 95% critical value is larger for small samples – 2.262 for n = 10, 2.045 for n = 30 – and approaches 1.96 as n grows.
The preset level is 95%, but you can change it, switch between mean and proportion, or enter a standard error directly if a paper or software output already gives you one.
How to use the 95% Confidence Interval Calculator
- Calculate a confidence interval for
- Sample mean (x̄)
- Spread you know
- Standard deviation or standard error
- Sample proportion (p̂)
- Sample size (n)
- Confidence level
- Custom confidence level
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Divide the standard deviation by √n to get the standard error, multiply by 1.96 (or by t* with n − 1 degrees of freedom when s is a sample statistic) to get the margin of error, and take the mean minus and plus that margin.
Worked example
Sixty-four commuters report a mean one-way commute of 27.3 minutes with s = 9.8 minutes. The standard error is 9.8 ÷ 8 = 1.225 min. With 63 degrees of freedom t* = 1.9983, so the margin of error is 2.45 min and the 95% confidence interval is 24.85 to 29.75 minutes. Using z* = 1.96 would give almost the same 24.90 to 29.70 minutes, because with 64 observations t and z barely differ.
The 95% interval and the 5% test
A two-sided hypothesis test at α = 0.05 rejects H₀: μ = μ₀ exactly when μ₀ lies outside the 95% interval. In the commuting example, a claimed average of 30 minutes would be rejected (30 is above 29.75), while 28 minutes would not.
Formula
95% CI = x̄ ± 1.96 × σ ÷ √n (σ known) · x̄ ± t*₀.₉₇₅,ₙ₋₁ × s ÷ √n (s from the sample) · α = 0.05, 2.5% in each tail
Worked example
Inputs — Calculate a confidence interval for: A population mean; Sample mean (x̄): 27.3; Spread you know: Sample standard deviation s (t interval); Standard deviation or standard error: 9.8; Sample size (n): 64; Confidence level: 95%.
Results — 95% confidence interval = 24.852 to 29.748 (27.3 ± 2.448); Lower bound = 24.852; Upper bound = 29.748; Margin of error (±) = 2.448; Standard error = 1.225; Critical value = 1.9983.
Key terms
- 95% confidence interval
- Interval from a procedure that captures the true value in 95% of samples.
- 1.96
- The 97.5th percentile of the standard normal distribution.
- Two-standard-error rule
- Approximating a 95% interval as estimate ± 2 SE.
Frequently asked questions
Why is 1.96 used for a 95% confidence interval?
95% of a standard normal distribution lies between −1.96 and 1.96, leaving 2.5% in each tail.
How do I calculate a 95% confidence interval by hand?
Mean ± 1.96 × SD ÷ √n. With mean 27.3 min, SD 9.8 and n = 64: SE = 1.225, margin = 2.40, interval 24.90 to 29.70 (z) or 24.85 to 29.75 with t* = 1.998.
Is it correct to say there is a 95% chance the true mean is in my interval?
Strictly no – the true mean is fixed. The 95% describes the method: 95 out of 100 such intervals would capture it.
What does it mean if two 95% intervals overlap?
Not necessarily that the difference is non-significant. Compare the groups with a proper two-sample test or an interval for the difference.
What is the 95% confidence interval for a proportion?
p̂ ± 1.96 √(p̂(1 − p̂)/n). For 44% of 64 people: 44% ± 12.2 percentage points.
How large a sample do I need for a ±2 margin?
n = (1.96 × σ ÷ 2)². With σ = 9.8 that is 92.2, so 93 observations.