Enter the sample mean, standard deviation and sample size to get the confidence interval — the range the true mean likely sits in.
What Is a Confidence Interval?
A confidence interval calculator gives the range a true mean likely falls in: mean 50, SD 10, n = 100 at 95% → 48.04 to 51.96 (margin ±1.96). The 95% means: if you repeated the sampling many times, 95% of such intervals would contain the true mean. Bigger samples shrink the interval — precision grows with √n.
How to Use
- Enter the sample mean.
- Enter the standard deviation.
- Enter the sample size.
- Pick the confidence level.
- Click “Calculate CI”.
Formula
CI = mean ± z × (SD ÷ √n). z = 1.96 for 95%, 2.576 for 99%, 1.645 for 90%.
Worked Example
Mean 50, SD 10, n = 100, 95%: SE = 1.00; margin = 1.96; CI = 48.04 to 51.96.
How It Works, In Plain Words
A sample mean is one draw from a noisy process, so the tool refuses to report it naked — it wraps it in a margin that reflects how much sampling luck could be inside it. The steps run in order. First, the standard error: SE = SD ÷ √n, here 10 ÷ √100 = 1.00. The √n is the whole story of precision: the mean of 100 readings wobbles far less than a single reading, and the shrink FAQ quantifies it — 4× the sample halves the width. Second, the confidence level picks z: 1.96 for 95%, 2.576 for 99%, 1.645 for 90%, straight from the formula line. Third, margin = z × SE = 1.96, and the interval is mean ± margin: 48.04 to 51.96.
Read the answer the way the introduction defines it: if you repeated the sampling many times, 95% of intervals built this way would contain the true mean. The level is a dial, not a virtue: 90% buys a narrower interval with weaker coverage, while 99% demands a wider one — the level FAQ calls 95% the standard default and 99% wider and more conservative. The tool will not run with a standard deviation or sample size of zero, because either would pretend the mean is known exactly, which is the one thing a confidence interval exists to deny.
Real-World Uses
A survey analyst reporting an average score of 50 from 100 responses uses the tool to attach the honest uncertainty: not “50” but 48.04 to 51.96 at 95%, with the margin of error ±1.96 quoted beside it — exactly the pairing the benefits list promises. A student checking a hand calculation works the worked example backwards — SE = 1.00, margin = 1.96 — and finds where their own attempt diverged, which is usually the √n step. An A/B tester comparing two variants uses the interval as a reality check on a tempting difference: if the plausible range for the mean still straddles the value that would change the decision, the data has not settled the question yet, however clean the single mean looked.
Benefits
- CI and margin of error together;
- 90/95/99% levels selectable;
- The √n precision idea explained;
- Useful for surveys and A/B tests.
Features of Our Tool
- Instant results in your browser;
- No sign-up, no download, 100% free;
- Works on mobile, tablet and desktop;
- Clear step-by-step explanation below the tool;
- Private — your entries never leave your device.
Common Mistakes
- Reading “95% confident” as “95% chance the true value is in THIS interval” — it is about the procedure;
- Using it on tiny or biased samples;
- Confusing the interval with a prediction interval for individual values.
Common Mistakes to Avoid
The mistakes list covers the interpretation traps: reading 95% as a probability about this one interval, using tiny or biased samples, and confusing the interval with a prediction interval for individual values. In prose, three more. Do not enter the standard error where the standard deviation belongs: the tool divides by √n itself, so pre-dividing shrinks the interval twice and flatters your precision. Do not raise the level to 99% and go on quoting the narrower width from memory: the wider, more conservative interval is the price of the stronger claim, and the tool recomputes it honestly. And do not treat a narrow interval as proof the sample was good — the biased-sample warning stands however tight the arithmetic looks.
FAQ
What is a 95% confidence interval?
Mean ± 1.96 × SE — here 48.04 to 51.96.
How do I shrink the interval?
Increase the sample size — precision grows with √n (4× the sample halves the width).
95% vs 99% — which to use?
95% is the standard default; 99% is wider and more conservative.
What is margin of error?
The ± part: z × SE — here ±1.96.
Does it need normal data?
For large n the central limit theorem covers most cases; small samples need the t-distribution.
Why does the interval use √n and not just n?
Because averaging cancels noise with the square root, not one-for-one: the standard error is SD ÷ √n, so the worked example’s √100 turns an SD of 10 into an SE of 1.00. The shrink FAQ states the consequence plainly — 4× the sample halves the width — which is why big precision gains get expensive quickly.
Can I use this for a percentage or proportion?
Not directly. The three inputs are a mean, a standard deviation and a sample size, and the formula CI = mean ± z × (SD ÷ √n) is the mean version. A proportion needs its own standard error, so do not force a percentage into the mean box and trust the output — the numbers will compute, but they will not mean what you need.
Related Calculators
Z-Score Calculator; P-Value Calculator; Sample Size Calculator