Calculator
What the Confidence Interval Calculator does
Confidence Interval Calculator: calculates Lower Bound, Upper Bound and Margin of Error from Sample Mean, Sample Std Dev and Sample Size n. Example: sample mean 50, sample std dev 10 and sample size n 36 gives lower bound 46.7333.
This calculator reads Sample Mean, Sample Std Dev, Sample Size n and z Critical (1.96≈95%) and produces Lower Bound, Upper Bound and Margin of Error. Values update as you type, so comparing options is a matter of editing one field.
Typical users are anyone verifying a hand calculation. Seeing the intermediate steps is usually more useful than the final number alone.
Because it is all local, the tool is as fast as your device and as private as your own notes.
Each field is explained further down, along with the method and a worked example that uses the default values.
What do the Confidence Interval Calculator fields mean?
The Confidence Interval Calculator uses 4 inputs. Sensible defaults are loaded up front, which means you can change one number at a time instead of filling the whole form first.
| Field | What to enter | Default |
|---|---|---|
| Sample Mean | required | 50 |
| Sample Std Dev | required | 10 |
| Sample Size n | required | 36 |
| z Critical (1.96≈95%) | required | 1.96 |
How does the Confidence Interval Calculator work?
Each output is derived from the inputs above in a single pass; there is no hidden state carried between runs, so the same inputs always give the same calculation.
Validation runs before the calculation, which is why a mistyped value shows an error rather than a plausible-looking result.
Worked example
These are the values the Confidence Interval Calculator loads by default, and the result it produces from them. These are the exact values loaded when the page opens, and the answer they produce.
| Sample Mean | 50 |
|---|---|
| Sample Std Dev | 10 |
| Sample Size n | 36 |
| z Critical (1.96≈95%) | 1.96 |
| Lower Bound | 46.7333 |
|---|---|
| Upper Bound | 53.2667 |
| Margin of Error | 3.2667 |
| What this means | CI ≈ [46.733333333333334, 53.266666666666666] |
Beyond the formula: confidence intervals
A 95% confidence interval does not mean there’s a 95% chance the true value falls inside this particular interval. It means that if you repeated the sampling process many times, about 95% of the intervals built this way would contain the true value. The uncertainty is in the sampling process, not in this one already-computed result.
Common mistakes and sanity checks
- This calculator uses a z-based interval, which assumes either a large sample size or a known population standard deviation. For small samples (roughly under 30) with an estimated standard deviation, a t-distribution-based interval is more accurate and slightly wider.
- A wider interval isn’t a worse result: it’s an honest reflection of a smaller sample or more variable data. Narrowing it requires more data or a lower confidence level (90% instead of 95%), not a different formula.
- The margin of error shrinks with the square root of sample size, not proportionally to it: quadrupling the sample size only halves the margin of error, which is why polls rarely bother collecting far more than a few thousand responses.
How to use it
- Set Sample Mean.
- Put Sample Std Dev.
- Key in Sample Size n.
- Fill in z Critical (1.96≈95%).
- Results update continuously rather than on submit.
- Check Lower Bound, Upper Bound, Margin of Error and What this means on the right.
- Print gives a tidy version without the navigation, which is handy for attaching to a file.
What this tool does not do
- Very large or very small values reach the limits of floating-point precision, about 15 significant digits.
- Displayed values are rounded; the underlying calculation keeps full precision.
- It handles the common case well and the unusual case literally.
Frequently asked questions
Sample Mean, Sample Std Dev, Sample Size n and z Critical (1.96≈95%). That is the whole form. Nothing starts empty, so the quickest approach is to replace the one number you actually care about.
With the values loaded when the page opens, lower bound comes out as 46.7333. Overwrite any of the starting values and the result follows.
It never leaves the tab. The page does the work itself rather than asking a server to.
No payment and no login. Advertising covers the running costs.
It follows the standard definition for this operation, so the result matches what a reference implementation would give.