What is a confidence interval, in plain terms?

It is a range built from your sample that is meant to capture the unknown true value. A 95 percent interval means that if you repeated the sampling many times, about 95 percent of the intervals built this way would contain the real mean or proportion. So with a sample mean of 100 and a 95% interval of [95.84, 104.16], the honest reading is that the method is right 95% of the time, not that there is a 95% chance the truth sits in this one range.

Why does 95 percent confidence use z = 1.96?

A 95% two-sided interval leaves 2.5% of the normal curve in each tail, so you need the z that puts 97.5% of the area to its left. That value is 1.95996, which everyone rounds to 1.96. By the same logic 90% uses 1.645 and 99% uses 2.576. This tool computes z from the inverse normal rather than reading a table, so a custom level like 92% returns its exact 1.7507.

What is the formula for a confidence interval of the mean?

CI = x̄ ± z·(s/√n), where x̄ is the sample mean, s is the standard deviation, n is the sample size and z is the critical value for your confidence level. The s/√n piece is the standard error of the mean. Example with x̄ = 100, s = 15, n = 50 at 95%: the margin is 1.96 × 15/√50 = 4.16, so the interval runs from 95.84 to 104.16.

How is a proportion confidence interval different?

For a proportion you use p ± z·√(p(1−p)/n), where p is the sample fraction (enter 0.45 for 45%). The spread depends on p itself and is widest near 0.5. Example with p = 0.45 and n = 100 at 95%: the margin is 1.96 × √(0.45×0.55/100) = 0.0975, giving [35.2%, 54.8%]. This tool clamps the bounds to 0 and 100 percent since a proportion cannot leave that range.

Why does the interval shrink as the sample size grows?

Both formulas divide by √n inside the margin, so quadrupling n only halves the margin. With s = 15 at 95%, n = 30 gives a margin of about 5.37, while n = 300 cuts it to about 1.70. That is why precision is expensive: to halve your error you need four times the data. The tool makes this visible by recomputing the moment you change n.

Should I use z or t for the mean?

Use z when the population standard deviation is known or the sample is large (n ≥ 30 is the common rule of thumb). Use the t-distribution when n is small and you only have the sample standard deviation, because t has fatter tails and gives a wider, more cautious interval. This tool uses z and shows a note when n drops below 30 so you know to reach for a t-table for a small-sample mean.

Report a survey result with a margin of error

You polled 100 people and 45 said yes. Switch to proportion mode, enter 0.45 and n = 100 at 95%, and the tool returns [35.2%, 54.8%] with a margin of about 9.8 points. Now your write-up reads "45% support, ±9.8 points at 95% confidence" instead of a bare number, which is exactly how pollsters and reporters quote results.

Check homework on confidence intervals

Working through a stats problem set, you compute an interval by hand and want to confirm it before submitting. Punch in the same x̄, s, n and confidence level, compare bounds to the fourth decimal, and you catch a slipped √n or a wrong z before it costs you points. The critical-z readout doubles as a quick table lookup.

Size a sample before you collect data

You need a mean estimate within ±2 units at 95%. Try a sample size, read the margin, and bump n until the margin drops under 2. Because the margin shrinks with √n, this trial-and-error converges fast and tells you roughly how much data to budget before you start collecting anything.

Put an honest range on a financial estimate

Average order value across a sample of recent orders is 84 dollars with a standard deviation of 30 over 60 orders. A 95% interval gives roughly [76.4, 91.6], so instead of presenting a single 84 you can tell stakeholders the true average is very likely between 76 and 92. The range is what makes the number defensible in a review.

Confidence Interval Calculator: Mean and Proportion

Mean and proportion confidence intervals with margin of error, exact z critical values, one-click copy, browser-only

Runs locally
Category Calculator
Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

Sample mean (x̄)

Std deviation (s)

Sample size (n)

Confidence level

Confidence interval

[95.8423, 104.1577]

We are 95% confident the true mean lies between 95.8423 and 104.1577.

Lower bound

95.8423

Upper bound

104.1577

Margin of error

±4.1577

Critical z: 1.96

What this tool does

A free confidence interval calculator for a population mean or a population proportion. For a mean it computes CI = x̄ ± z·(s/√n) from your sample mean, standard deviation and sample size; for a proportion it computes p ± z·√(p(1−p)/n). Pick 90, 95 or 99 percent, or type any custom level, and the tool returns the exact critical z from the inverse normal (95% gives 1.95996, not a rounded 1.96), the lower and upper bounds, and the margin of error on its own line. A small worked example x̄ = 100, s = 15, n = 50 at 95% lands on [95.84, 104.16] with a margin of 4.16, which you can reproduce here in seconds. When n is below 30 the tool flags that a t-distribution is the more honest choice for a mean. Everything runs in your browser, with a shareable URL that reopens the exact interval and a copy button for your report. 100% client-side, no upload, no sign-in.

Tool details

Input: Numbers; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Copy; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Shareable URL state; Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget: Initial JS <= 9 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Calculator · Student; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How Confidence Interval Calculator fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

Getting a realistic range before a purchase, plan, workout, or schedule decision.
Comparing scenarios by changing one input at a time.
Turning rough assumptions into a number you can discuss.

Calculation checks

Double-check units, dates, rates, and rounding assumptions.
Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Report a survey result with a margin of error
You polled 100 people and 45 said yes. Switch to proportion mode, enter 0.45 and n = 100 at 95%, and the tool returns [35.2%, 54.8%] with a margin of about 9.8 points. Now your write-up reads "45% support, ±9.8 points at 95% confidence" instead of a bare number, which is exactly how pollsters and reporters quote results.
Check homework on confidence intervals
Working through a stats problem set, you compute an interval by hand and want to confirm it before submitting. Punch in the same x̄, s, n and confidence level, compare bounds to the fourth decimal, and you catch a slipped √n or a wrong z before it costs you points. The critical-z readout doubles as a quick table lookup.
Size a sample before you collect data
You need a mean estimate within ±2 units at 95%. Try a sample size, read the margin, and bump n until the margin drops under 2. Because the margin shrinks with √n, this trial-and-error converges fast and tells you roughly how much data to budget before you start collecting anything.
Put an honest range on a financial estimate
Average order value across a sample of recent orders is 84 dollars with a standard deviation of 30 over 60 orders. A 95% interval gives roughly [76.4, 91.6], so instead of presenting a single 84 you can tell stakeholders the true average is very likely between 76 and 92. The range is what makes the number defensible in a review.

Common pitfalls

Reading a 95% interval as "95% chance the truth is in this range". The truth is fixed; it is the method that succeeds 95% of the time. Any single interval either contains the true value or it does not. State the confidence as a property of the procedure, not of one result.
Entering a proportion as a percentage instead of a fraction. The proportion field wants 0.45, not 45. Typing 45 is out of the 0 to 1 range and the tool will reject it. Convert percent to a decimal first, then read the bounds back as percentages.
Using z for a small sample mean. With n below 30 and only the sample standard deviation, z gives an interval that is too narrow. Switch to a t-distribution, which has fatter tails. The tool shows a note when n drops under 30 for exactly this reason.

Privacy

Every step here, the critical z, the standard error, the bounds and the margin of error, is plain JavaScript that runs in your browser tab. Your sample mean, standard deviation, proportion and sample size never leave the page and nothing is logged. The one caveat is the shareable URL, which encodes your inputs in the query string, so a link pasted into chat will record those numbers in the recipient server access log. For sensitive figures, use the copy button and paste the text instead of sharing the URL.

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Confidence Interval Calculator: Mean and Proportion

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How Confidence Interval Calculator fits into your work

Calculation jobs

Calculation checks

Good next steps

Real-world use cases

Report a survey result with a margin of error

Check homework on confidence intervals

Size a sample before you collect data

Put an honest range on a financial estimate

Common pitfalls

Privacy

FAQ

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