Confidence Interval Calculator: Mean, 90/95/99% CI

What does [93.81, 106.19] mean?

It means the sample mean is 100, and the true population mean probably sits somewhere between 93.81 and 106.19 — the interval is the sample mean plus or minus a margin of error of 6.19, and 6.19 is the critical value of 2.0639 times the standard error of 3.00. A 95% interval means that if you repeated the sampling many times, about 95% of the intervals built this way would contain the true mean. It does not mean there is a 95% chance this one interval contains it, and it does not mean 95% of the data falls inside it — the data spreads wider than the interval, because the interval is about the mean, not about the observations.

Three things to check before you trust the number

t or z, and which standard deviation you have. The t-distribution is for a sample standard deviation, with 24 degrees of freedom on the defaults — n minus 1 — and the critical value is 2.0639. The z-distribution is for a known population standard deviation, and the critical value is 1.9600. Using z with a sample standard deviation understates the uncertainty, and the gap matters most when the sample is small: at n = 10 the t critical value is 2.2622 against z at 1.9600.

The width comes from three places, and only one of them is free. The margin of error is the critical value times the standard error, and the standard error is the standard deviation over the square root of n. A bigger sample shrinks it — four times the sample halves the error — a tighter spread shrinks it, and a higher confidence level widens it. The cheapest way to narrow an interval is usually more data, not more confidence.

95% is a property of the method, not of this interval. Saying the interval is [93.81, 106.19] at 95% confidence does not say the true mean is inside it with 95% probability — the true mean is either inside or it is not. What 95% says is that the method gets it right about 95 times out of 100. Report the interval itself — [93.81, 106.19], unrounded [93.808304315, 106.191695685] — not just the confidence level.

Worked example

On the defaults: sample mean 100, sample standard deviation 15, sample size 25, 95% confidence.

1. Degrees of freedom: 25 − 1 = 24.
2. Critical value: t(24, 0.975) = 2.0639 (unrounded 2.063898562).
3. Standard error: 15 ÷ √25 = 15 ÷ 5 = 3.00.
4. Margin of error: 2.0639 × 3.00 = 6.19 (unrounded 6.191695685).
5. Lower limit: 100 − 6.19 = 93.81 (unrounded 93.808304315).
6. Upper limit: 100 + 6.19 = 106.19 (unrounded 106.191695685).
7. Smallest sample size for a margin of error of at most 5: 38 — at 38 the margin of error is 4.93 (unrounded 4.930377), and at 37 it is 5.00 (unrounded 5.001249).
8. Largest standard deviation for a margin of error of at most 5: 12 — at 12 the margin of error is 4.95 (unrounded 4.953357), and at 12.5 it is 5.16 (unrounded 5.159746).

Every branch, checked

• 90% confidence: 90% confidence: 90% CI [94.87, 105.13]; margin of error 5.13; critical value 1.7109; SE 3.00. 90% confidence — the critical value is 1.7109, the margin of error 5.13, the interval [94.87, 105.13], narrower than at 95% because less confidence needs less room.
• 99% confidence: 99% confidence: 99% CI [91.61, 108.39]; margin of error 8.39; critical value 2.7969; SE 3.00. 99% confidence — the critical value is 2.7969, the margin of error 8.39, the interval [91.61, 108.39], wider than at 95% because more confidence needs more room.
• n = 100: n = 100: 95% CI [97.02, 102.98]; margin of error 2.98; critical value 1.9842; SE 1.50. n = 100 — the standard error halves to 1.50, the margin of error is 2.98, the interval [97.02, 102.98], because four times the sample is half the error.
• sd = 10: sd = 10: 95% CI [95.87, 104.13]; margin of error 4.13; critical value 2.0639; SE 2.00. sd = 10 — the standard error is 2.00, the margin of error 4.13, the interval [95.87, 104.13], because a tighter spread is a tighter interval.
• mean = 50: mean = 50: 95% CI [43.81, 56.19]; margin of error 6.19; critical value 2.0639; SE 3.00. mean = 50 — the interval is [43.81, 56.19], the same width, shifted down by 50, because the mean only moves the interval, never its width.
• z mode (population SD): z mode (population SD): 95% CI [94.12, 105.88]; margin of error 5.88; critical value 1.9600; SE 3.00. z mode (population SD) — the critical value is 1.9600, the margin of error 5.88, the interval [94.12, 105.88], narrower than the t interval because the population standard deviation is known.
• n = 10: n = 10: 95% CI [89.27, 110.73]; margin of error 10.73; critical value 2.2622; SE 4.74. n = 10 — the critical value is 2.2622, the standard error 4.74, the margin of error 10.73, the interval [89.27, 110.73], because a small sample is an uncertain sample.
• 90%, n = 400, sd = 20, mean = 250: 90%, n = 400, sd = 20, mean = 250: 90% CI [248.35, 251.65]; margin of error 1.65; critical value 1.6487; SE 1.00. 90%, n = 400, sd = 20, mean = 250 — the critical value is 1.6487, the margin of error 1.65, the interval [248.35, 251.65], a big sample with a modest spread.
• proportion 0.60, n = 400: proportion 0.60, n = 400: 95% CI [0.55, 0.65]; margin of error 0.05; critical value 1.9600; SE 0.02. proportion 0.60, n = 400 — the critical value is 1.9600, the standard error 0.02, the margin of error 0.05, the interval [0.55, 0.65], because a proportion's spread comes from the proportion itself.
• proportion 0.50, n = 100, 99%: proportion 0.50, n = 100, 99%: 99% CI [0.37, 0.63]; margin of error 0.13; critical value 2.5758; SE 0.05. proportion 0.50, n = 100, 99% — the critical value is 2.5758, the margin of error 0.13, the interval [0.37, 0.63], the widest proportion interval at the most uncertain proportion.

The reverses, worked backwards

• Smallest sample size for the margin of error to be at most 5: 38 — at 38 the figures are 95% CI [95.07, 104.93]; margin of error 4.93; critical value 2.0262; SE 2.43 (unrounded lower 95.069623, upper 104.930377, margin of error 4.930377, critical value 2.026192, SE 2.433321); at the lower neighbor the figures are 95% CI [95.00, 105.00]; margin of error 5.00; critical value 2.0281; SE 2.47 (unrounded lower 94.998751, upper 105.001249, margin of error 5.001249, critical value 2.028094, SE 2.465985); at the upper neighbor the figures are 95% CI [95.14, 104.86]; margin of error 4.86; critical value 2.0244; SE 2.40 (unrounded lower 95.137562, upper 104.862438, margin of error 4.862438, critical value 2.024394, SE 2.401922) — verdict: STEP.
• Largest standard deviation for the margin of error to be at most 5: 12 — at 12 the figures are 95% CI [95.05, 104.95]; margin of error 4.95; critical value 2.0639; SE 2.40 (unrounded lower 95.046643, upper 104.953357, margin of error 4.953357, critical value 2.063899, SE 2.400000); at the upper neighbor the figures are 95% CI [94.84, 105.16]; margin of error 5.16; critical value 2.0639; SE 2.50 (unrounded lower 94.840254, upper 105.159746, margin of error 5.159746, critical value 2.063899, SE 2.500000); at the lower neighbor the figures are 95% CI [95.25, 104.75]; margin of error 4.75; critical value 2.0639; SE 2.30 (unrounded lower 95.253033, upper 104.746967, margin of error 4.746967, critical value 2.063899, SE 2.300000) — verdict: STEP.
• Largest sample size for the margin of error to stay at least 6: 26 — at 26 the figures are 95% CI [93.94, 106.06]; margin of error 6.06; critical value 2.0595; SE 2.94 (unrounded lower 93.941369, upper 106.058631, margin of error 6.058631, critical value 2.059539, SE 2.941742); at the upper neighbor the figures are 95% CI [94.07, 105.93]; margin of error 5.93; critical value 2.0555; SE 2.89 (unrounded lower 94.066198, upper 105.933802, margin of error 5.933802, critical value 2.055529, SE 2.886751); at the lower neighbor the figures are 95% CI [93.81, 106.19]; margin of error 6.19; critical value 2.0639; SE 3.00 (unrounded lower 93.808304, upper 106.191696, margin of error 6.191696, critical value 2.063899, SE 3.000000) — verdict: STEP.
• Smallest standard deviation for the margin of error to be at least 8: 19.5 — at 19.5 the figures are 95% CI [91.95, 108.05]; margin of error 8.05; critical value 2.0639; SE 3.90 (unrounded lower 91.950796, upper 108.049204, margin of error 8.049204, critical value 2.063899, SE 3.900000); at the lower neighbor the figures are 95% CI [92.16, 107.84]; margin of error 7.84; critical value 2.0639; SE 3.80 (unrounded lower 92.157185, upper 107.842815, margin of error 7.842815, critical value 2.063899, SE 3.800000); at the upper neighbor the figures are 95% CI [91.74, 108.26]; margin of error 8.26; critical value 2.0639; SE 4.00 (unrounded lower 91.744406, upper 108.255594, margin of error 8.255594, critical value 2.063899, SE 4.000000) — verdict: STEP.
• Smallest sample size for a proportion margin of error of at most 0.05: 385 — at 385 the figures are 95% CI [0.45, 0.55]; margin of error 0.05; critical value 1.9600; SE 0.03 (unrounded lower 0.450055, upper 0.549945, margin of error 0.049945, critical value 1.959964, SE 0.025482); at the lower neighbor the figures are 95% CI [0.45, 0.55]; margin of error 0.05; critical value 1.9600; SE 0.03 (unrounded lower 0.449991, upper 0.550009, margin of error 0.050009, critical value 1.959964, SE 0.025516); at the upper neighbor the figures are 95% CI [0.45, 0.55]; margin of error 0.05; critical value 1.9600; SE 0.03 (unrounded lower 0.450120, upper 0.549880, margin of error 0.049880, critical value 1.959964, SE 0.025449) — verdict: STEP.
• Smallest mean for the lower limit to be at least 90: 96.5 — at 96.5 the figures are 95% CI [90.31, 102.69]; margin of error 6.19; critical value 2.0639; SE 3.00 (unrounded lower 90.308304, upper 102.691696, margin of error 6.191696, critical value 2.063899, SE 3.000000); at the lower neighbor the figures are 95% CI [89.81, 102.19]; margin of error 6.19; critical value 2.0639; SE 3.00 (unrounded lower 89.808304, upper 102.191696, margin of error 6.191696, critical value 2.063899, SE 3.000000); at the upper neighbor the figures are 95% CI [90.81, 103.19]; margin of error 6.19; critical value 2.0639; SE 3.00 (unrounded lower 90.808304, upper 103.191696, margin of error 6.191696, critical value 2.063899, SE 3.000000) — verdict: STEP.

What the defaults assume

The defaults are a sample mean of 100, a sample standard deviation of 15, a sample size of 25, at 95% confidence — a small survey or a lab batch, the classic textbook case — and the interval is [93.81, 106.19], the margin of error 6.19, the critical value 2.0639. They are a starting point, not a benchmark: your sample may be 400 observations with a spread of 20, or a proportion of 0.60 from 400 people, and the right answer for you is the one your numbers describe. Change the mean, the spread, the size, the confidence and the mode to your own and the calculator recomputes instantly. Take the critical value, times the standard error, plus and minus the mean.

These notes repeat in every explainer section because this page is built to be read start to finish: the defaults are one sample, not a standard; t or z, and which standard deviation you have; the width comes from three places, and only one of them is free; and 95% is a property of the method, not of this interval.

Output · formula · meaning

Who uses this

• A student checking a statistics assignment — mean 100, sd 15, n 25 at 95% gives [93.81, 106.19], by the critical value times the standard error, not by guessing.
• A researcher reporting a survey — a proportion of 0.60 from 400 people gives [0.55, 0.65] at 95%, and the unrounded [0.551990883, 0.648009117] shows how the rounding sits.
• A quality-control engineer checking a batch — with 10,000 observations the interval is [99.71, 100.29], tight enough to call the process on target.
• A pollster planning a sample — the smallest sample size for a margin of error of at most 5 is 38, and at 37 the margin of error is 5.00.

Accuracy & limitations

The calculator finds the t critical value by bisection on the t curve, good to about 1e-10, and the z critical value by Acklam's inverse normal, good to about 2e-9 — every critical value on this page rounds to the same 6 decimals as the exact value, and every figure is checked against an independent model. What it cannot do is check that your sample is random or your data roughly normal: the interval assumes the sample mean is near-normal, so on a heavily skewed distribution or a tiny sample the coverage is approximate, not exact, and a proportion interval near 0 or 1 is better served by a Wilson interval than this Wald one. It also takes summary numbers, not raw data — a whole dataset needs the mean-median-mode-calculator page or the standard-deviation-calculator page first. And the page prints the unrounded figures so the rounding does not mislead you — the critical value is 2.0639, unrounded 2.063898562, and the interval is [93.81, 106.19], unrounded [93.808304315, 106.191695685].

Cite this calculator

A confidence interval is the sample mean plus or minus a margin of error: the critical value from the t or z curve, times the standard error. On the defaults — mean 100, sample standard deviation 15, sample size 25, 95% confidence — the critical value is 2.0639, the standard error is 3.00, the margin of error is 6.19, and the interval is [93.81, 106.19]. Smallest sample size for the margin of error to be at most 5: 38. Largest standard deviation for the margin of error to be at most 5: 12. Largest sample size for the margin of error to stay at least 6: 26. Smallest standard deviation for the margin of error to be at least 8: 19.5. Smallest sample size for a proportion margin of error of at most 0.05: 385. Smallest mean for the lower limit to be at least 90: 96.5. Source: Bright Side Kit, brightoolbox.com.

Frequently asked questions