How do you calculate a 95% confidence interval?

Sample mean ± 1.96 × (standard deviation ÷ √sample size). A mean of 50 with SD 12 and n = 30: 12 ÷ √30 = 2.19, 1.96 × 2.19 = 4.29, so the interval is 50 ± 4.29 = (45.71, 54.29). Enter the three numbers above and the page shows each step.

What is the margin of error in a confidence interval?

The half-width: z × σ ÷ √n, the amount added and subtracted from the sample mean. OpenStax calls it the error bound (EBM). In the 90% example it is 0.8225. The margin-of-error calculator on this site computes it on its own.

When should I use t instead of z?

When the population standard deviation is unknown and estimated from a sample of fewer than about 30. The t multiplier is larger than z for small samples (2.26 instead of 1.96 at n = 10 for 95%), so the interval is wider, which is honest about the extra uncertainty. For large samples the two agree.

Does a wider confidence interval mean more confidence?

A wider interval at the same data means a higher confidence level: to be 99% sure rather than 95%, the interval has to reach further (2.576 instead of 1.96 standard errors). At the same confidence level, a wider interval means a smaller or more variable sample.

Confidence Interval Calculator for a Mean (z, Known σ)

A 90%, 95% or 99% confidence interval for a population mean from a sample mean, standard deviation and sample size, working shown.

Inputs

Sample mean

-1,000,000,0001,000,000,000

Std dev

01,000,000,000

Sample size

21,000,000,000

Confidence

Include CalculateEveryday mark

90% CI

[67.1775, 68.8225]

Working: standard error = 3 ÷ √36 = 0.5; error bound = 1.645 × 0.5 = 0.8225; interval = 68 ± 0.8225
Margin of error (error bound): ±0.8225
Standard error: 0.5
z for 90%: 1.645
Width of the interval: 1.645 — to halve it, quadruple n to 144

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What each Confidence Interval for a Mean input means

Sample mean: The sample mean, x̄ — the average of your observations.
Std dev: The population standard deviation σ if known; otherwise the sample's, which is fine for samples of about 30 or more.
Sample size: The number of observations in the sample.
Confidence: The confidence level: 90%, 95% or 99%, with z = 1.645, 1.96 or 2.576. 95% is what nearly every report means.

Confidence Interval for a Mean formula

CI = x̄ ± z × σ ÷ √n · z = 1.645 (90%), 1.96 (95%), 2.576 (99%) · error bound EBM = z × σ ÷ √n

\bar{x} \pm z_{\alpha/2}\, \frac{\sigma}{\sqrt{n}}

A sample mean is an estimate of the population mean, and a confidence interval is that estimate with its uncertainty attached. The uncertainty comes from the standard error, σ ÷ √n — sample means scatter less than individual values, and the scatter shrinks with the square root of the sample size. Multiplying the standard error by a z value gives the error bound: 1.96 standard errors either side captures the true mean in 95% of samples, 1.645 in 90%, 2.576 in 99%. The interval is the sample mean plus and minus that bound. This is the z-interval, which assumes the population standard deviation is known or the sample is large enough (about 30 or more) that the sample standard deviation stands in for it; with a small sample and an estimated σ, the t-interval is wider and is the right one.

Confidence Interval for a Mean worked example

OpenStax's Example 8.2: exam scores with σ = 3, a sample of 36 with mean 68, at 90% confidence. The standard error is 3 ÷ √36 = 0.5; the error bound is 1.645 × 0.5 = 0.8225; the interval is 68 ± 0.8225 = (67.18, 68.82). At 95% the bound would be 1.96 × 0.5 = 0.98, giving (67.02, 68.98) — more confidence costs a wider interval. Doubling the sample to 144 halves the standard error and the bound.

More Confidence Interval for a Mean examples

A sample of 30 with mean 50 and standard deviation 12, at 95%: the standard error is 12 ÷ √30 = 2.1909, the bound is 1.96 × 2.1909 = 4.2941, and the interval is [45.7059, 54.2941].

Heights of 100 adults with mean 172 cm and standard deviation 8 cm, at 99%: 2.576 × 0.8 = 2.0608, giving [169.9392, 174.0608] — about 170 to 174 cm.

Where the Confidence Interval for a Mean calculator can give the wrong answer

'95% confidence' is about the procedure, not this interval: 95% of intervals built this way contain the true mean. It does not mean there is a 95% probability the true mean is in this particular interval, though that is how everyone reads it.
With the standard deviation estimated from a small sample (under about 30), use a t-interval instead; this page's z-interval is too narrow in that case, and noticeably so below n = 15.
The formula assumes a random sample and, for small n, a roughly normal population. A convenience sample or a heavily skewed variable breaks the guarantee regardless of the arithmetic.
The width shrinks with the square root of n, so it takes four times the sample to halve the interval — which is why the page shows the n that would do it.
Only 90%, 95% and 99% are offered because those are the z values every textbook tabulates; other levels need the inverse normal, and 95% is what nearly every report uses.

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Confidence Interval for a Mean reference table

Error bound for a standard deviation of 10, by sample size and confidence level
Sample size	Standard error	90%	95%	99%
10	3.162	±5.20	±6.20	±8.15
30	1.826	±3.00	±3.58	±4.70
100	1.000	±1.65	±1.96	±2.58
400	0.500	±0.82	±0.98	±1.29
1,000	0.316	±0.52	±0.62	±0.81

Source: Arithmetic: z × σ ÷ √n with σ = 10; scale the bounds in proportion for any other standard deviation

Confidence Interval for a Mean FAQ

: Sample mean ± 1.96 × (standard deviation ÷ √sample size). A mean of 50 with SD 12 and n = 30: 12 ÷ √30 = 2.19, 1.96 × 2.19 = 4.29, so the interval is 50 ± 4.29 = (45.71, 54.29). Enter the three numbers above and the page shows each step.
: The half-width: z × σ ÷ √n, the amount added and subtracted from the sample mean. OpenStax calls it the error bound (EBM). In the 90% example it is 0.8225. The margin-of-error calculator on this site computes it on its own.
: When the population standard deviation is unknown and estimated from a sample of fewer than about 30. The t multiplier is larger than z for small samples (2.26 instead of 1.96 at n = 10 for 95%), so the interval is wider, which is honest about the extra uncertainty. For large samples the two agree.
: A wider interval at the same data means a higher confidence level: to be 99% sure rather than 95%, the interval has to reach further (2.576 instead of 1.96 standard errors). At the same confidence level, a wider interval means a smaller or more variable sample.