What does a 95% confidence interval actually mean?

It means that if you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true value. It describes the long-run reliability of the method — not the probability that one particular interval is right. Any single interval either contains the parameter or it doesn't.

When should I use z instead of t?

Use the z-distribution when the population standard deviation is known or the sample is large (commonly n ≥ 30). Use the t-distribution for small samples with an unknown standard deviation; its heavier tails give a slightly wider, more honest interval. This tool's Auto setting picks t for n < 30 and z otherwise.

How do I find a confidence interval for a proportion?

Switch the tool to Proportion mode and enter the number of successes and the sample size. It computes p̂ = successes / n, the standard error √(p̂(1−p̂)/n), multiplies by the z critical value, and adds and subtracts that margin from p̂ — the standard Wald interval used for poll results.

Does a higher confidence level give a wider or narrower interval?

Wider. Asking for more confidence (say 99% instead of 95%) requires a larger critical value, which stretches the margin of error. There is always a trade-off: more confidence means a less precise, wider range from the same data.

How does sample size affect the interval?

The standard error falls with the square root of the sample size, so quadrupling n only halves the margin of error. Diminishing returns set in quickly — going from 1,000 to 4,000 respondents in a poll roughly halves the margin, while the first few hundred do most of the work.

What assumptions does the interval rely on?

A random sample, independent observations, and an approximately normal sampling distribution of the estimate. For means that follows from the Central Limit Theorem at moderate sample sizes; for proportions a common rule of thumb is at least 10 successes and 10 failures before the normal approximation is trustworthy.

Statistics

Confidence Interval Calculator

Build a confidence interval for a mean or a proportion from your sample. Pick any confidence level, let it choose z or t automatically, and see the standard error, margin of error, and interval worked out — updating as you type.

Your sample statistics

x̄Sample mean

The average of your sample

sStandard deviation

Spread of the data

nSample size

Number of observations (≥ 2)

Confidence level

% custom

Distribution

Auto uses t for small samples (n < 30) and z otherwise.

95% confidence interval

93.8083 to 106.1917

centred on 100, margin of error ±6.1917

Using the t-distribution· df = 24

93.808

106.192

100

Std. error

Critical value

2.064

Margin

±6.1917

Width

12.3834

Try an example

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The basics

What is a confidence interval?

A sample gives you a single estimate — an average, a percentage — but it never lands exactly on the true population value. A confidence interval wraps that estimate in a range and attaches a confidence level, so you report not just a best guess but how much room for error comes with it. Every interval is built from the same two pieces.

estimate ± margin of error

The shape of every interval

Take your point estimate and add and subtract a margin of error. The midpoint is the estimate; the half-width is the margin. That is the whole structure, whether you are estimating a mean or a proportion.

100 ± 5.88 → [94.12, 105.88]

margin = critical value × standard error

What sets the width

The standard error measures how much the estimate would bounce around across samples; the critical value stretches it to match the confidence level you ask for. Multiply them for the margin.

1.96 × 3 = 5.88

Method

How to build one by hand

Four steps for a mean. A proportion follows the same arc with its own standard-error formula.

1
Compute the standard error
Divide the standard deviation by the square root of the sample size. This is how far the sample mean x̄ typically strays from the true mean.
SE = s / √n
2
Look up the critical value
Pick z* from the normal table for large samples, or t* at n − 1 degrees of freedom for small ones. It grows with the confidence level.
z* (95%) = 1.96
3
Multiply for the margin of error
The critical value times the standard error gives the half-width of the interval.
margin = z* · SE
4
Add and subtract from the estimate
Centre the interval on the estimate: subtract the margin for the lower bound, add it for the upper.
x̄ ± margin

Worked examples

See it across the three cases

The same recipe, computed live for a large-sample mean, a small-sample mean, and a poll proportion.

Given large sample, σ known: x̄ = 100, s = 15, n = 50, 95%

SE = 15 / √50 = 2.1213

z* (95%) = 1.96

margin = 1.96 × 2.1213 = 4.1578

100 ± 4.1578

95% CI = [95.8422, 104.1578]

z or t?

Which distribution to use

Both give a critical value; they differ in how much they trust a small sample to estimate the spread.

z* · large n or known σ

The normal distribution

Use z when the population standard deviation is known, or the sample is large enough that estimating it adds little extra uncertainty. The critical value does not depend on sample size.

z* = 1.96 at 95%, for any n

t* · small n, unknown σ

The Student-t distribution

Use t for small samples with an unknown standard deviation. Its fatter tails widen the interval to pay for the guesswork in s, and it tightens toward z as n grows.

t* = 2.20 at 95%, df = 11

Reference

Critical values at a glance

The two-sided z* for the standard confidence levels — the numbers you plug into the margin of error.

Confidence level	Tail area (α/2)	z* critical value
80%	0.100	1.282
85%	0.075	1.44
90%	0.050	1.645
95%	0.025	1.96
98%	0.010	2.326
99%	0.005	2.576
99.9%	0.0005	3.291

For small samples, the t* value also depends on the degrees of freedom (df = n − 1). It starts well above z* and shrinks toward it as the sample grows.

df	t* (90%)	t* (95%)	t* (99%)
1	5.572	9.711	28.471
2	2.885	4.171	8.761
3	2.347	3.159	5.646
4	2.130	2.769	4.547
5	2.014	2.568	4.010
8	1.859	2.306	3.352
10	1.812	2.228	3.168
15	1.753	2.131	2.946
20	1.725	2.086	2.845
30	1.697	2.042	2.750
50	1.676	2.009	2.678
100	1.660	1.984	2.626
∞ (z*)	1.645	1.960	2.576

Quick chart

Margin of error by sample size

The classic poll question — “what is the margin of error for n respondents?” — answered for every common sample size. These are the widest-case margins (at 50/50), the figure news outlets cite.

Confidence level

Worst-case margin at p̂ = 0.5

n = 50

±13.9%

n = 100

±9.8%

n = 200

±6.9%

n = 300

±5.7%

n = 400

±4.9%

n = 500

±4.4%

n = 750

±3.6%

n = 1000

±3.1%

n = 1500

±2.5%

n = 2000

±2.2%

n = 3000

±1.8%

n = 5000

±1.4%

Worth knowing

Rules and pitfalls

Six things that trip people up — and the facts that keep your intervals honest.

not P(parameter in interval) = 95%

It's about the method

A 95% interval does not mean a 95% chance the true value sits inside this one range. The true value is fixed; it is the procedure that succeeds 95% of the time.

margin ∝ 1/√n

Diminishing returns

Halving the margin of error takes four times the sample. The first few hundred observations buy most of the precision; the rest get expensive fast.

higher confidence → wider

Confidence costs width

Demanding more confidence stretches the critical value and the interval with it. There is no free lunch — precision and confidence pull in opposite directions.

np ≥ 10 and n(1−p) ≥ 10

Check the proportion rule

The normal-based interval for a proportion needs roughly ten successes and ten failures. With tiny counts, use an exact (Wilson or Clopper–Pearson) method instead.

random + independent

Garbage in, garbage out

The math assumes a random, independent sample. A biased sample produces a tidy interval around the wrong number — the formula cannot detect bad data.

overlap ≠ no difference

Overlap is not a test

Two overlapping confidence intervals do not prove the groups are equal. To compare them properly, build an interval for the difference and see if it contains zero.

Where it shows up

Where intervals matter

Anywhere a number comes from a sample, a confidence interval is the honest way to report it.

52% ± 3%

Polling & surveys

Election polls and market research report a proportion with a margin of error so readers can judge whether a lead is real or within the noise.

effect ± CI

Medicine & trials

Clinical studies report treatment effects as intervals; if the interval for a difference excludes zero, the effect is statistically significant.

x̄ ± margin

Quality & A/B testing

Manufacturing tolerances and conversion-rate experiments use intervals to decide whether a measured change is signal or sampling chance.

FAQ

Common questions

A confidence interval is a range of values, built from sample data, that is likely to contain the true population value — a mean or a proportion. Instead of a single point estimate, it reports a plausible range together with a confidence level, so you can see both the estimate and its uncertainty.

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