Confidence Interval Calculator

Q: What does a 95% confidence level actually mean?

It refers to the long-run frequency of intervals containing the true parameter, not the probability that any single interval contains it. Once you've computed the interval, the parameter is either in it or not — it's not random.

Q: How does sample size affect the interval width?

Width shrinks as 1/√n. Quadrupling your sample size halves the margin of error. This is why doubling sample size has diminishing returns past a point.

Q: Can I compute a confidence interval for a proportion?

Yes. The formula is p̂ ± z_(α/2) · √(p̂(1−p̂)/n). The calculator switches between mean and proportion modes — pick the one that matches your data.

Q: What's the relationship between CI and hypothesis testing?

If a 95% CI excludes the null value, the corresponding two-tailed test rejects at α = 0.05. CIs convey effect size and uncertainty in one shot — they're more informative than a bare p-value.

Q: Why do common confidence levels stop at 99%?

Higher levels (99.9%, 99.99%) require dramatically wider intervals for marginal gains in coverage probability. 90/95/99 covers most practical decisions; beyond that, the interval becomes too wide to be useful.

Calculate confidence intervals for means and proportions.

Interval Type

Sample Mean (x̄)

Sample Size (n)

Population Std Dev (σ)

Confidence Level

What is a Confidence Interval?

A confidence interval provides a range of plausible values for an unknown population parameter based on sample data. Instead of giving a single estimate, it quantifies the uncertainty in your estimate.

The interval has the form: point estimate ± margin of error

Confidence Level

The confidence level (commonly 90%, 95%, or 99%) represents how often the interval would contain the true parameter if we repeated the sampling process many times.

A 95% confidence interval does NOT mean there’s a 95% probability the parameter is in the interval. It means 95% of similarly constructed intervals would contain the parameter.

Types of Confidence Intervals

Z-Interval ($ \sigma $ known)

When the population standard deviation is known:

$$ \bar{x} \pm z^* \cdot \frac{\sigma}{\sqrt{n}} $$

T-Interval ($ \sigma $ unknown)

When using the sample standard deviation:

$$\bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}}$$

The t-distribution has heavier tails, giving wider intervals to account for the extra uncertainty.

Proportion Interval

For estimating population proportions:

$$\hat{p} \pm z^* \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Critical Values

Confidence	$ z^* $	$ t^* $ $df=29$
90%	1.645	1.699
95%	1.960	2.045
99%	2.576	2.756

Factors Affecting Interval Width

Sample Size

Larger samples give narrower intervals ($ \sqrt{n} $ in the denominator).

Confidence Level

Higher confidence requires wider intervals.

Variability

More variable data gives wider intervals.

Interpreting the Interval

Correct: “We are 95% confident that the true population mean is between 45 and 55.”
Incorrect: “There is a 95% probability that the population mean is between 45 and 55.”

The parameter is fixed (not random). The interval either contains it or doesn’t.

Sample Size Determination

To achieve a desired margin of error ( E ):

$$n = \left( \frac{z^* \cdot \sigma}{E} \right)^2$$

Double the precision requires four times the sample size.

Assumptions

For Mean Intervals

Random sample
Normal population OR large sample ($ n \geq 30 $)
For t-interval: no extreme outliers

For Proportion Intervals

Random sample
$ np \geq 10 $ and $ n(1-p) \geq 10 $ (success/failure condition)

Frequently Asked Questions

What is a confidence interval?

A confidence interval is a range of values that’s likely to contain the true population parameter. A 95% confidence interval means that if you repeated the sampling 100 times, about 95 of the resulting intervals would contain the true value.

What’s the formula for a confidence interval for the mean?

CI = x̄ ± z_(α/2) · (σ / √n) when the population standard deviation σ is known, or x̄ ± t_(α/2, n-1) · (s / √n) when only the sample standard deviation s is available.

When do I use z vs t distribution?

Use z when σ is known or n ≥ 30. Use t when σ is unknown and n < 30. The t-distribution has heavier tails to account for extra uncertainty from estimating σ from a small sample.

What does a 95% confidence level actually mean?

It refers to the long-run frequency of intervals containing the true parameter, not the probability that any single interval contains it. Once you’ve computed the interval, the parameter is either in it or not — it’s not random.

How does sample size affect the interval width?

Width shrinks as 1/√n. Quadrupling your sample size halves the margin of error. This is why doubling sample size has diminishing returns past a point.

Can I compute a confidence interval for a proportion?

Yes. The formula is p̂ ± z_(α/2) · √(p̂(1−p̂)/n). The calculator switches between mean and proportion modes — pick the one that matches your data.

What’s the relationship between CI and hypothesis testing?

If a 95% CI excludes the null value, the corresponding two-tailed test rejects at α = 0.05. CIs convey effect size and uncertainty in one shot — they’re more informative than a bare p-value.

Why do common confidence levels stop at 99%?

Higher levels (99.9%, 99.99%) require dramatically wider intervals for marginal gains in coverage probability. 90/95/99 covers most practical decisions; beyond that, the interval becomes too wide to be useful.