Confidence Interval Calculator

Math

Calculate confidence intervals for a population mean (using Student’s t-distribution or standard normal Z) or population proportion (with Wald and Wilson Score intervals).

Presets:

95% Confidence Interval

[46.2659, 53.7341]

Statistical Interpretation: We can be 95% confident that the true population mean lies between 46.27 and 53.73.

Sample Mean (x̄) 50

Margin of Error (±E) 3.7341

Critical Value (t* (df = 29)) 2.0452

Standard Error (SE) 1.8257

Step-by-Step Statistical Derivations

1 1. Standard Error of the Mean (SE)

SE = s / √n = 10 / √30 = 10 / 5.4772 = 1.82574

Measures sampling variability of the mean across repeated independent samples.

2 2. Critical Value (t* (df = 29))

For 95% confidence level (α = 0.05), t* (df = 29) = 2.0452

Based on Student's t-distribution with n - 1 degrees of freedom because population standard deviation is unknown.

3 3. Margin of Error (E)

E = Critical × SE = 2.0452 × 1.82574 = 3.73406

Maximum expected difference between the sample estimate and the true population parameter.

4 4. Confidence Interval Bounds

CI = x̄ ± E = 50 ± 3.7341 = [46.2659, 53.7341]

Lower bound = x̄ - E, Upper bound = x̄ + E.

How This Calculator Works

A confidence interval is a range of estimates for an unknown population parameter, constructed so that a designated percentage (e.g. 95%) of such calculated intervals will contain the true parameter across repeated hypothetical samplings.

📐 Confidence Interval Formulas

Mean (t): x̄ ± t*(s / √n) | Mean (Z): x̄ ± z*(σ / √n) | Proportion: p̂ ± z*√[p̂(1 - p̂)/n]

x̄: Sample mean (point estimate)

s, σ: Sample standard deviation s or known population standard deviation σ

n: Sample size

t*, z*: Critical values from Student's t or standard normal distributions

SE: Standard error of the estimate

E: Margin of error = Critical Value × SE

Key Mathematical Rules & Properties

When to Use t vs z Distribution for Mean

Use the Student's t-distribution whenever the population standard deviation σ is unknown and must be estimated using the sample standard deviation s (which is almost always the case in real-world research).

Wald vs Wilson Score for Proportions

The traditional Wald interval can perform poorly when sample sizes are small or when the observed proportion is very close to 0 or 1. The Wilson Score interval adjusts the center toward 0.5 to ensure accurate coverage.

✎

Worked Example: 95% Confidence Interval for Mean

A random sample of 25 battery lives has a mean of 40 hours with s = 5 hours. Find the 95% confidence interval.

1. Sample mean x̄ = 40, s = 5, n = 25, df = 25 - 1 = 24.
2. For 95% confidence with df = 24, t* ≈ 2.064.
3. Standard error SE = 5 / √25 = 5 / 5 = 1.0 hour.
4. Margin of error E = 2.064 × 1.0 = 2.064 hours.
5. Interval = 40 ± 2.064 = [37.936, 42.064] hours.

Final Answer: [37.94, 42.06] hours

Frequently Asked Questions

Does a 95% confidence interval mean there is a 95% chance the true mean is inside this specific interval?

Strictly speaking in frequentist statistics, the true parameter is a fixed constant, not a random variable. The 95% refers to the long-run success rate of the estimation procedure: if 100 independent samples are drawn, approximately 95 of the resulting intervals will capture the true mean.

Understanding your calculation

Method, example and limitations

How the calculation works

An interval for a mean uses the estimate plus or minus a critical value times its standard error. Known population spread and estimated sample spread require different choices of z or t.

Example to check

With mean 100, known population standard deviation 10 and n = 100, a 95% normal interval is 100 ± 1.96, or 98.04 to 101.96.

Before using the result

A 95% procedure covers the fixed true parameter in about 95% of repeated samples under its assumptions. It does not assign a 95% probability to a fixed parameter after observing this interval.

NIST: Confidence limits for the mean · Calculation methodology · Report a result that needs review