Confidence Interval Calculator
Calculate the confidence interval for a population mean from sample data.
How to use this tool
- Enter sample mean (x̄), sample standard deviation (s), sample size (n) and z* (confidence level) in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your lower bound and the full breakdown beneath it.
Calculate the confidence interval: CI = x̄ ± z* × (s / √n). Enter your sample mean, standard deviation, size, and z-value.
Formula
Standard error: SE = s / √n
Margin of error: ME = z* × SE
Confidence interval: [x̅ − ME, x̅ + ME]
How it works
This calculator constructs a confidence interval for an unknown population mean using sample mean x̅, sample standard deviation s, and sample size n. The user supplies the critical value z* directly (e.g. 1.96 for 95% confidence, 2.576 for 99%). The formula assumes the sampling distribution of the mean is approximately normal, which is reasonable for large samples (n ≥ 30) by the Central Limit Theorem or when the population itself is normal.
Worked example
Worked example
- Inputs: x̅ = 50, s = 10, n = 100, z* = 1.96 (95% confidence level).
- Standard error: SE = 10 / √100 = 10 / 10 = 1.0.
- Margin of error: ME = 1.96 × 1.0 = 1.96.
- Confidence interval: [50 − 1.96, 50 + 1.96] = [48.04, 51.96].
Lower bound = 48.04; upper bound = 51.96; margin of error = 1.96.
Key terms
- Confidence interval
- A range of values computed from sample data that is expected to contain the true population parameter a specified percentage of the time (e.g. 95%) under repeated sampling.
- Standard error (SE)
- The standard deviation of the sampling distribution of the mean, equal to s/√n. It quantifies how much the sample mean is expected to vary from sample to sample.
- Margin of error (ME)
- Half the width of the confidence interval; the maximum expected difference between the sample mean and the true population mean at the chosen confidence level.
- Critical value (z*)
- The z-score from the standard normal distribution corresponding to the desired confidence level; common values are 1.645 (90%), 1.96 (95%), and 2.576 (99%).
- Central Limit Theorem
- The statistical result stating that the sampling distribution of the mean approaches a normal distribution as n increases, justifying the use of z-based intervals for large samples.
Frequently asked questions
- How do I choose the z* value?
- Common critical z-values: 1.645 for 90% confidence, 1.960 for 95%, 2.576 for 99%. These assume a large enough sample (n ≥ 30) so the Central Limit Theorem applies.