📊 Confidence Interval Calculator
Compute confidence intervals for a population mean or proportion.
What Is a Confidence Interval?
A confidence interval (CI) is a range of values that is likely to contain an unknown population parameter (like a mean or proportion) with a certain degree of confidence. Instead of giving a single estimate, a CI provides an interval estimate that communicates the uncertainty around a sample statistic.
For example, a 95% confidence interval for a population mean tells you: “If we repeated this sampling process many times, about 95% of the intervals constructed would capture the true population mean.”
How to Use This Calculator
- Select the type: Mean (for continuous data) or Proportion (for yes/no data).
- Enter your sample statistics:
- Mean: sample mean, standard deviation, sample size.
- Proportion: number of successes and total sample size.
- Choose the confidence level (90%, 95%, 99%, or enter a custom value).
- For the mean, select whether you are using the sample standard deviation (t‑distribution) or the population standard deviation (Z‑distribution).
- Click Calculate. The results show the margin of error, lower and upper bounds, and the critical value used.
Confidence Interval Formulas
For a Population Mean
When population standard deviation (σ) is known (Z‑interval):
CI = x̄ ± Zα/2 × (σ / √n)
When only sample standard deviation (s) is known (t‑interval):
CI = x̄ ± tα/2, df × (s / √n), where df = n − 1.
For a Population Proportion
CI = p̂ ± Zα/2 × √(p̂(1 − p̂) / n)
The calculator uses the standard (Wald) method. For very small samples the Wilson score interval is preferred, but Wald works well for moderate/large n.
Understanding Confidence Levels
The confidence level (e.g., 95%) indicates how often the true parameter would be captured if the study were repeated many times. Higher confidence levels yield wider intervals (more certainty about capturing the true value), while lower levels give narrower intervals but less certainty.
When to Use t‑Distribution vs Z‑Distribution
- Z‑distribution: used when the population standard deviation (σ) is known, or the sample size is large (n > 30).
- t‑distribution: used when σ is unknown and estimated by the sample standard deviation (s). The t‑distribution accounts for extra uncertainty, especially with small samples.
This calculator applies the t‑distribution automatically when you choose “Sample Std Dev”. For custom confidence levels with the t‑distribution, it approximates the critical value using the Z‑distribution (a valid approximation for larger samples).
Example Calculation
Mean example: Sample mean = 68.5, sample standard deviation = 7.2, n = 40, 95% confidence (t‑distribution).
Standard error = 7.2/√40 ≈ 1.138, t0.025,39 ≈ 2.023, Margin of error ≈ 2.30. CI = [66.20, 70.80].
Proportion example: 113 successes out of 200, 95% confidence.
p̂ = 0.565, standard error ≈ 0.0351, Z = 1.96, margin ≈ 0.0688, CI = [0.4962, 0.6338] (or 49.62% to 63.38%).
Frequently Asked Questions
What does a 95% confidence interval really mean?
It means that if you took 100 different random samples and computed a 95% CI from each, roughly 95 of those intervals would contain the true population parameter. It is not a 95% probability that the true value lies inside a single computed interval.
Can I use this calculator for very small samples?
Yes, but when using the t‑distribution for the mean, the table of critical values covers degrees of freedom down to 1. For proportions, the Wald method may be unreliable if np̂ < 5 or n(1 – p̂) < 5. In such cases consider the Wilson or Clopper‑Pearson method.
Why does the interval get wider with a higher confidence level?
A higher confidence level (e.g., 99% vs 95%) requires a larger critical value, which increases the margin of error. This gives a wider range to ensure the true parameter is captured with greater reliability.
What’s the difference between standard deviation and standard error?
Standard deviation measures the spread of individual data points. Standard error measures the precision of the sample mean (how much the sample mean would vary if you repeated the sampling). The standard error shrinks as the sample size increases.
Use this free confidence interval calculator to quickly find the range for your population parameter. Bookmark it for your statistics homework or data analysis projects!
