Confidence Interval Calculator
Calculate confidence intervals for population means, proportions, and differences. Estimate required sample sizes and determine statistical significance.
Confidence Interval Calculator
Calculate confidence interval for a population mean from sample data.
How to Use the Confidence Interval Calculator
Choose the type of confidence interval you need: mean (for continuous data), proportion (for success/failure data), difference (comparing two groups), or sample size (planning studies).
Confidence Interval Types
Mean Confidence Interval
Estimates the range where the true population mean likely falls:
- Formula: x̄ ± t × (s/√n)
- Uses t-distribution for small samples (n < 30)
- Uses z-distribution for large samples or known σ
Proportion Confidence Interval
Estimates the range for a population proportion using the Wilson score interval:
- More accurate than the Wald interval for extreme proportions
- Works well even with small sample sizes
- Never produces intervals outside [0, 1]
Difference of Means
Uses Welch's t-interval for comparing two independent groups:
- Doesn't assume equal variances
- More robust than pooled t-test
- If CI includes 0, difference is not significant
Sample Size Estimation
Calculate how many observations you need for a desired precision:
- Smaller margin of error → larger sample needed
- Higher confidence level → larger sample needed
- Requires estimated standard deviation
Interpreting Confidence Intervals
- 95% CI: If we repeated the study many times, 95% of CIs would contain the true value
- Width: Narrower intervals indicate more precise estimates
- Significance: If a CI for difference doesn't include 0, the difference is significant
Common Confidence Levels
- 90%: z = 1.645 (less strict)
- 95%: z = 1.96 (most common)
- 99%: z = 2.576 (more conservative)
- 99.9%: z = 3.291 (very conservative)
Important Considerations
- Sample size matters: Larger samples produce narrower intervals
- Assumptions: Most CIs assume random sampling
- Not probability: The CI either contains the true value or it doesn't
- Margin of error: Half the width of the confidence interval
Applications
- Survey research and polling
- Clinical trials and medical studies
- Quality control and manufacturing
- A/B testing and marketing experiments
- Scientific research and hypothesis testing
Worked Example: 95% CI for a Mean
Suppose you measure the weights of n = 25 packages and find a sample mean of x̄ = 502 g with a sample standard deviation of s = 10 g. Because n < 30 and σ is unknown, use the t-distribution with 24 degrees of freedom (t = 2.064 at 95%):
- Standard error: s/√n = 10/√25 = 2 g
- Margin of error: 2.064 × 2 = 4.13 g
- 95% CI: 502 ± 4.13 → (497.87 g, 506.13 g)
If your target fill weight is 500 g, the interval contains 500, so the data do not provide evidence (at the 5% level) that the machine is over- or under-filling.
Frequently Asked Questions
What does a 95% confidence interval actually mean?+
It means that if you repeated the same sampling procedure many times and computed an interval each time, about 95% of those intervals would contain the true population value. It does not mean there is a 95% probability that the true value lies in your one specific interval — the true value is fixed; it is the interval that varies from sample to sample.
When should I use the t-distribution instead of the z-distribution?+
Use the t-distribution whenever the population standard deviation is unknown and you are estimating it from the sample, which is almost always the case in practice. The calculator applies the t-distribution automatically for mean intervals; for large samples (roughly n greater than 30) the t and z values are nearly identical, so the distinction matters most for small samples.
Why does this calculator use the Wilson score interval for proportions?+
The textbook Wald interval (p ± z√(p(1−p)/n)) breaks down for small samples or proportions near 0 or 1 — it can even produce bounds below 0% or above 100%. The Wilson score interval stays within [0, 1], has much better coverage accuracy, and is the interval recommended in most modern statistics references.
How do I reduce the margin of error?+
The two levers are sample size and confidence level. Quadrupling the sample size halves the margin of error (it shrinks with √n), while lowering the confidence level from 99% to 95% or 90% also narrows the interval. Use the calculator's sample size mode to find exactly how many observations you need for a target margin of error.
Is this confidence interval calculator free to use?+
Yes. It is completely free, requires no sign-up, and all calculations run entirely in your browser, so your data never leaves your device. It works on phones, tablets, and desktops, making it convenient for coursework, research, and quick checks during data analysis.
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