π Statistics & Probability
Confidence Interval Calculator
Calculate 90%, 95%, 99% confidence intervals for a mean or proportion. Free online CI calculator.
What This Calculator Tells You
A confidence interval gives a range of plausible values for a population average based on a sample. Instead of saying 'the average order is $84.50', you can say 'we are 95% confident the true average is between about $77 and $92' β a far more honest statement.
Enter your sample mean, standard deviation, sample size and confidence level. The calculator uses the t-distribution, which is the right choice when the population standard deviation is unknown.
What to Enter
- Sample mean β the average of your sample.
- Std deviation β the sample standard deviation (get it from the standard deviation calculator).
- Sample size β number of observations (at least 2).
- Confidence (%) β commonly 95; 90 and 99 are also used.
Worked Examples
1. Average order value
40 orders, mean $84.50, SD $22, 95% confidence.
- Standard error β $3.48, margin β Β±$7.04.
- 95% CI β $77.46 to $91.54.
2. App rating survey
120 users, mean rating 3.2, SD 1.1.
- 95% CI β 3.00 to 3.40.
3. A small, high-stakes sample
15 test batches with mean yield 650 units and SD 180, at 99% confidence.
- 99% CI β 512 to 788 β a wide range, because the sample is small and confidence is high.
How It’s Calculated
- CI: Confidence Interval
- x: Observed value
- z: Z-score (standard normal)
- s: Sample standard deviation
- n: Number of periods or sample size
- p: Sample proportion / p-value
How to Read Your Result
- Narrow interval = precise estimate; wide interval = collect more data.
- Halving the margin needs roughly four times the sample.
- Comparing groups: if two intervals barely overlap, the difference may be significant; use a proper test to confirm.
- Small differences from other tools in the last decimal come from how t-critical values are approximated.
Common Mistakes to Avoid
- Saying there is a 95% chance the true mean is in this interval. Strictly, 95% of intervals built this way would contain it.
- Using population SD formulas on a sample.
- Applying it to very skewed data with small samples, such as revenue per user with a few huge accounts.
- Ignoring that the sample must be random and representative.
When to Use This Calculator
- Report averages from surveys, experiments or samples with a range.
- Decide whether more data is needed before making a call.
- Compare performance between products, regions or periods.
- Present research results the way academic and business reviewers expect.
Frequently Asked Questions
What is the confidence interval formula?
CI = mean Β± t Γ (SD Γ· βn), where t is the critical value from the t-distribution with n β 1 degrees of freedom.
Why t instead of z?
When the population standard deviation is unknown, which is almost always, the t-distribution accounts for extra uncertainty. With large samples t and z are nearly identical.
What confidence level should I use?
95% is the standard in most business and academic work. Use 90% for quick, low-risk decisions and 99% when mistakes are costly.
How is this different from the margin of error calculator?
This one is for averages (means). The margin of error calculator is for proportions such as survey percentages.
Do you store the values I enter?
No. All calculations happen in your browser. Your data never leaves your device.
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