Confidence Interval Calculator
Calculate confidence intervals for population mean.
Sample Statistics
Average of your sample observations. A value of 0 is valid.
Population σ (use z) or sample s (use t for small samples). Must be greater than 0.
Number of observations
Any custom level from 50% to 99.99%
Using standard normal (z)
Confidence Interval
Confidence Interval
[94.63, 105.37]
At 95% confidence level
Margin of Error
±5.37
Lower Bound
94.63
Upper Bound
105.37
Parameters:
Mean: 100
SD: 15
Sample Size: 30
Distribution: Standard normal (z)
z-score: 1.960
About This Calculator
The Confidence Interval Calculator helps you calculate a confidence interval for a population mean based on a sample. A confidence interval provides a range of values within which the true population mean is likely to fall with a specified level of confidence.
Enter your sample mean, standard deviation, sample size, and desired confidence level (typically 95%). The calculator will determine the margin of error and the confidence interval bounds. Wider intervals indicate more uncertainty, while narrower intervals indicate more precision.
Formula & Calculation Method
Confidence Interval Formula:
Confidence Interval:
CI = x̄ ± E
CI = x̄ ± (c × s / √n)
Margin of Error:
E = c × (s / √n)
Bounds:
Lower Bound = x̄ - E
Upper Bound = x̄ + E
where:
x̄ = sample mean
c = critical value — z (standard normal) when σ is known or n ≥ 30, otherwise t with df = n − 1
s = standard deviation (population σ or sample s)
n = sample size
The confidence level (e.g., 95%) means that if you repeated the sampling process many times, 95% of the calculated intervals would contain the true population mean. Higher confidence levels result in wider intervals.
How to Use This Calculator
- Enter Sample Mean: Input the mean of your sample.
- Enter Standard Deviation: Input the population standard deviation (or sample SD if population is unknown).
- Enter Sample Size: Input the number of observations in your sample.
- Select Confidence Level: Choose your desired confidence level (typically 95%).
- View Results: See the confidence interval, margin of error, and bounds.
Frequently Asked Questions
What is a confidence interval?
A confidence interval is a range of values that is likely to contain the true population parameter (e.g., mean) with a specified level of confidence. A 95% confidence interval means that 95% of such intervals would contain the true value.
What does 95% confidence mean?
A 95% confidence level means that if you repeated the sampling process 100 times, about 95 of the calculated confidence intervals would contain the true population mean. It does not mean there's a 95% chance the true mean is in your specific interval.
How does sample size affect the confidence interval?
Larger sample sizes result in narrower confidence intervals (more precision) because the margin of error decreases. Smaller samples give wider intervals (less precision). The relationship is inverse: doubling sample size reduces margin of error by about 30%.
When should I use t-distribution instead of z-distribution?
Use t-distribution when the population standard deviation is unknown and sample size is small (typically n < 30). Use z-distribution when population standard deviation is known or sample size is large (n ≥ 30). This calculator selects automatically based on sample size (and lets you force z or t), computing the exact critical value for any confidence level.
NumCalculators Editorial Team
Multi-disciplinary Expert Team
The NumCalculators Editorial Team consists of subject matter experts across various fields who review and verify the accuracy of our calculators. Each calculator undergoes rigorous testing and validation before publication.