Confidence Interval Calculator
Calculate confidence intervals for a population mean. Breaks out the Margin of Error (MOE), critical Z-value, and standard error. Supports 90%, 95%, and 99% confidence levels.
Confidence Interval
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Sample Size
100 (large sample)
Formula
CI = x̄ ± Z × (s/√n) = 50 ± 1.645 × (10/√100) = [48.355, 51.645]
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Confidence Interval
[48.355, 51.645]
Lower
48.355
Margin of Error
±1.645
Upper
51.645
Critical Z
1.645 (90% confidence)
Std Error
1
MOE = Z × SE
±1.645
Formula Application
CI = x̄ ± Z × (s/√n) = 50 ± 1.645 × (10/√100) = [48.355, 51.645]
The Formula
A confidence interval provides a range of plausible values for an unknown population parameter. The margin of error (MOE) is the critical value (Z) times the standard error (SE). The standard error measures how much the sample mean is expected to vary from the true population mean.
Variable Definitions
Sample Mean
The average of your sample data. The point estimate of the population mean.
Critical Value (Z-score)
1.645 for 90%, 1.96 for 95%, 2.576 for 99% confidence. Higher confidence requires a larger Z-value and produces a wider interval.
Standard Error & Margin of Error
SE = σ/√n measures sampling variability. MOE = Z × SE is the ± range around the mean that creates the confidence interval.
How to Use This Calculator
- 1
Enter your sample mean, sample size, and standard deviation.
- 2
Select the desired confidence level (95% is the standard for most published research).
- 3
View the confidence interval range, margin of error, and critical Z-value used.
- 4
The MOE is explicitly broken out — researchers often need this for reporting in papers and tables.
- 5
A note indicates whether your sample is small (<30), moderate, or large to guide interpretation.
A confidence interval provides a range of plausible values for a population parameter
Understanding the Concept
Confidence intervals are fundamental to inferential statistics. They express the uncertainty inherent in estimating a population parameter from a sample. A 95% confidence interval means: if you repeated the sampling process 100 times under identical conditions, approximately 95 of those intervals would contain the true population mean. It does NOT mean there is a 95% probability that the true mean lies within your specific interval — the true mean either is or is not in that interval, and you do not know which. The margin of error (MOE) is the key number reported in opinion polls, market research, and clinical studies — it communicates the precision of the estimate. The standard error (SE) shrinks as sample size increases, which means larger samples produce narrower intervals with more precise estimates. Doubling the sample size reduces the SE by about 29% (1/√2). For small samples (n < 30), the t-distribution should be used instead of the normal distribution because the sample standard deviation s is an imperfect estimate of the true population σ. The t-distribution has heavier tails, producing wider (more conservative) intervals that account for this additional uncertainty.
Frequently Asked Questions
Sources & References
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