Confidence Interval Calculator
Confidence Interval
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Margin of Error
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Standard Error
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Z/T Score Used
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Understanding Confidence Intervals
A confidence interval gives a range of values likely to contain the true population parameter. A 95% confidence interval means that if you repeated your study 100 times, approximately 95 of those intervals would contain the true value. The formula is: CI = Mean plus or minus Z * (Standard Deviation / sqrt of n).
The margin of error is the half-width of the confidence interval. It depends on three factors: the confidence level (higher confidence = wider interval), the standard deviation (more variability = wider interval), and the sample size (larger samples = narrower interval). Doubling the sample size reduces the margin of error by about 29%.
For large samples (n >= 30), the Z-score is used. For smaller samples, the t-distribution provides more accurate intervals that account for the additional uncertainty from estimating the population standard deviation. This calculator automatically switches to approximate t-values for small samples.
Frequently Asked Questions
What is the difference between confidence level and confidence interval?
The confidence level (e.g., 95%) is the probability that the interval contains the true parameter. The confidence interval is the actual range of values (e.g., [68.1, 76.9]). Higher confidence levels produce wider intervals.
Why do larger samples give narrower intervals?
Standard error (SE = SD/sqrt(n)) decreases as n increases. Since margin of error is proportional to SE, larger samples produce smaller margins of error and thus narrower, more precise confidence intervals.
When should I use a t-distribution vs z-distribution?
Use the t-distribution when the sample size is small (n < 30) and the population standard deviation is unknown (estimated from the sample). For large samples, the t and z distributions are nearly identical.
How do I interpret a 95% confidence interval?
If you calculated a 95% CI of [68, 77] for a mean, it means you are 95% confident the true population mean falls between 68 and 77. It does NOT mean there is a 95% probability the true mean is in this range for this specific interval.