Confidence Interval Calculator

Calculate confidence intervals estimating the range where the population mean is likely to fall based on sample data. Used for statistical inference and decision making.

Confidence Interval

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Lower Bound

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Upper Bound

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Standard Error

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Margin of Error

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CI = x̄ ± (Z × SE) SE = s / √n

GUIDE

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What is a Confidence Interval?

A confidence interval is a range in which a population parameter (mean, proportion, etc.) is estimated to fall. For example, a 95% confidence interval of [48, 52] means that if you conducted 100 sample surveys using the same method, about 95 would contain the true population mean between 48 and 52.

Meaning of Confidence Level

The confidence level represents the probability that the confidence interval contains the true parameter. Commonly 90%, 95%, or 99% is used, with 95% being most widely used. Higher confidence levels produce wider intervals, lower levels produce narrower intervals. High confidence is more certain but less precise, low confidence is more precise but less certain.

Standard Error and Margin of Error

Standard error (SE) is the standard deviation of the sample mean, calculated as SE = s/√n. Larger sample sizes result in smaller standard errors. Margin of error is the radius of the confidence interval, calculated by multiplying the standard error by the Z-value. The confidence interval is expressed as sample mean ± margin of error.

How to Calculate Confidence Intervals

Confidence intervals are calculated using CI = x̄ ± Z × (s/√n). Where x̄ is the sample mean, Z is the Z-value for the confidence level (1.96 for 95%), s is the sample standard deviation, and n is the sample size. For sample sizes less than 30 when the population is not normally distributed, use the t-distribution instead of Z.

Applications of Confidence Intervals

Confidence intervals are used in various fields including opinion polls, clinical trials, quality control, and marketing research. For example, they are used to evaluate the effectiveness of new drugs, estimate average product lifespan, or predict voter support rates. Narrower confidence intervals indicate more precise estimates, while wider intervals indicate greater uncertainty.

Cautions in Interpreting Confidence Intervals

A 95% confidence interval of [48, 52] does not mean there is a 95% probability that the population mean is in this range. The population mean is a fixed value that either is or is not in the interval. The correct interpretation is "if we repeated the survey using the same method, 95% of the confidence intervals would contain the true population mean."

Frequently asked questions

What changes if I switch the confidence level from 95% to 99%?

The Z-value increases from 1.96 to 2.576, which increases the margin of error and widens the interval. In exchange, the interval is more likely to actually contain the true population mean.

Does increasing the sample size narrow the confidence interval?

Yes. Since SE = s/√n, a larger n produces a smaller standard error, so with everything else equal, a bigger sample gives a narrower interval and a more precise estimate.

Can I still use this calculator if my sample size is under 30?

If the sample is small and the population is not known to be normally distributed, a t-distribution is technically more accurate than the Z-distribution used here. Treat results from small samples as an approximation.

How is the margin of error related to the confidence interval?

The margin of error is the standard error multiplied by the Z-value, and the interval bounds are the sample mean plus or minus that margin. In short, confidence interval = sample mean ± margin of error.

What does a "95% confidence interval" actually mean?

It does not mean a specific interval has a 95% chance of containing the population mean. It means that if you repeated the same sampling procedure many times, about 95% of the resulting intervals would contain the true population mean.