Enter your sample mean, standard deviation and sample size to calculate the margin of error and confidence interval at your chosen confidence level.
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The margin of error is calculated as the z-score for your chosen confidence level multiplied by the standard error (standard deviation ÷ square root of sample size). The confidence interval is then the sample mean plus and minus that margin of error, giving a range you can be reasonably confident contains the true population mean.
For a sample mean of 100, standard deviation of 15, sample size of 50, at a 95% confidence level (z ≈ 1.96): the standard error is 15 ÷ √50 ≈ 2.1213. Margin of error = 1.96 × 2.1213 ≈ 4.1578. The confidence interval is 100 ± 4.1578, giving a lower bound of about 95.84 and an upper bound of about 104.16.
Interpreting a 95% confidence interval as 'a 95% chance the true mean is in this specific range', that's a common but technically incorrect interpretation, the 95% describes the long-run reliability of the method across repeated sampling, not the probability for this one interval.
Using a higher confidence level (like 99%) without realizing it widens the interval, higher confidence requires a larger margin of error to maintain that confidence, there's a direct tradeoff between confidence level and precision.
Assuming a bigger sample always dramatically narrows the interval, since margin of error scales with the square root of sample size, quadrupling the sample size only halves the margin of error, not a proportional four-times reduction.
It means that if you repeated your sampling process many times, about 95% of the resulting intervals would contain the true population mean. It doesn't mean there's a 95% chance the true mean falls in this specific interval.
The margin of error divides by the square root of sample size, so larger samples produce smaller standard errors and, therefore, narrower, more precise confidence intervals.
Z-Score Calculator
Calculate a z-score and percentile from a value, mean and standard deviation.
Sample Size Calculator
Calculate the survey sample size needed for a target confidence level and margin of error.
Standard Deviation Calculator
Calculate mean, variance and standard deviation from a data set.
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