Standard Error & Confidence Interval Calculator

Standard Error Guide

Standard Error vs Standard Deviation

Standard deviation (SD or s): measures spread of individual data points around the mean. How variable are the measurements? Standard error (SE or SEM): measures precision of the sample mean as an estimate of the population mean. How reliable is our estimate? SE = s / √n. Key insight: as sample size increases, the standard error decreases (we get a better estimate of the true mean). For n=4: SE = s/2. For n=100: SE = s/10. For n=10,000: SE = s/100. Example: measuring plant heights. s = 5cm (natur

Confidence Intervals

A 95% confidence interval: if we repeated the experiment many times, 95% of the resulting confidence intervals would contain the true population mean. Formula: CI = x̄ ± z × SE. z values: 90% CI: z=1.645. 95% CI: z=1.960. 99% CI: z=2.576. Example: mean=14.5, SE=1.2, 95% CI. Lower = 14.5 - 1.960×1.2 = 12.15. Upper = 14.5 + 1.960×1.2 = 16.85. Interpretation: we are 95% confident the true population mean lies between 12.15 and 16.85. Common misconception: 95% CI does NOT mean 'there is a 95% probab

Sample Size and Precision

To achieve a desired margin of error (E) at 95% confidence: n = (z × s / E)². Example: survey with known SD of 10. Want margin of error ±2 at 95% confidence. n = (1.96 × 10 / 2)² = 9.8² = 96.04 → round up to 97 participants. To halve the margin of error: must quadruple the sample size. This inverse square root relationship means large precision improvements require dramatically more data. Practical implication: doubling sample size from 100 to 200 reduces SE by 29% (from s/10 to s/14). Doubling

t-Distribution for Small Samples

For small samples (n < 30), the z-distribution is replaced by the t-distribution, which has heavier tails. t-distribution accounts for uncertainty about the population standard deviation. As n increases, the t-distribution approaches the z (normal) distribution. For n=10, 95% CI: use t=2.262 instead of z=1.960. For n=5, 95% CI: use t=2.776. The t-distribution is why small clinical trials have wider confidence intervals — more uncertainty from less data. In practice: always report both the confid