Standard deviation, standard error & 95% confidence intervals

First, separate spread from uncertainty

Sample standard deviation (SD) describes how spread out the observed values are around their sample mean. It uses the same units as the measurements. Use the sample SD calculated with n minus 1 in the variance denominator; do not enter a variance or an already-calculated standard error. NIST’s measures of scale explains this distinction.

Standard error (SE) estimates how much a sample mean would vary across repeated samples under the sampling model. For independent, identically distributed observations with finite variance, its usual estimate is sample SD divided by the square root of sample size. SD concerns the observations; SE concerns the estimated mean.

A worked example: n = 25, mean = 50, SD = 10

Imagine 25 independent measurements in a fictional production check, summarized by a mean of 50 units and sample SD of 10 units. Assume an appropriate random sample from one normal population, with unknown population standard deviation. Those assumptions are part of the example, not facts the calculator can verify.

1. Standard error: 10 / square root of 25 = 10 / 5 = 2 units.
2. Degrees of freedom: 25 minus 1 = 24.
3. Critical value: for a two-sided 95% interval, use Student-t’s 0.975 quantile with 24 degrees of freedom: about 2.0639.
4. Margin of error: 2.06389856 times 2 = about 4.1278 units.
5. Endpoints: 50 minus 4.1278 and 50 plus 4.1278 give approximately 45.8722 to 54.1278 units.

The standard error and 95% confidence interval calculator starts with these values. It retains full precision internally and rounds the display. NIST’s Student-t table gives the same critical value rounded to 2.064.

What the interval means

The interval estimates the population mean. If we repeated the same valid sampling process and interval construction many times, about 95% of the resulting intervals would contain that fixed mean. It does not mean that 95% of individual measurements fall between these endpoints, or assign a 95% probability to the fixed mean being inside this one completed interval. See NIST’s confidence limits for the mean.

With the same SD of 10 and a hypothetical sample size of 100, SE would be 1. Keeping the mean at 50, the t interval would be about 48.0158 to 51.9842. This is an illustration with SD held fixed; collecting more observations does not guarantee the next sample’s SD or mean stays unchanged.

Checks the three summary numbers cannot do

• The normal-population t interval is exact under its model. With only 25 observations, strong skew, heavy tails, or outliers deserve particular attention. Inspect the original data.
• Repeated measurements of the same item or clustered observations may not be independent. Counting them as independent can make SE misleadingly small.
• A larger sample does not repair selection bias, a changing process, or poor measurements.
• Zero sample SD produces a zero-width arithmetic interval, but it does not establish zero population variability.

This tool is for one population mean. A proportion, a future individual observation, paired data, or a comparison between groups needs a method suited to that question. NIST also provides a separate worked t-interval example for further practice.