# Standard Deviation of the Sample Mean Calculator URL: https://speedycalculator.com/calculators/statistics/standard-deviation-sample-mean/ Last updated: 2026-10-30 > The standard deviation of the sample mean calculator finds σx̄ = σ ÷ √n, the spread of the sampling distribution of the mean, from the population standard deviation and sample size – optionally with the finite population correction – plus the variance of the mean and the 95% range of sample means. ## Inputs - Population standard deviation (σ) - Sample size (n) - Population size - Population size (N) ## Results - Finite population correction factor - Standard deviation of the sample mean σx̄ - Variance of the sample mean σx̄² - 95% of sample means within ± (1.96σx̄) - 99% of sample means within ± (2.576σx̄) - Reduction compared with one observation - Sample size needed to halve σx̄ ## Formula ``` σx̄ = σ ÷ √n · with a finite population of N: σx̄ = (σ ÷ √n) × √((N − n) ÷ (N − 1)) · 95% of sample means lie within μ ± 1.96σx̄ ``` ## Explanation Draw many samples of the same size from a population and compute the mean of each. Those means will not all be equal – they form their own distribution, the sampling distribution of the mean. Its standard deviation, σx̄, tells you how far a single sample mean is likely to fall from the true population mean. The rule is simple and powerful: σx̄ = σ ÷ √n. Averages vary less than individual values, and the larger the sample the less they vary. Quadrupling the sample size halves σx̄; a hundred-fold increase shrinks it to a tenth. By the central limit theorem the sampling distribution is approximately normal for samples of about 30 or more even when the population is not, so about 95% of sample means fall within 1.96σx̄ of μ. When you sample a sizeable share of a small population without replacement, the means vary even less. The finite population correction √((N − n) ÷ (N − 1)) accounts for this; it matters once the sample is more than about 5% of the population. When σ is unknown and estimated by the sample SD s, the same quantity is called the standard error of the mean. ### How to use the Standard Deviation of the Sample Mean Calculator - **Population standard deviation (σ)** - **Sample size (n)** - **Population size** - **Population size (N)** Results update as you type, and everything is calculated in your browser. ### How it works Enter the population standard deviation σ and the sample size n. Choose a finite population and give its size N if you sample without replacement from a small population. The calculator divides σ by √n and applies the correction if chosen. ### Worked example IQ scores have σ = 15. For random samples of n = 36 people, σx̄ = 15 ÷ √36 = 2.5 points, so 95% of sample means fall within ±1.96 × 2.5 = ±4.9 points of the population mean. If the 36 are drawn without replacement from a school of 500 students, the correction √(464 ÷ 499) = 0.9643 reduces σx̄ to 2.41. To halve σx̄ to 1.25 you would need 144 people. ### How sample size shrinks σx̄ (σ = 15) nσx̄95% range of sample means115.0±29.447.5±14.7253.0±5.91001.5±2.94000.75±1.5 ### Inputs explained FieldWhat to enterPopulation standard deviation (σ)A numberSample size (n)A numberPopulation sizeChoose one: Very large or sampling with replacement, Finite – apply the correctionPopulation size (N)A number ### What each result means - **Standard deviation of the sample mean σx̄** = [Population standard deviation (σ)]÷sqrt([Sample size (n)])×[Finite population correction factor] - **Variance of the sample mean σx̄²** = [Standard deviation of the sample mean σx̄]^2 - **95% of sample means within ± (1.96σx̄)** = 1.959964×[Standard deviation of the sample mean σx̄] - **99% of sample means within ± (2.576σx̄)** = 2.575829×[Standard deviation of the sample mean σx̄] - **Reduction compared with one observation** = [Standard deviation of the sample mean σx̄]÷[Population standard deviation (σ)]×100 - **Sample size needed to halve σx̄** = 4×[Sample size (n)] ## Worked example Inputs — Population standard deviation (σ): 15; Sample size (n): 36; Population size: Very large or sampling with replacement. Results — **Standard deviation of the sample mean σx̄ = 2.5**; Variance of the sample mean σx̄² = 6.25; 95% of sample means within ± (1.96σx̄) = 4.89991; 99% of sample means within ± (2.576σx̄) = 6.439573; Reduction compared with one observation = 16.67 %; Sample size needed to halve σx̄ = 144. ## Key terms - **Sampling distribution**: The distribution of a statistic, such as the mean, over all possible samples of a given size. - **Central limit theorem**: The result that sample means are approximately normal for large samples, whatever the population shape. - **Finite population correction**: The factor √((N − n)/(N − 1)) that reduces the spread when sampling without replacement. ## FAQ **What is the standard deviation of the sample mean?** The standard deviation of all possible sample means of size n: σx̄ = σ ÷ √n. With σ = 7.1 cm and n = 100 it is 0.71 cm. **Is it the same as the standard error?** Conceptually yes. "Standard deviation of the sample mean" usually uses the known population σ; the "standard error" uses the sample SD s as an estimate. **Why does a larger sample give a smaller value?** Random highs and lows cancel out more in a larger average. Because of the square root, four times the sample size is needed to halve σx̄. **When do I need the finite population correction?** When sampling without replacement and n is more than about 5% of N. Sampling 36 from 500 gives a factor of 0.964, reducing σx̄ from 2.5 to 2.41. **What is the difference between the sample distribution and the sampling distribution?** The sample distribution is the spread of individual values in one sample; the sampling distribution is the spread of a statistic (such as the mean) across many samples. **Does the population have to be normal?** No. For n of about 30 or more the central limit theorem makes the sampling distribution approximately normal; for very skewed populations a larger n is safer. ## Sources - [OpenStax – Introductory Statistics 2e](https://openstax.org/details/books/introductory-statistics-2e) - [Penn State STAT 414 – Introduction to Probability Theory](https://online.stat.psu.edu/stat414/) - [Khan Academy – Statistics and probability](https://www.khanacademy.org/math/statistics-probability)