dopecalc

Standard Error Calculator

Calculate the standard error of the mean (SE) from a sample standard deviation and sample size. Determine the margin of error at the 95% confidence level and the relative standard error.

About this Calculator

Calculate the standard error of the mean (SE) from a sample standard deviation and sample size. Determine the margin of error at the 95% confidence level and the relative standard error.

Formula & Calculations

Formula

SE = σ / √n; Margin of Error (95%) = 1.96 × SE; Relative SE = SE / σ
Where:
  • σ=Sample standard deviation
  • n=Sample size (number of observations)
  • SE=Standard error of the mean (precision of the sample mean estimate)
  • MOE=Margin of error at the 95% confidence level
  • RSE=Relative standard error (SE standardized by the sample standard deviation)

Assumptions

  • Samples are independent and randomly selected from the population.
  • The standard error decreases as sample size increases (inverse square root relationship).
  • The 95% margin of error uses z = 1.96, appropriate for large samples or known population parameters.
  • For small samples (n < 30), consider using a t-distribution for more accurate MOE.

Calculation Examples

Example 1

Inputs:Standard Deviation = 15, Sample Size = 25
Result:SE = 3.0, MOE (95%) = 5.88, Relative SE = 0.20

SE = 15 / √25 = 15 / 5 = 3.0. MOE = 1.96 × 3.0 = 5.88. RSE = 3.0 / 15 = 0.20.

Example 2

Inputs:Standard Deviation = 8, Sample Size = 100
Result:SE = 0.8, MOE (95%) = 1.568, Relative SE = 0.10

Increasing the sample size from 25 to 100 (4x) reduces the SE by a factor of 2.

Frequently Asked Questions

What is the difference between standard deviation and standard error?

Standard deviation (σ) measures the spread or variability within a single sample. Standard error (SE = σ/√n) measures the precision of the sample mean as an estimate of the population mean. As sample size increases, SE decreases, meaning your estimate becomes more precise.

How large should my sample size be to get a small standard error?

Since SE decreases with the square root of n, quadrupling your sample size halves the standard error. For precise estimates, larger samples are always better, but diminishing returns set in — going from n=100 to n=400 may only halve your SE.