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Sample-Mean Expectation & Standard Error

Find the expected sample mean and its standard error for independent observations with a common finite mean and variance.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

For a population mean of 50, standard deviation 10 and 100 independent observations, the expected sample mean is 50 and its standard error is 1.

E(sample mean) = population mean; SE(sample mean) = population SD / √n

Worked example

Enter these known values and leave the other values blank.

Population mean
50
Population standard deviation
10
Independent observation count
100
Expected sample mean
50
Standard error of the sample mean
1

Assumptions and limitations

  • Observations are independent and identically distributed with finite variance. Sample size is a positive whole number; standard deviations are nonnegative.
  • The expected sample mean is not the mean of a particular observed dataset. Standard error describes variability across repeated sample means.
  • These expectation and variance identities hold for positive sample sizes. The central limit theorem concerns a limiting distribution; thirty observations is not a universal guarantee of a good normal approximation.
  • Finite-population corrections, dependence, weights and an estimated population standard deviation are outside this model.

Common questions

Why does the calculator accept fewer than thirty observations?

The expectation and standard-error identities still apply under the stated sampling assumptions. Approximating the sampling distribution by a normal curve is a separate question.

Is standard error the same as the spread of individual observations?

No. Here individual observations have population SD σ, while their mean has standard error σ/√n.

References

Bookify allows every positive whole sample size for the expectation and standard-error identities; a normal approximation is not guaranteed by a sample-size cutoff.

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