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Standard Error Calculator

Calculate the standard error of the mean (SEM) to measure sampling precision

What this tool does

Calculate the standard error of the mean (SEM) to measure sampling precision. You fill in these fields: Data values. It shows standard error of the mean, sample standard deviation, mean, and sample size (n).

Inputs

  • Data values: Longer text, up to 20000 characters. Separate numbers with commas, spaces, semicolons or new lines. At least 2 values.

How to use

  1. Type the data values.
  2. Read the result: standard error of the mean, sample standard deviation, mean, and sample size (n). It appears once every required field has a valid value.

Worked examples

Example 1. Inputs:

  • Data values: "2, 4, 4, 4, 5, 5, 7, 9"

Results:

  • Standard error of the mean: 0.755929 (sample standard deviation divided by the square root of n)
  • Sample standard deviation: 2.13809
  • Mean: 5
  • Sample size (n): 8

How this is worked out: n = 8, sum = 40, mean = 5. Squared deviations 9, 1, 1, 1, 0, 0, 4, 16 add to 32. Sample variance = 32 / (8 - 1) = 4.571429, s = 2.138090. SE = s / sqrt(n) = 2.138090 / 2.828427 = 0.755929 (equivalently sqrt(32/7/8) = sqrt(4/7)).

Example 2. Inputs:

  • Data values: "3 7"

Results:

  • Standard error of the mean: 2 (sample standard deviation divided by the square root of n)
  • Sample standard deviation: 2.828427
  • Mean: 5
  • Sample size (n): 2

How this is worked out: n = 2, mean = 5. Deviations are -2 and 2, squares 4 and 4, sum 8. Sample variance = 8 / (2 - 1) = 8, s = sqrt(8) = 2.828427. SE = sqrt(8) / sqrt(2) = 2.

Example 3. Inputs:

  • Data values: "5 5 5 5"

Results:

  • Standard error of the mean: 0 (sample standard deviation divided by the square root of n)
  • Sample standard deviation: 0
  • Mean: 5
  • Sample size (n): 4

How this is worked out: Identical values: every deviation from the mean 5 is 0, so the sum of squares is 0, s = 0 and SE = 0 / sqrt(4) = 0.

Example 4. Inputs:

  • Data values: "-1.5; 0; 2.5; 3"

Results:

  • Standard error of the mean: 1.06066 (sample standard deviation divided by the square root of n)
  • Sample standard deviation: 2.12132
  • Mean: 1
  • Sample size (n): 4

How this is worked out: Negative and decimal values, semicolons: n = 4, sum = 4, mean = 1. Deviations -2.5, -1, 1.5, 2; squares 6.25, 1, 2.25, 4 add to 13.5. Sample variance = 13.5 / (4 - 1) = 4.5, s = sqrt(4.5) = 2.1213203. SE = s / sqrt(4) = 2.1213203 / 2 = 1.0606602.

Limitations

  • Needs at least 2 values. A single value has no spread to measure.
  • Treats the values as a simple random sample. It does not correct for bias in how the data was collected.
  • A comma always separates numbers, so 1,000 is read as the two numbers 1 and 0. Type 1000 without a comma.
  • Shows the standard error only. It does not build a confidence interval.

FAQs

Q: What is the standard error of the mean?
A: It estimates how far a sample mean is likely to fall from the true mean of the population the sample came from. It is the sample standard deviation divided by the square root of the number of values.
Q: Why does a larger sample give a smaller standard error?
A: The standard error divides by the square root of n, so more values shrink it. Quadrupling the sample size halves the standard error, if the spread of the data stays the same.
Q: Is this the same as the standard deviation?
A: No. The standard deviation describes how spread out the individual values are. The standard error describes how precisely the sample mean is known. The tool shows both.
Q: Which standard deviation does the tool use?
A: The sample standard deviation, which divides the sum of squared deviations by n - 1. It is the usual estimate of spread when the population standard deviation is not known.
Q: What happens if I type something that is not a number?
A: The tool names the entry it could not read and shows no result, rather than guessing. For example, 3abc is refused instead of being read as 3.

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