Free · Hypothesis-test statistics

One-Sample t Statistic from Summary Data

Calculate a one-sample t statistic from a sample mean, hypothesized mean, sample standard deviation and sample size.

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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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Formula and method

With sample mean 12, hypothesized mean 10, sample SD 4 and n = 16, the t statistic is 2. A one-sample t reference distribution would have 15 degrees of freedom.

t = (sample mean − hypothesized mean) ÷ (sample SD ÷ √n)

Worked example

Enter these known values and leave the other values blank.

Sample mean
12
Hypothesized population mean
10
Sample standard deviation
4
Sample size
16
One-sample t statistic
2

Assumptions and limitations

  • Sample size is a whole number of at least two and sample SD is positive. All measurement values use the same units.
  • The usual one-sample t inference assumes independent observations and an appropriate sampling distribution; for small samples, population normality matters.
  • The reference degrees of freedom are n − 1. This tool does not compute a p-value, confidence interval or significance decision.
  • A two-sample or paired analysis requires the corresponding model. For a paired one-sample analysis, use the mean and SD of paired differences.

Common questions

Does t = 2 mean the result is significant?

Not by itself. You also need degrees of freedom, one- or two-sided testing and the chosen significance level.

Can the statistic be negative?

Yes. It is negative when the sample mean is below the hypothesized mean.

References

A one-sample t statistic based on sample standard deviation requires an integer sample size of at least two.

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