One-sample t statistic and interpretation
The t statistic is the mean difference expressed in standard-error units. Its sign indicates the direction of the difference. Degrees of freedom are n − 1; no p value or significance classification is provided.
To calculate the confidence interval supported by its documented input assumptions, use the Confidence Interval Calculator and follow its own input conventions.
The formula
Calculate the standard error as sample deviation divided by the square root of sample size, then divide the sample-minus-hypothesized mean difference by that standard error.
Verified worked example
Sample mean 12, hypothesized mean 10, sample standard deviation 4, and n = 16 give standard error 1 and t = 2, with 15 degrees of freedom.
How to use this calculator
- Enter the sample mean and the population mean proposed under the null hypothesis.
- Enter a positive sample standard deviation and a whole sample size of at least two.
- Read t, degrees of freedom, and standard error; use appropriate separate critical values or p-value analysis if needed.
T Statistic input conventions
The t statistic expresses the mean difference in estimated standard-error units. A positive result places the sample mean above the hypothesized mean and a negative result below it. Enter the sample standard deviation rather than the standard error. Using the latter would divide by the sample-size factor a second time and change the statistic.
Interpreting one-sample t statistic
Inference based on the Student t distribution requires suitable sampling conditions, including independent observations and an appropriate distributional justification. Outliers or a small skewed sample can undermine a simple test. The selected alternative hypothesis also matters when interpreting a critical value or p value. This page reports the statistic and n minus one degrees of freedom; it does not calculate a p value or make a significance decision.
Assumptions & limitations
What this calculation assumes
- The supplied summary describes one independent sample with positive sample standard deviation.
What to keep in mind
- One-sample t statistic only; no p value, critical values, or paired/two-sample modes.
Common questions
Does a t statistic of 2 mean significance?
That depends on degrees of freedom, the significance level, and whether the alternative is one-sided or two-sided. The statistic alone does not supply that decision.
Can I compare two independent samples here?
No. Two-sample and paired-sample tests use different inputs or difference summaries. This tool is specifically the one-sample summary-statistic calculation.
Can the t statistic be negative?
Yes. It is negative when the sample mean is below the hypothesized mean. The sign alone does not determine statistical significance.
Sources & further reading
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