Enter your sample mean, population mean, sample standard deviation, and sample size to compute the one-sample t-statistic, degrees of freedom, and standard error.
Estimates only.
The one-sample t-test determines whether a sample mean is significantly different from a known or hypothesized population mean. It is appropriate when the population standard deviation is unknown and you are estimating it from the sample.
For the defaults (x-bar = 52, mu = 50, s = 5, n = 30): SE = 5 / sqrt(30) = 0.9129; t = (52 - 50) / 0.9129 = 2.191; df = 29. With 29 degrees of freedom, a t of 2.191 exceeds the critical value of 2.045 at alpha = 0.05 (two-tailed), so the result is statistically significant.
A one-sample t-test compares the mean of a single sample to a hypothesized or known population mean. It answers the question: is my sample's average significantly different from what I expect? For example, if a factory claims widgets weigh 50 g on average, you can test a sample of 30 widgets and use a one-sample t-test to decide whether the observed mean differs significantly from 50 g.
A t-statistic is compared to a critical value from the t-distribution at your chosen significance level (typically 0.05) and degrees of freedom. If the absolute value of t exceeds the critical value, the result is significant. For large samples (df > 30), the critical value at alpha = 0.05 two-tailed is about 2.042. The larger the absolute t-statistic, the stronger the evidence against the null hypothesis.
The t-statistic measures how many standard errors the sample mean is from the hypothesized population mean. A t-statistic of 2.191 means the sample mean is 2.191 standard errors above the population mean. Large absolute values suggest the difference is unlikely due to chance.
Degrees of freedom equal n minus 1 for a one-sample t-test. With fewer degrees of freedom (smaller samples), the t-distribution has heavier tails, so a larger t-statistic is required to reach significance. As the sample size grows, the t-distribution approaches the standard normal distribution and critical values converge toward 1.96 (for alpha = 0.05 two-tailed).