# Student's t-Test Calculator Run a one-sample Student's t-test against a hypothesized mean and get the t-statistic, degrees of freedom, and two-tailed p-value instantly. --- - **Canonical URL:** https://dothecalculation.com/calculators/t-test-calculator - **Category:** Math calculators - **Publisher:** Do The Calculation (https://dothecalculation.com) - **Cost:** Free, no account or sign-up required - **Privacy:** Runs entirely in the browser; inputs are never sent to a server - **Methodology:** https://dothecalculation.com/methodology --- ## Student's t-Test Calculator Perform a one-sample Student's t-test against a hypothesized mean to evaluate statistical significance. - One-sample t-test vs. a hypothesized mean - Two-tailed significance (p-value) estimate - Interactive t-distribution plot ## Introduction to Student's t-Tests The Student's t-test is one of the most widely used statistical tools for hypothesis testing. Introduced by William Sealy Gosset in 1908 under the pseudonym "Student," it determines whether a sample mean differs significantly from a reference value. It's ideal for scenarios with small sample sizes (\(N < 30\)) where the population standard deviation is unknown. T-tests are generally categorized into three variants: one-sample, independent two-sample, and paired (dependent). A one-sample t-test compares a single sample mean to a known or hypothesized population mean. An independent two-sample t-test compares the means of two unrelated groups (e.g., treatment vs. placebo). A paired t-test compares means from the same group at different times (e.g., pre-test vs. post-test). This calculator computes the one-sample t-statistic, degrees of freedom (\(df\)), and a two-tailed p-value, letting you check whether your sample mean differs significantly from a hypothesized value. It does not compute the two-sample or paired variants — see the note below. ## Core t-Test Formula **One-sample t-test formula (what this calculator computes):** $$t = \frac{\bar{x} - \mu_0}{s / \sqrt{N}}$$ where \(\bar{x}\) is the sample mean, \(\mu_0\) is the hypothesized mean, \(s\) is the sample standard deviation, and \(N\) is the sample size. Degrees of freedom are \(df = N - 1\). **Independent two-sample t-test (reference formula — not computed by this tool):** $$t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{N_1} + \frac{1}{N_2}}}$$ where \(s_p\) is the pooled standard deviation. **Paired t-test (reference formula — not computed by this tool):** $$t = \frac{\bar{d}}{s_d / \sqrt{N}}$$ where \(\bar{d}\) is the average of the paired differences and \(s_d\) is their standard deviation. ## How to Use This Calculator Enter your sample as a comma-separated list of numbers and set the hypothesized mean \(\mu_0\) you want to test against. The calculator computes the sample mean, standard deviation, standard error, t-statistic, degrees of freedom, and an approximate two-tailed p-value. ## Worked Example: {14, 18, 16, 15, 17, 21, 19} vs. μ₀ = 15 Using the calculator's default dataset (\(N = 7\)): $$\bar{x} = 17.1429, \qquad s = 2.4103, \qquad SE = \frac{s}{\sqrt{7}} = 0.9110$$ $$t = \frac{17.1429 - 15}{0.9110} \approx 2.352, \qquad df = 6$$ For a two-tailed test at \(df = 6\), the critical value at \(\alpha = 0.05\) is about 2.447, so \(t \approx 2.352\) falls just short of significance at the standard 5% threshold — matching what the calculator returns for its default inputs. ## Related Calculators For a goodness-of-fit test on categorical count data, use the [chi-square test calculator](/calculators/chi-square-test-calculator). To estimate a range for the population mean instead of testing a specific hypothesis, see the [confidence interval calculator](/calculators/confidence-interval-calculator). ## Assumptions and Interpretation of t-Tests Before relying on a t-test, several statistical assumptions should hold: the data should be continuous, sampled randomly, and approximately normally distributed. When these conditions are satisfied, a low p-value (typically \(p < 0.05\)) indicates the observed difference from the hypothesized mean is unlikely to have occurred by random chance, justifying rejection of the null hypothesis. ## Frequently asked questions ### What is a t-test? A t-test is an inferential statistic used to determine if there is a significant difference between a sample mean and a reference value, or between the means of two groups. ### What is a one-sample t-test? A test comparing the mean of a single sample group to a hypothesized population mean. This is the test this calculator computes. ### Does this calculator perform two-sample or paired t-tests? No — it computes the one-sample t-test only, testing one dataset against a hypothesized mean you enter. The two-sample and paired formulas are provided above as reference material, but aren't calculated here. ### What is an independent two-sample t-test? A test comparing the means of two independent (unrelated) groups. Not computed by this calculator. ### What is a paired t-test? A test comparing the means of two related groups (e.g., the same individuals tested before and after a treatment). Not computed by this calculator. ### Why is it called Student's t-test? Because its creator, William Gosset, published it under the pen name "Student" while working at the Guinness Brewery. ### What is the null hypothesis of a one-sample t-test? The null hypothesis (H₀) states that the sample mean equals the hypothesized mean (x̄ = μ₀). ### What is a pooled standard deviation? A weighted average of the standard deviations from two sample groups, used in the independent two-sample t-test when group variances are assumed equal. ### What is a one-tailed vs. two-tailed test? A one-tailed test checks for a difference in a specific direction (greater or less). A two-tailed test, which is what this calculator reports, checks for any difference in either direction. ### What does degrees of freedom (df) mean here? For a one-sample t-test, df = N - 1, where N is the sample size. It shapes the t-distribution used to compute the p-value. ### What happens if my data is not normally distributed? If data severely violates normality, non-parametric alternatives like the Wilcoxon signed-rank test should be used instead of a t-test. ### What p-value threshold counts as statistically significant? The conventional threshold is p < 0.05, though the appropriate threshold depends on your field and study design. ## Related concepts - **P-Value** — The probability of observing a result at least as extreme as the sample, assuming the null hypothesis is true. - **Degrees of Freedom** — The N - 1 parameter shaping the t-distribution used in this calculator's significance test. - **Standard Error** — The standard deviation of the sample mean, used to scale the t-statistic. ## Related guides - [Standard Deviation Guide: Measure Data Spread Clearly](https://dothecalculation.com/blog/math/standard-deviation-explained) — Understand variance, population vs sample formulas, and data spread with worked examples tied to the live DTC standard deviation calculator. ## Related calculators - [Chi-Square Test Calculator](https://dothecalculation.com/calculators/chi-square-test-calculator) — Run a Chi-Square goodness-of-fit test on observed vs. expected category counts and get the test statistic, degrees of freedom, and p-value. - [Average Calculator](https://dothecalculation.com/calculators/average-calculator) — Use our free average calculator to find the mean, weighted average, and GPA. Shows step-by-step work for any set of numbers. - [Binomial Distribution Probability Calculator](https://dothecalculation.com/calculators/binomial-distribution-calculator) — Compute binomial probability mass function, cumulative distribution, mean, and variance instantly with this free statistics calculator. - [Percentile & Percentile Rank Calculator](https://dothecalculation.com/calculators/percentile-calculator) — Find the percentile value of a dataset or compute the percentile rank of a specific data point instantly with this free tool. - [Standard Deviation Calculator](https://dothecalculation.com/calculators/standard-deviation-calculator) — Calculate mean, variance, and standard deviation for any dataset instantly with this free step-by-step statistics calculator. - [Combination Calculator](https://dothecalculation.com/calculators/combination-calculator) — Calculate combinations (nCr) instantly with step-by-step work, with or without repetition. Perfect for lottery odds, poker hands, and team selection. --- _This mathematical solver is designed for academic, engineering, and educational analysis. Rounding errors, numerical tolerances, or algorithm constraints might apply near poles, boundary conditions, or complex coordinate spaces. Always verify critical computations independently._ --- _Source: [Do The Calculation](https://dothecalculation.com/calculators/t-test-calculator). Quote freely with attribution and a link to this page._