t-test Calculator

A t-test evaluates whether two sets of data are significantly different from one another. The fundamental mechanics rely on calculating a t-score, which measures the ratio of the departure of the estimated value from its hypothesized value to its standard error. For a one-sample t-test, the t-score is computed using the formula: t = (x̄ - μ) / (s / √n), where x̄ is the sample mean, μ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size. For independent two-sample tests assuming equal variances, the pooled standard deviation is integrated into the denominator to weigh sample sizes accurately. Degrees of freedom (df) are calculated simultaneously to reference the correct t-distribution curve, generating precise p-values and critical regions based on your chosen significance level (α).

Imagine you are testing a new study method for college students. You take a random sample of 16 students (n = 16) who use the method and find a sample mean score (x̄) of 85, with a sample standard deviation (s) of 6. You want to test if this is significantly higher than the historical population average (μ) of 80, using a significance level of 0.05. First, calculate the standard error: 6 / √16 = 6 / 4 = 1.5. Next, find the t-score: t = (85 - 80) / 1.5 = 5 / 1.5 = 3.33. With 15 degrees of freedom (16 - 1), a t-score of 3.33 yields a one-tailed p-value well below 0.05, allowing you to reject the null hypothesis and conclude that the study method produces a statistically significant improvement.

Before running your analysis, always verify that your data meets the core assumptions of a t-test: continuous measurement, random sampling, independence of observations, and approximate normal distribution for smaller sample sizes. Pay close attention to whether your test requires a one-tailed or two-tailed hypothesis; selecting the wrong tail direction can misinterpret your significance thresholds. Finally, when comparing two independent samples, check if your variances are roughly equal to decide whether to use standard pooled variance or Satterthwaite's approximation for unequal variances.