Loading Calculator...
Please wait a moment
Please wait a moment
Perform statistical t-tests with instant p-value calculations
t-statistic: t = (x̄ - μ) / (s / √n)
Degrees of Freedom: df = n - 1
A t-test is a statistical test used to determine if there's a significant difference between the means of two groups or between a sample mean and a known value. It's widely used in research, quality control, and data analysis to test hypotheses about population means when the population standard deviation is unknown. T-tests are particularly useful when working with small sample sizes (typically n < 30).
The three main types are: (1) One-sample t-test - compares a sample mean to a known or hypothesized population mean, (2) Two-sample (independent) t-test - compares means between two separate groups, and (3) Paired t-test - compares means from the same group at two different times or under two different conditions. Each type has specific assumptions and use cases depending on your research design.
Focus on two key values: the t-statistic and the p-value. The t-statistic measures how many standard errors the sample mean is from the hypothesized mean. The p-value indicates the probability of obtaining results as extreme as observed if the null hypothesis is true. If p < 0.05 (common threshold), reject the null hypothesis and conclude there's a statistically significant difference. Also check confidence intervals and effect size for practical significance.
The p-value represents the probability of obtaining test results at least as extreme as observed, assuming the null hypothesis is true. A p-value of 0.03 means there's a 3% chance the observed difference occurred by random chance alone. Commonly, p < 0.05 is considered statistically significant, meaning you reject the null hypothesis. However, statistical significance doesn't always imply practical importance - consider effect size too.
T-tests assume: (1) data is continuous and approximately normally distributed (especially important for small samples), (2) independence of observations (each data point is independent), (3) for two-sample t-tests, homogeneity of variance (similar variability in both groups), and (4) random sampling from the population. Violations of these assumptions can affect the validity of results, though t-tests are relatively robust to minor departures from normality with larger samples.
Use a t-test when the population standard deviation is unknown and you're estimating it from sample data, especially with small samples (n < 30). Use a z-test when the population standard deviation is known and sample size is large (n ≥ 30). In practice, t-tests are more common because population parameters are rarely known. As sample size increases, the t-distribution approaches the normal distribution, making results similar.
Degrees of freedom (df) represent the number of independent pieces of information available for estimating a parameter. For a one-sample t-test, df = n - 1. For a two-sample t-test, df ≈ n₁ + n₂ - 2 (for equal variances). For a paired t-test, df = n - 1 where n is the number of pairs. Degrees of freedom affect the shape of the t-distribution and critical values used to determine statistical significance.
Yes, you can perform a two-sample t-test with unequal sample sizes using Welch's t-test (also called unequal variances t-test), which adjusts the degrees of freedom calculation. This is actually preferred over the standard t-test when variances are unequal between groups. The test remains valid as long as other assumptions are met, though having similar sample sizes generally provides better statistical power.