1.25
24
1.6
0.25
1.25
24
1.6
0.25
The One-Sample T-Test Calculator tests whether a sample mean significantly differs from a hypothesized population mean when the population standard deviation is unknown and must be estimated from the sample. This is the most commonly used version of the t-test and is appropriate for small to moderate sample sizes.
Developed by William Sealy Gosset (publishing under the pseudonym 'Student') in 1908, the t-test accounts for the additional uncertainty introduced by estimating the standard deviation from the sample itself. This extra uncertainty is reflected in the heavier tails of the t-distribution compared to the standard normal, resulting in larger critical values especially for small samples.
Applications range from clinical research (testing if a treatment produces a different outcome than a known baseline), to quality control (checking if a process mean has drifted), to psychology (comparing a group's performance to a population norm). This calculator provides the t-statistic, degrees of freedom, standard error, and Cohen's d effect size.
The one-sample t-test evaluates:
Test Statistic:
$$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$$
where $s$ is the sample standard deviation.
Degrees of Freedom:
$$df = n - 1$$
The degrees of freedom determine the shape of the t-distribution. With more degrees of freedom, the t-distribution becomes closer to the standard normal.
Standard Error:
$$SE = \frac{s}{\sqrt{n}}$$
Cohen's d (Effect Size):
$$d = \frac{|\bar{x} - \mu_0|}{s}$$
Cohen's d measures the standardized difference between the sample mean and the hypothesized mean. Guidelines: d = 0.2 (small), d = 0.5 (medium), d = 0.8 (large effect).
The T-Statistic measures how many standard errors the sample mean is from the hypothesized mean. Compare |t| to the critical t-value from a t-table with the given degrees of freedom and your chosen α level.
Cohen's d provides a measure of practical significance that is independent of sample size. Even a statistically significant result (small p-value) may have a negligible effect size, indicating little practical importance.
Inputs
Results
A class of 20 students with mean score 78 vs. national average 75. t=1.34 with 19 df. Cohen's d=0.3 indicates a small-to-medium effect.
Inputs
Results
A drug trial with 16 patients showing mean BP of 120 vs. baseline 130. t=-2.67, d=0.67. This is a medium-to-large effect suggesting clinically meaningful reduction.
Both test a sample mean against a hypothesized value. The z-test uses the known population standard deviation (σ), while the t-test uses the sample standard deviation (s). The t-test produces wider confidence intervals and requires more extreme values for significance, especially with small samples.
Key assumptions: (1) observations are independent, (2) data are approximately normally distributed (especially important for small n), and (3) the variable is measured on an interval or ratio scale. The t-test is moderately robust to violations of normality for n > 30.
Degrees of freedom (df = n - 1) represent the number of independent pieces of information available for estimating variability. One degree is 'used up' by estimating the mean. Higher df means the t-distribution is closer to the normal, giving smaller critical values.
Cohen's d is a standardized effect size measuring the magnitude of the difference in standard deviation units. Unlike the p-value, Cohen's d is not affected by sample size. A significant p-value with a tiny d suggests the difference, while real, may not be practically important.
For moderate to large samples (n ≥ 30), the t-test is robust to non-normality due to the Central Limit Theorem. For small samples from non-normal populations, consider the Wilcoxon signed-rank test as a non-parametric alternative.
Computing an exact t-distribution p-value requires the incomplete beta function, which is complex for AST-based computation. Use the t-statistic and df to look up the p-value in a t-table or use the P-Value Calculator on this site for an approximation.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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