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24
3
0.3333
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24
3
0.3333
The T-Test Calculator (one-sample) determines whether a sample mean differs significantly from a hypothesized population mean when the population standard deviation is unknown. The t-test is one of the most widely used statistical tests, developed by William Sealy Gosset under the pseudonym "Student" in 1908. It is essential when working with small samples where the normal distribution assumption of the z-test is too liberal.
Unlike the z-test, the t-test uses the sample standard deviation as an estimate of the population standard deviation and accounts for the additional uncertainty through the t-distribution, which has heavier tails than the normal distribution. This calculator computes the t-statistic, degrees of freedom, standard error, and Cohen's d effect size. The effect size is crucial because statistical significance alone does not indicate practical importance; a large sample can make a trivially small difference statistically significant.
The one-sample t-statistic is calculated as:
$$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$$
Where x̄ is the sample mean, μ₀ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size. The degrees of freedom are df = n - 1.
The standard error $$SE = \frac{s}{\sqrt{n}}$$ estimates the variability of the sample mean. Cohen's d is computed as:
$$d = \frac{|\bar{x} - \mu_0|}{s}$$
Cohen's guidelines classify effects as small (d = 0.2), medium (d = 0.5), or large (d = 0.8). Compare the calculated t-statistic to critical values from the t-distribution with df degrees of freedom to determine significance.
A larger absolute t-statistic indicates stronger evidence against the null hypothesis. Compare your t-statistic to critical values: for df = 24, the two-tailed critical values are approximately ±2.064 at α = 0.05 and ±2.797 at α = 0.01. If |t| exceeds the critical value, reject the null hypothesis. Also consider Cohen's d: values below 0.2 indicate a negligible effect regardless of statistical significance, while values above 0.8 indicate a large, practically meaningful difference.
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Results
T = 3.20 with 35 df is highly significant (p < 0.005). Cohen's d = 0.53 indicates a medium effect size.
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Results
T = 1.26 with 9 df is not significant at α = 0.05 (critical value ≈ 2.262). The 0.02 deviation from target is within acceptable variation.
Use a t-test when the population standard deviation is unknown and you estimate it from your sample data (which is almost always the case in practice). The t-test is especially important for small samples (n < 30) where the t-distribution's heavier tails properly account for the uncertainty in estimating the standard deviation. For very large samples, the t and z distributions converge.
A one-sample t-test (this calculator) compares a sample mean to a known or hypothesized value. A two-sample t-test compares the means of two independent groups to each other. A paired t-test compares means from the same group measured at two different times. Each has a different formula for the standard error.
The t-statistic is influenced by sample size: even a tiny difference can produce a large t-value with a big enough sample. Cohen's d is independent of sample size and measures the magnitude of the difference in standard deviation units. This makes it possible to compare effect sizes across studies with different sample sizes.
The t-test assumes: (1) the data is a random sample from the population, (2) the population is approximately normally distributed (less important for n > 30 due to the Central Limit Theorem), and (3) observations are independent of each other. The test is fairly robust to moderate departures from normality, especially with larger samples.
The p-value requires looking up the t-statistic in a t-distribution table with the appropriate degrees of freedom, or using statistical software. For a rough estimate with large df (>30), the t-distribution approximates the standard normal, so you can use our P-Value Calculator with the t-statistic as the z-score.
Yes. A negative t-statistic means the sample mean is below the hypothesized mean. For a two-tailed test, only the absolute value matters. For a one-tailed test, the sign determines which tail of the distribution you evaluate. The magnitude (absolute value) determines the strength of evidence.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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