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You have just the raw data? use this calculator instead: One-Sample T-Test for raw data calculator
The one-sample t-test is used to determine whether the mean of a sample differs significantly from a known or hypothesized population mean. This test is particularly useful when the population standard deviation is unknown and the sample size is small.
t = (x̄ - μ₀) / (s / √n)
Where:
x̄ = sample mean μ₀ = hypothesized population mean s = sample standard deviation n = sample size
If the calculated t-value exceeds the critical value from the t-distribution (based on degrees of freedom and significance level), we reject the null hypothesis, indicating a significant difference.
To calculate a one-sample t-test, use the formula: t = (x̄ - μ₀) / (s / √n), where x̄ is your sample mean, μ₀ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size. Our calculator automates this process—simply enter your sample statistics and hypothesized mean, and it instantly computes the t-statistic, p-value, and provides a visual distribution plot.
A one-sample t-test is used to determine whether your sample mean significantly differs from a known or hypothesized population mean. It's commonly applied when testing if a new treatment produces results different from a standard value, or if a manufacturing process meets specifications. This calculator helps you make this determination by comparing your data against any hypothesized mean you specify.
Use a one-sample t-test (like this calculator provides) when your population standard deviation is unknown and you're working with a sample standard deviation instead. Use a z-test only when you know the true population standard deviation, which is rare in practice. The t-test is more appropriate for most real-world scenarios, especially with smaller sample sizes.
The p-value represents the probability of obtaining your results (or more extreme) if the null hypothesis is true. If the p-value is less than your significance level (typically 0.05), you reject the null hypothesis, indicating a significant difference. Our calculator automatically provides this interpretation, clearly stating whether to reject or fail to reject the null hypothesis, along with a confidence interval for the population mean.
The one-sample t-test formula is: t = (x̄ - μ₀) / (s / √n). Here, the numerator (x̄ - μ₀) measures how far your sample mean deviates from the hypothesized mean, while the denominator (s / √n), called the standard error, accounts for sample variability and size. The calculator applies this formula and compares the resulting t-statistic against the t-distribution with n-1 degrees of freedom.
Use a two-tailed test (μ ≠ μ₀) when testing if your mean differs in either direction from the hypothesized value. Use a one-tailed test when you have a specific directional hypothesis: left-tailed (μ < μ₀) if testing whether your mean is less than the hypothesized value, or right-tailed (μ > μ₀) if testing whether it's greater. Our calculator offers all three options and adjusts the critical values and p-value calculation accordingly.
Technically, you need at least 2 observations (n ≥ 2) to calculate a sample standard deviation and perform the test. However, for reliable results, aim for n ≥ 30 for non-normal data, or n ≥ 15-20 if your data is approximately normal. Smaller samples (n < 15) require data to be very close to normally distributed. Our calculator accepts any sample size of 2 or greater and automatically adjusts the degrees of freedom.
The one-sample t-test assumes: (1) data are randomly sampled and independent, (2) the population is approximately normally distributed (or n ≥ 30 for Central Limit Theorem), and (3) the population standard deviation is unknown. While this calculator performs the mathematical computations for any input, you should verify these assumptions hold for your data before interpreting the results as valid.
For a one-sample t-test, degrees of freedom (df) equals n - 1, where n is your sample size. For example, if you have 25 observations, df = 24. Degrees of freedom represent the number of independent pieces of information available to estimate variability. Our calculator automatically computes this value and uses it to determine the appropriate critical values from the t-distribution, displayed in both the results and the distribution plot.
Statistical significance means your sample provides strong enough evidence to conclude the population mean differs from the hypothesized value, rather than the difference being due to random chance. When p-value < α (commonly 0.05), the result is statistically significant. This calculator clearly indicates significance by comparing your p-value to your chosen alpha level and providing a clear "reject" or "fail to reject" decision with color-coded conclusions.