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One Sample T-Test Calculator (With Steps)

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One Sample T-Test Calculator

The one-sample t-test checks whether the mean of a population differs from a hypothesised value μ₀ when the population standard deviation is unknown, which is the usual situation. Paste the raw data, or leave the data box empty and type the sample mean, standard deviation and size, then choose the alternative hypothesis and significance level.

You get the sample statistics, the standard error s/√n, the t statistic with n − 1 degrees of freedom, the exact p-value, the critical t value, a confidence interval for the mean and Cohen’s d. Every step shows your numbers substituted into the formula, and the conclusion is written the way an exam answer or lab report should read.

The test assumes the data are roughly normal or the sample is not tiny; with n of 15 or more it is robust to moderate skew. For paired before-and-after measurements use the paired t-test, and for two independent groups the two-sample t-test.

Separate values with commas, spaces or new lines. Leave blank to use the summary boxes.

Result and step-by-step solution

t(9) = 2.2119
p = 0.0543
Fail to reject H₀ at α = 0.05. The result is not statistically significant: the data do not give enough evidence at the 5% level to conclude that the population mean differs from 12.
  1. State the hypotheses
    H₀: μ = 12   H₁: μ ≠ 12 (two-tailed test)
  2. Sample statistics
    Sample (n = 10): x̄ = Σx ÷ n = 124.5 ÷ 10 = 12.45; s = √[Σ(x − x̄)² ÷ (n − 1)] = √(3.725 ÷ 9) = 0.6433.
  3. Standard error
    SE = s ÷ √n = 0.6433 ÷ √10 = 0.2034
  4. Test statistic
    t = (x̄ − μ₀) ÷ SE = (12.45 − 12) ÷ 0.2034 = 2.2119 with df = n − 1 = 9
  5. p-value
    2 × P(T ≥ |2.2119|) = 0.0543 from the t distribution with 9 degrees of freedom
  6. Decision
    p = 0.0543 ≥ α = 0.05. Critical value approach: critical values ±2.2622 (reject when |t| > 2.2622).
95% confidence interval for μ11.9898 to 12.9102
Effect size (Cohen’s d)0.699

Frequently asked questions

What are the degrees of freedom for a one-sample t-test?
n − 1, where n is the sample size. With 10 observations, df = 9.
How do I interpret the result?
If the p-value is below your significance level, reject H₀ and conclude the population mean differs from μ₀ (in the direction of the alternative). If not, the data are consistent with μ₀.
Can I enter summary statistics instead of raw data?
Yes. Leave the data box empty and fill in the mean, standard deviation and sample size. The calculator uses those directly.
How large does the effect need to be?
Cohen’s d = (x̄ − μ₀)/s. Around 0.2 is small, 0.5 medium and 0.8 large. A significant p-value with a tiny d usually just means the sample was large.

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Last reviewed: September 25, 2026. Calculations run in your browser and were validated against SciPy.

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