One Sample T Test Calculator

Select null and alternative hypotheses:

Population Mean (μ₀)

The claimed or hypothesized population mean.

Sample Mean (X̄)

The average value observed in your sample.

Sample Standard Deviation (s)

How spread out your sample's values are.

Sample Size (n)

Total number of people or items in your sample.

Significance Level

Usually 0.05 — your risk tolerance for a false positive.

About This Test

What Is the One-Sample T-Test?

The one-sample t-test checks whether your sample's mean is significantly different from a claimed or hypothesized population mean. It uses the t-distribution instead of the normal distribution because the population's true standard deviation is unknown — you're estimating it from your own sample, which adds extra uncertainty the t-distribution accounts for.

When to Use It

Testing whether a product's actual average matches its labeled specification
Testing whether a sample's average differs from a historical or claimed benchmark
Any single sample compared against one specific target value, with unknown population standard deviation

The Formula

\[ t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}} \]

Worked Example

A cereal box is labeled to contain 500g. A sample of 20 boxes has a mean weight of 495g with a standard deviation of 8g. At α = 0.05, is there evidence the true average differs from 500g?

\( \mu_0 = 500, \ \bar{X} = 495, \ s = 8, \ n = 20 \)
Degrees of freedom: \( df = n - 1 = 19 \)
\( t \approx -2.795 \), two-tailed critical value ≈ ±2.093
p-value ≈ 0.0115
Since 0.0115 < 0.05, we reject H₀ — the boxes' true average weight is likely different from 500g.

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One Sample T Test Calculator

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