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One-sample, two-sample and paired t-tests on your raw data, worked line by line.
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The Coddy math course teaches the method itself - you work each step on an interactive board and get told exactly where a move went wrong.
Every t-test has the same shape. There is a difference you care about: a sample mean against a claimed value, two group means against each other, or the average change across pairs. There is a standard error: how much that difference would wobble from one sample to the next, estimated from the spread of the data. The t statistic is the first divided by the second, so t = 3 means the difference is three standard errors wide, and the t curve for the right degrees of freedom turns that into a p-value.
The three tests differ only in what the difference and the standard error are. One-sample: the mean minus the hypothesised value, over s divided by root n. Paired: the same thing applied to the list of differences, one per pair, which is why pairing is so powerful when each pair shares a lot of noise. Two-sample: the gap between the means over a standard error built from both spreads. For two samples the page defaults to Welch's version, which does not assume the two groups have the same variance and is what every modern statistics package defaults to; the pooled Student version is a switch away for when a course requires it.
The page keeps what can be kept exact. A mean of 26 is printed as 26 and a variance of 32/3 as 32/3, not 10.67; the standard error is shown as a radical, since it is a square root; the t statistic is the first decimal, because dividing by a root leaves nothing rational to keep. Underneath, every step is listed so the sample summary table can be checked against your own working.
One sample against a claimed mean; two independent samples against each other; paired when every value in one list has a partner in the other.
Commas, spaces or new lines all work. For a paired test the two lists must be the same length and in the same order.
Two-tailed for "different", one-tailed for a direction fixed in advance. Alpha of 0.05 is the convention.
Check that the means and variances match what you expected. Most wrong t-tests are wrong because a value was typed into the wrong list.
The verdict says whether the difference cleared the bar; the interval says how big it probably is. Report both.
What each one divides by what.
| Test | Difference | Standard error | Degrees of freedom |
|---|---|---|---|
| One sample | x̄ − μ₀ | s / √n | n − 1 |
| Paired | mean of the differences d̄ | s_d / √n | n − 1 |
| Two sample, pooled | x̄₁ − x̄₂ | √(s²_p (1/n₁ + 1/n₂)) | n₁ + n₂ − 2 |
| Two sample, Welch | x̄₁ − x̄₂ | √(s₁²/n₁ + s₂²/n₂) | from the two variances, often not whole |
20, 22, 24, 25, 26, 27, 28, 29, 30, 29 against μ₀ = 24The ten values have mean 26 and sample variance 32/3. The standard error is the root of 32/30, about 1.033, so t is about 1.94 with 9 degrees of freedom and a two-tailed p of about 0.085. Not significant at 5%: a mean of 26 is within what a sample of ten from a population with mean 24 would produce about one time in twelve.
84, 79, 91, 88, 76, 85, 90 against 78, 81, 74, 80, 77, 83, 79The first group averages about 84.7, the second about 78.9. Welch's test gives t of about 2.46 on about 9 degrees of freedom and p around 0.036; the pooled version gives the same t on 12 degrees of freedom and p around 0.030. Significant either way, and the interval for the difference runs from roughly 0.5 to 11.2.
72, 68, 75, 80, 66, 71 before; 75, 70, 79, 84, 69, 74 afterEvery pair went up by 2 to 4 points. The differences average 19/6, about 3.17, with a small spread, so the paired t is large, about 10.3 on 5 degrees of freedom, and p is about 0.0001. A two-sample test on the same numbers would have found far less, because the spread between people swamps the change within each.
