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t-Statistic — Definition, Formula & Examples

The t-statistic (also called a t score) is a number that tells you how many standard errors a sample mean lies above or below the mean claimed in the null hypothesis. You use it instead of a z-score when the population standard deviation is unknown and has to be estimated from the sample.

For a simple random sample of size nn with mean xˉ\bar{x} and sample standard deviation ss, the one-sample t-statistic is t=xˉ−μ0s/nt = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}. If H0:μ=μ0H_0: \mu = \mu_0 is true and the population is approximately normal (or nn is large), this statistic follows a t-distribution with n−1n - 1 degrees of freedom. That distribution has heavier tails than the standard normal and approaches it as nn grows.

Key Formula

t=xˉ−μ0s/n,df=n−1t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}, \qquad df = n - 1
Where:
  • xˉ\bar{x} = sample mean
  • μ0\mu_0 = population mean stated in the null hypothesis
  • ss = sample standard deviation
  • nn = sample size
  • s/ns/\sqrt{n} = standard error of the sample mean
  • dfdf = degrees of freedom used to read the t-table

Worked Example

Problem: A company claims its batteries last 50 hours on average. You test a random sample of 16 batteries and find a mean life of 55 hours with a sample standard deviation of 8 hours. Compute the t-statistic for H0:μ=50H_0: \mu = 50 versus Ha:μ≠50H_a: \mu \neq 50. Then compare it with the two-sided critical value at α=0.05\alpha = 0.05.
Step 1: Find the standard error of the mean.
SE=sn=816=84=2SE = \frac{s}{\sqrt{n}} = \frac{8}{\sqrt{16}} = \frac{8}{4} = 2
Step 2: Subtract the hypothesized mean and divide by the standard error.
t=55−502=52=2.5t = \frac{55 - 50}{2} = \frac{5}{2} = 2.5
Step 3: Find the degrees of freedom. For a two-sided test at α=0.05\alpha = 0.05 with 15 degrees of freedom, the t-table gives t∗≈2.131t^* \approx 2.131. Your statistic is larger.
df=16−1=15,2.5>2.131df = 16 - 1 = 15, \qquad 2.5 > 2.131
Answer: t=2.5t = 2.5 with df=15df = 15. Because 2.5>2.1312.5 > 2.131, you reject H0H_0 at the 0.05 level. The sample gives evidence that the mean battery life differs from 50 hours.

Why It Matters

Every inference procedure for means in AP Statistics depends on the t-statistic. That includes one-sample t-tests, paired t-tests, two-sample t-tests, and the t-intervals built from them. Clinical researchers use the same calculation to decide whether a drug changes average blood pressure more than chance would explain.

Common Mistakes

Mistake: Dividing by the sample standard deviation ss instead of the standard error s/ns/\sqrt{n}.
Correction: The t-statistic measures distance in standard errors of the mean, not in individual-data standard deviations. In the example, dividing by 8 gives 0.625 instead of 2.5. That error would lead you to the wrong conclusion.

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