Understanding the T-Test
The t-test is a statistical hypothesis test used to determine if there is a significant difference between the means of groups. It was developed by William Sealy Gosset under the pseudonym "Student," which is why it's often called Student's t-test. The test uses the t-distribution, which accounts for the additional uncertainty when working with small sample sizes.
Types of T-Tests
One-Sample T-Test
Compares a sample mean to a known or hypothesized population mean. For example, testing if the average score of students differs from the national average of 100.
t = (x̄ - μ₀) / (s / √n)Two-Sample (Independent) T-Test
Compares the means of two independent groups. For example, comparing test scores between two different teaching methods. Can use pooled variance (equal variances assumed) or Welch's method (unequal variances).
t = (x̄₁ - x̄₂) / SEPaired T-Test
Compares measurements from the same subjects at two different times or conditions. For example, blood pressure before and after medication. Uses the mean of differences.
t = d̄ / (sᵈ / √n)Key Statistics
- t-Statistic: Measures how many standard errors the sample mean is from the hypothesized mean.
- Degrees of Freedom (df): Affects the shape of the t-distribution. More df = closer to normal distribution.
- p-Value: Probability of observing results as extreme as calculated, if the null hypothesis is true.
- Confidence Interval: Range of plausible values for the true mean or difference.
Effect Size (Cohen's d)
Measures the practical significance of the difference, independent of sample size.
- |d| < 0.2: Negligible effect
- |d| ≈ 0.2: Small effect
- |d| ≈ 0.5: Medium effect
- |d| ≥ 0.8: Large effect
Assumptions of the T-Test
- Normality: Data should be approximately normally distributed. Less critical for large samples (n > 30) due to Central Limit Theorem.
- Independence: Observations should be independent of each other (except in paired t-test where pairs are related).
- Equal Variances (for pooled t-test): Groups should have similar variances. Use Welch's t-test if variances are unequal.
- Continuous Data: The dependent variable should be measured on a continuous scale (interval or ratio).
Practical Tips
- • Always report both p-value AND effect size for complete interpretation
- • A statistically significant result may not be practically significant
- • Larger samples detect smaller differences—consider if the difference matters
- • When in doubt about equal variances, use Welch's t-test (it's more robust)
- • Check your data for outliers before running the test
- • Consider the direction of the hypothesis when choosing one-tailed vs two-tailed
Frequently Asked Questions
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