t-test
The t-test is a statistical hypothesis test used to determine whether there is a significant difference between the means of two groups. It is widely used when the sample size is small and the population standard deviation is unknown.1. What is a t-test?
A t-test compares the means of one or two samples to determine whether the observed difference is statistically significant or likely due to random chance.
It is especially useful when:
- The sample size is small
- The population standard deviation is unknown
- The data is approximately normally distributed
- One-sample t-test: Compares a sample mean to a known value.
- Independent two-sample t-test: Compares means of two independent groups.
- Paired t-test: Compares means from the same group at two different times.
3. Types of t-Tests: Mathematical Explanation
The t-test is used to determine whether there is a statistically significant difference between means. Depending on the structure of the data and the research question, three main types of t-tests are used.
1. One-Sample t-Test
The one-sample t-test is used when we want to compare the mean of a single sample to a known or hypothesized population mean.
Hypotheses
H₀: μ = μ₀ (sample mean equals population mean)
H₁: μ ≠ μ₀ (sample mean is different from population mean)
Test Statistic
The t-statistic is computed as:
$$t= \frac{\bar{x}-\mu_0}{s/\sqrt{n}}$$where:
- x̄ = sample mean
- μ₀ = hypothesized population mean
- s = sample standard deviation
- n = sample size
This statistic measures how many standard errors the sample mean is away from the hypothesized population mean.
2. Independent (Two-Sample) t-Test
The independent t-test compares the means of two independent groups to determine whether their population means differ significantly.
Hypotheses
H₀: μ₁ = μ₂
H₁: μ₁ ≠ μ₂
Test Statistic (Equal Variance Assumption)
When the population variances are assumed equal, a pooled variance is used.
$$ t= \frac{\bar{x}_1-\bar{x}_2}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}} $$where the pooled standard deviation is:
$$ s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}} $$Degrees of Freedom
df = n₁ + n₂ − 2
Unequal Variance Case (Welch’s t-test)
When variances are not equal, Welch’s t-test is preferred:
$$ t=\frac{\bar{x}_1-\bar{x}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}} $$Degrees of freedom are approximated using the Welch–Satterthwaite equation.
3. Paired Sample t-Test
The paired t-test is used when observations come in pairs — for example, before-and-after measurements on the same subjects.
Hypotheses
H₀: μd = 0 (mean difference is zero)
H₁: μd ≠ 0
Test Statistic
First compute the difference for each pair:
di = x1i − x2i
Then compute:
$$ t = \frac{\bar{d}}{s_d/\sqrt{n}} $$where:
- \(\bar{d}\) = mean of the differences
- sd = standard deviation of the differences
- n = number of paired observations
Summary Table
| Test Type | Used When | Key Assumption |
|---|---|---|
| One-sample t-test | Compare sample mean to known value | Normal population |
| Independent t-test | Compare two independent groups | Independent samples, normality |
| Paired t-test | Compare before–after or matched data | Differences are normally distributed |
Key Takeaway
The t-test family provides a powerful framework for comparing means under different experimental conditions. Choosing the correct version depends on the structure of the data and the relationship between observations.
Interpretation
After calculating the t-value and degrees of freedom, the p-value can be obtained from the t-distribution table. If the p-value is less than the significance level, then the null hypothesis is rejected, indicating that there is a significant difference between the means of the two groups. If the p-value is greater than the significance level, then the null hypothesis cannot be rejected, indicating that there is not a significant difference between the means of the two groups.