Yes, you can compare two samples with different sizes. The common t-test is designed to handle unequal sample sizes and doesn’t require special interpretation of results due to this difference. Even comparing a single observation to a population with a known distribution is possible. For instance, an IQ of 130 is demonstrably higher than 97.7% of the population’s IQ scores.
Understanding Statistical Power and Sample Size
While comparing samples with different sizes is statistically valid, the power of the test is affected. Power refers to the probability of finding a statistically significant difference when a true difference exists. For a given total sample size (N), power is maximized when the sample sizes in each group (n) are equal. Unequal group sizes reduce the additional information gained with each new observation, thus decreasing the power.
To illustrate this concept, consider a simple simulation:
This R code simulates 10,000 t-tests comparing two normally distributed samples with different means but equal standard deviations. Three scenarios are tested, each with a total sample size of 100 but varying group sizes:
- Scenario 1: n1 = 50, n2 = 50
- Scenario 2: n1 = 75, n2 = 25
- Scenario 3: n1 = 90, n2 = 10
The simulation reveals:
- Scenario 1 (equal sizes): The test was significant approximately 70% of the time.
- Scenario 2 (unequal sizes): Power dropped to 56%.
- Scenario 3 (highly unequal sizes): Power decreased further to 33%.
This demonstrates that even with the same underlying difference between groups, unequal sample sizes reduce the likelihood of detecting that difference.
Analogy for Understanding Power and Sample Size
Think of calculating the area of a rectangle with a fixed perimeter. The area is maximized when the length and width are equal (a square). As the sides become increasingly unequal (a more elongated rectangle), the area shrinks. Similarly, as sample sizes diverge, the power of the statistical test diminishes.
Conclusion: Balancing Sample Sizes for Optimal Power
While comparing samples with different sizes is possible with a t-test, achieving equal sample sizes maximizes the statistical power of the analysis. This increases the probability of detecting a true difference between the groups. When designing studies, aim for balanced group sizes whenever feasible. However, if unequal sample sizes are unavoidable, the t-test remains a valid method for comparing group means.
