Welch's T-Test Calculator

Historical Background

Welch's T-Test, developed by Bernard Lewis Welch, is an adaptation of the Student’s t-test. It is specifically designed for use when two samples have unequal variances and possibly different sample sizes. This method has become a staple in statistical analysis since it provides a more reliable result than the standard t-test when the assumption of equal variances does not hold.

Calculation Formula

The formula for Welch's t-statistic is:

\[ t = \frac{M_1 - M_2}{\sqrt{\frac{S_1^2}{N_1} + \frac{S_2^2}{N_2}}} \]

Where:

• \(M_1, M_2\) = Means of Sample 1 and Sample 2
• \(S_1^2, S_2^2\) = Variances of Sample 1 and Sample 2
• \(N_1, N_2\) = Sample sizes of Sample 1 and Sample 2

The degrees of freedom (df) are calculated as:

\[ df = \frac{\left(\frac{S_1^2}{N_1} + \frac{S_2^2}{N_2}\right)^2}{\frac{\left(\frac{S_1^2}{N_1}\right)^2}{N_1 - 1} + \frac{\left(\frac{S_2^2}{N_2}\right)^2}{N_2 - 1}} \]

Example Calculation

Suppose Sample 1 has a mean of 50, variance of 20, and size of 30. Sample 2 has a mean of 45, variance of 25, and size of 25.

1. Calculate the numerator: \(50 - 45 = 5\).
2. Calculate the variance sum: \(\frac{20}{30} + \frac{25}{25} = 0.6667 + 1 = 1.6667\).
3. Calculate the t-statistic:

\[ t = \frac{5}{\sqrt{1.6667}} \approx 3.87 \]

4. Calculate degrees of freedom:

\[ df = \frac{(1.6667)^2}{\frac{(0.6667)^2}{29} + \frac{(1)^2}{24}} \approx 46.15 \]

Importance and Usage Scenarios

Welch’s T-Test is critical when comparing two independent groups, especially in cases where the assumption of equal variances does not hold. It is widely used in various fields such as psychology, medicine, and economics to test hypotheses about differences between groups.

Common FAQs

1. When should I use Welch's T-Test?

   • Use Welch's T-Test when the two samples have unequal variances and/or different sample sizes.

2. What is the difference between a standard t-test and Welch's t-test?

   • Welch's T-Test does not assume equal variances between the two samples, making it more robust in cases of heteroscedasticity (unequal variances).

3. Can Welch's T-Test be used for small sample sizes?

   • Yes, Welch's T-Test can be used for small sample sizes, but like all statistical tests, its power increases with larger samples.

This calculator simplifies the process of performing Welch's T-Test, providing the t-statistic and degrees of freedom based on the