## Testing differences between two samples with different variance

If the two variance are significantly different, we test them with this procedure.

### Testing the difference between two means

The means of two samples can be tested for difference by:

$$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}}$$

You will need to calculate the degrees of freedom for the critical value with the following formula:

$$v = \frac{\left(\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2} \right)^2}{\frac{\left( \frac{s^2_1}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s^2_2}{n_2} \right)^2}{n_2 - 1}}$$

If this formula produces a decimal value truncate the value to an integer.

Also See:

Chapter 9 - Significance of a Difference between Two Means pages 108-124 in:

Phillips, J. L. 2000. How to think about statistics. W. H. Freeman and Co. New York. 202 pp. ISBN 0-7167-3654-3

Chapter 9 - Two-Sample Hypotheses pages 126-130 in:

Zar, J. H. 2007. Biostatistical Analysis. Prentice-Hall, Inc. Englewood Cliffs, New Jersey. 718 pp.