The formula for the combined standard deviation (SD) of two groups with different means and standard deviations is:

$s_c = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2 + \frac{n_1 n_2}{n_1 + n_2} (\bar{x}_1 - \bar{x}_2)^2}{n_1 + n_2 - 1}}$

Where:

• $s_c$ = Combined standard deviation
• $s_1, s_2$ = Standard deviations of the two groups
• $n_1, n_2$ = Sample sizes of the two groups
• $\bar{x}_1, \bar{x}_2$ = Means of the two groups

This formula accounts for both within-group variation and the difference in means between the groups. If the means are the same, it simplifies to the pooled standard deviation:

$s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}$

Would you like a worked-out example to illustrate this?

Related Questions:

1. How do you calculate the standard deviation for more than two groups?
2. What is the difference between combined SD and pooled SD?
3. How does sample size affect the combined standard deviation?
4. When should you use the combined standard deviation formula?
5. Can you use combined standard deviation for weighted datasets?

Tip: Standard deviation is sensitive to outliers, so always check for extreme values before calculating.