You will not find activity variance in the PMBOK. Despite that omission, activity variance has a nice easy formula, it's just SD2. This of course uses the PM forula for SD which is P-O/6. The only comlexity arises when adding the SD of a set of activities. When adding up the activity variance of a series of tasks you don't just sum the figures.
AV is the square root of the sum of the total activity variance. The standard deviation for the project as a whole is the sum of all the actvity variances.
Take for example:
Task 1 If SD = 33, then 332 = 1,089
Task 2 If SD = 100, then = 1002 = 10,000
Task 3 If SD = 133, then = 1332 = 17,689
Task 4 If SD = 166, then = 1662 = 2,7556
If you sum those figures 1089 + 10,000 + 17,689 + 2,7556 = 56,334.
AV = √ 56,334 or 237.34
The AV is not the mean. The mean would be 56,334 ÷ 4 = 14,083 The AV is also not the mean of the sum of the SD. That would be 33 + 100 + 133 + 166 = 432 ÷ 4 = 108
But this is not exactly the way were supposed to calculate the SD of a string. The method used by most PMs is not quite what is reccomended in statistics. The "correct" method is to calculate the standard deviation of the figures then calculate the difference of each data point from the mean. Then square the result of each. Then compute the average of these values, and take the square root.
So given the set above:
Task 1 If SD = 33, then 33 - 100 = -662=4,356
Task 2 If SD = 100, then 100 - 100 = 02=0
Task 3 If SD = 133, then 133 - 100 = 332=1,089
Task 4 If SD = 166, then 166 - 100 = 662=4,356
So then 4,356 + 0 + 1,089 + 4,356 = 9,801, and the AV = √9801 or 99.
You will note that this is not the same result as the previous method.
The obvious difference here is that the real AV is about half of the figure produced by the PM formula, it is a number very similar to the mean. There is a reason for that difference.