How to find a combined standard deviation of two groups of figures?

So I’ve got two groups of numbers, which I want to probabilistically add together. The question is how the variation found within each set will effect the variation in the combined sets.

Group 1 = 1,2,3,4
Average = 2.5
St. Dev. = 1.1180

And a 2nd smaller group made up of two numbers.

Group 2 = 7,8
Average = 7.5
St. Dev. = 0.5

This gives me a 3rd group of combined numbers.

Group 3 = (1+7), (2+7), (3+7), (4+7), (1+8), (2+8), (3+8), (4+8)
Group 3 = 8, 9, 10, 11, 9, 10, 11, 12
Average = (2.5+7.5) = 10
St. Dev. = 1.2247

So the question is how do I arrive at the standard deviation of the 3rd combined group using only the standard deviations of the simpler groups?

It turns out to be a very simple calculation, square the standard deviations, sum them together and then take the square root.

= SQRT(1.1180^2 + 0.5^2)
= SQRT(1.25 + .25)
= SQRT(1.5)
= 1.2247

So as long as my sets are independent then it is quite easy to find the combined variation. However as soon as their is any real link between the numbers, for example low numbers in the first set will lead to higher numbers in the 2nd set then the maths goes out the window.

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