Relative frequencies

Question

i am looking for a theorem to state if exists any for the following problem.

1. I have 4 different groups of different sizes with smokers and non smokers and I have calcualted relative frequencies for smokers in each group.

2. I combined four groups into one and calculated relative frequencies.

Now the question: are the average of realtive frequencies of 1 and relative frequenices of 2 the same?

If yes, which theorem explains that?

Thanks!

Answer 1

Let's simplify with two groups:

• group A: 2 people, both smoke, therefore 100% of smokers
• group B: 10 people, 1 smokes, therefore 10% of smokers

Now group A and B together have 3 smokers over 12, hence 25%.

How can you get to that 25% by averaging? Consider the relative weight of the two groups:

$$100\% * \frac{2}{12} + 10\% * \frac{10}{12} = 0,25 = 25\%$$

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