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?



1 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:

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

More info here.

  • $\begingroup$ Thanks @Ale, so weighted average would always be the same as the groups combined? $\endgroup$
    – sveer
    Commented Dec 15, 2020 at 13:12
  • 1
    $\begingroup$ Yes, you are weighting the result of each group by the share of each group in the total. $\endgroup$
    – Ale
    Commented Dec 15, 2020 at 13:19

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