Standard Deviation of Sets X & Y: s_{XY}=s_{X}\overline{Y}+s_{Y}\overline{X}?

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SUMMARY

The discussion centers on the validity of the formula for the standard deviation of sets X and Y, expressed as s_{XY}=s_{X}\overline{Y}+s_{Y}\overline{X}. Participants express skepticism about its general applicability, noting that the formula may only hold under specific conditions. Key points include the necessity for equal set sizes and the implications of the means and standard deviations of the sets involved. A counterexample is provided, demonstrating that the formula fails when standard deviations are zero, leading to a misleading combined standard deviation.

PREREQUISITES
  • Understanding of standard deviation and its calculation
  • Familiarity with statistical notation and terminology
  • Knowledge of sample versus population statistics
  • Basic concepts of set theory and unions
NEXT STEPS
  • Research the properties of standard deviation in relation to combined datasets
  • Study the implications of means and standard deviations in statistical formulas
  • Explore statistical textbooks for proofs related to standard deviation formulas
  • Examine specific cases where the formula may hold true
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Statisticians, data analysts, students studying statistics, and anyone interested in understanding the nuances of standard deviation in combined datasets.

amcavoy
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For sets X and Y, is it true that

s_{XY}=s_{X}\overline{Y}+s_{Y}\overline{X},​

where s represents the standard deviation and XY is the set containing x_{i}y_{i}?
 
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First off, I assume you mean sample deviation? Standard deviation is a constant.

I doubt the result is true in general. However, there may be a specific case where the formula holds.

Do you have any idea what X and Y might be distributed as? Where does the problem arise?
 
What are X and Y, and what does XY really mean? How can x_iy_i make sense, since this will completely depend on how one labels elements of the sets (apparently they're sets) X and Y? It also presupposes that |X|=|Y| too.
 
Hi there. Remember me? :)

I think it means that the standard deviation of the union of X and Y is equal to the standard deviation of x * the mean of Y + the standard deviation of y * the mean of X.

Uggh I don't know how to prove those - have to go to some statistics textbooks...

Anyways, if Y mean = 100 and Y SD = 0, and X mean = 0 and X SD = 0, then the formula would compute a combined SD of 0. But then your combined sample has both elements of 0 and 100, and it must have a standard deviation. So the formula is not universally true.
 
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