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

In summary, the conversation discusses whether the formula s_{XY}=s_{X}\overline{Y}+s_{Y}\overline{X} holds true for sets X and Y, where s represents the standard deviation and XY is the set containing x_{i}y_{i}. It is determined that the formula may not hold true in general, but there may be specific cases where it does. It is also mentioned that the formula relies on assumptions such as the distributions of X and Y and the size of the sets being equal. A counterexample is provided to show that the formula is not universally true.
  • #1
amcavoy
665
0
For sets X and Y, is it true that

[tex]s_{XY}=s_{X}\overline{Y}+s_{Y}\overline{X},[/tex]​

where [tex]s[/tex] represents the standard deviation and [tex]XY[/tex] is the set containing [tex]x_{i}y_{i}[/tex]?
 
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  • #2
possibly...
 
  • #3
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?
 
  • #4
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.
 
  • #5
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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1. What is the formula for calculating the standard deviation of sets X and Y?

The formula for calculating the standard deviation of sets X and Y is sXY = sX√Y + sY√X, where sX and sY are the standard deviations of sets X and Y, and √X and √Y are the means of sets X and Y, respectively.

2. How is the standard deviation of sets X and Y related to their individual standard deviations?

The standard deviation of sets X and Y, sXY, is calculated by adding the products of the individual standard deviations, sX and sY, and the means of the sets, √Y and √X. This formula takes into account the variation in both sets and their relationship with each other.

3. What does the standard deviation of sets X and Y represent?

The standard deviation of sets X and Y represents the measure of spread or variability of the data points in the sets. It shows how much the data points deviate from their mean values and how they are distributed around the mean.

4. Can the standard deviation of sets X and Y be negative?

No, the standard deviation of a set cannot be negative. It is always a positive value or zero, depending on the data points in the set. A negative value would not make sense as it would imply that the data points are less than the mean, which is not possible.

5. How is the standard deviation of sets X and Y affected by outliers?

The standard deviation of sets X and Y is greatly affected by outliers, which are extreme values that are significantly higher or lower than the rest of the data points. Outliers can increase the standard deviation, making it a less accurate measure of spread. Hence, it is important to identify and address outliers when calculating the standard deviation of a set.

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