Why do I get two different mean values with two different methods for the same sample?

Question

I have this dataframe where I created the 3rd column using the first two columns. Both ${Y}$ and ${X}$ are independent random variables.

$\bar{Y}$	$\bar{X}$	$\bar{Z} = 100\frac{\bar{X}-\bar{Y}}{\bar{Y}}$
13435	13502	0.4987
21847	22354	2.3207
15584	18014	15.5929
17121	16738	-2.2370
18758	19664	4.8299
18994	22234	17.0580
22864	23555	3.0222
30365	31452	3.5798
29850	34808	16.6097
38674	38791	0.3025
47803	39717	-16.9153
88777	91756	3.3556

When I manually calculate the expectation of $\bar{Z}$ using the data in the 3rd column (in the above table) and pass it statistics.mean() function in Python, I get $E[Z] = 0.04.$

On the other hand, I know that I can calculate $E[Z]$ also as follows:

$E[Z] = E[100\frac{X-Y}{Y}] = 100E[\frac{X}{Y}-1] =100E[\frac{X}{Y}]-100 =100E[X]E[\frac{1}{Y}] -100 $ (as according to this link https://en.wikipedia.org/wiki/Ratio_distribution I can write $E[\frac{X}{Y}] =E[X]E[\frac{1}{Y}]$ )

but then I get $E[Z] = 33.75$ (since the mean of $\bar{X}$ and $\frac{1}{\bar{Y}}$ are $E[\bar{X}] =31048.75 $ and $E[\frac{1}{\bar{Y}}] = 0.000043$, respectively.)

Apparently I get two very different mean values for $\bar{Z}$. When I do the same process for standart deviation, I again get very different values. So which approach should I use? Manually calculating using the 3rd row (by using Python's statistics library mean() and stdev() functions) or using the derivation formula?

In this study http://www.statistics.du.se/essays/D09_Zhang%20Ling%20&%20Han%20Kun.pdf, as far as I understood, they straightaway calculated mean and standard deviation by using the data in Percentage and Absolute columns.

I am really looking forward to some informative answers!

Thank you!

Answer 1

There are a couple of issues that can be addressed here.

First,

In general, the expected value of the product of two random variables does not need to be equal to the product of their expectation. This means that

$$ E(XY) \neq E(X)E(Y) $$

The equality only holds when the random variables are independent. But... This links directly to the second issue.

Second

Do not mix random variables with samples. The above property holds for independent random variables, but it is very much likely that the samples you display here are actually correlated, so you can't use above property on your samples.

Conclusion

For this reason, the correct result would be to compute the third column and then obtain the mean of that column. Or, if you want to use an alternative derivation, you could go as far as this:

$$E(z) = E(100\frac{x-y}{y}) = 100 E(\frac{x}{y}-\frac{y}{y})=100E(\frac{x}{y})-100$$

But $E(\dfrac{x}{y})\neq E(x)E(\dfrac{1}{y})$