Sum of variances law

Question

I am currently trying to grasp the concept of Variance Sum Law where

variance(x + y) = variance(x) + variance(y)

I am not even sure if I understand it correctly so I tried to do a little practical example.

Correct me where my reasoning is wrong.

For 2 independent variables & values the calculated variance is as follows:

for measurements of x [9,5,13] variance = 10.67 for measurements of y [1,14, 2] variance = 34.89

Now according the law I would expect for:

[10 (9+1) , 19 (5 + 14) , 15 (13 +2)] variance = 10.67 + 34.89 but after putting it to variance calculator I get the value 13.56.

Seems I am totally not getting what this is about. Can you show me where is my reasoning bad?

I don't understand your notation. Can you format your question a little better? Also, the second number in the sum should be `29`. — StasK, Jul 16 '15 at 16:41
That would be a problem since I am just beginning with statistics and do not now the standard notation. — ps-aux, Jul 16 '15 at 16:44
You are looking at a finite dataset & the correlation between these 2 vectors is not 0. — gung - Reinstate Monica, Jul 16 '15 at 16:49
Probably relevant: http://stats.stackexchange.com/questions/31177/ — Kyle Kanos, Jul 16 '15 at 19:02
Please tell us how you know the two variables are independent. (One necessary condition is that their covariance be zero: what is the covariance?) — whuber, Jul 16 '15 at 19:35

score 1 · Answer 1 · answered Jul 17 '15 at 01:31

As others have noted, $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)$ only holds for $X$ and $Y$ uncorrelated, but you're pairing them in a systematic way which isn't expected to give zero correlation. Try pairing each value from the set $x$ with every value from the set $y$, taking the sum and then find the variance of this distribution.

Sum of variances law

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