There are a load of discussions online about adding means and recalculating the standard deviation, but on none have I found answer to this question.
I have two sub groups with mean $x_1$ and $x_2$, and $s_1$ and $s_2$. I want to add the two means to create $x_1 + x_2 = x_3$. How do I calculate $s_3$. The two sub groups are additive.
The exact data I have is on herbicide and fungicide use on crops on four different farms, where herbicide + fungicide = pesticide.
Herbicide (kg): $2, 3, 1, 2$
Fungicide (kg): $4, 7, 3, 1$
Pesticide (kg): $6, 10, 4, 3$
Herbicide $x_1 = 2.0$; $s_1 = 0.8$
Fungicide $x_2 = 3.8$; $s_2 = 2.5$
Pesticide $x_3 = 5.8$; $s_3 = 3.1$
Unfortunately I don't have any of the underlying data. I have only been provided with herbicide and fungicide means and standard deviations, $x_1$, $x_2$, $s_1$ and $s_2$. Is calculating this possible?
Thank you!
Best Answer
The mean $E(X+Y)$ is equal to the sum of the means $E(X)$ and $E(Y)$, i.e., in your case $2+3.8=5.8$.
The standard deviation is the square root of the variance $Var(X+Y) = Var(X)+Var(Y)+2Cov(X,Y)$.
If you assume that the use of herbicide and fungicide are independent - a bold assertion, although I don't know much about agriculture - then this simplifies to $Var(X+Y)=Var(X)+Var(Y)$ and allows you to calculate the standard deviation by observing that
$Var(X)=0.82^2=0.6724$
$Var(Y)=2.5^2=6.25$
Which leads us to find that $\sigma_3=\sqrt{0.6724+6.25} \approx 2.631$