The standard deviation of a linear tranformation $Y=a+bX$ of a random variable $X$ is $\sigma_X= |b| \cdot \sigma_X$ so, for example, if $Y=4X$, then
$$
\sigma_{4X} = 4\cdot \sigma_X \tag{1}
$$
However, the standard deviation of a sum of random variables $X_1$ and $X_2$ is
$\sigma_{X_1+X_2} = \sqrt{\sigma_{X_1}^2+\sigma_{X_2}^2}$. It seems to me that if you take a sum of four random variables $X$, which are all the same, this formula would lead to
\begin{align*}
\sigma_{X+X+X+X} &= \sqrt{\sigma_{X}^2+\sigma_{X}^2+\sigma_{X}^2+\sigma_{X}^2} \\
&=\sqrt{4 \cdot \sigma_{X}^2} \\
&=2 \cdot \sigma_{X} \tag{2}
\end{align*}
but summing four random variables $X+X+X+X$ is simply $4X$, and we showed in equation (1) that the standard deviation of $Y=4X$ is actually $4 \sigma_X$, not $2 \
\sigma_X$. Is this a contradiction? Where did our calculations go wrong?
Best Answer
You have forgotten the covariances
$\sigma_{X_1+X_2+X_3+X_4}$
$= \sqrt{\sigma_{X_1}^2+\sigma_{X_2}^2+\sigma_{X_3}^2+\sigma_{X_4}^2+2cov(X_1,X_2)}$
$\overline{+2cov(X_1,X_3)+2cov(X_1,X_4)+2cov(X_2,X_3)+2cov(X_2,X_4)+2cov(X_3,X_4)}$
Let $X=X_1=X_2=X_3=X_4$, then we get
$\sigma_{X+X+X+X}=\sigma_{4X}$
$= \sqrt{\sigma_{X}^2+\sigma_{X}^2+\sigma_{X}^2+\sigma_{X}^2}$ $\overline{+2cov(X,X)+2cov(X,X)+2cov(X,X)+2cov(X,X)+2cov(X,X)+2cov(X,X)}$
$= \sqrt{\sigma_{X}^2+\sigma_{X}^2+\sigma_{X}^2+\sigma_{X}^2+2\sigma_{X}^2+2\sigma_{X}^2+2\sigma_{X}^2+2\sigma_{X}^2+2\sigma_{X}^2+2\sigma_{X}^2}=\sqrt{16\sigma^2_X}=4\sigma_X$