The variance of the sum of independent random variables is the sum of the variances of the random variables. This is the additive property of the variance. The standard deviation being the positive square root of the variance does not have that property.

you may want to see

Article TO DETERMINE SKEWNESS, MEAN AND DEVIATION WITH A NEW APPROAC...

Or to put Jochen's answer another way,

Total variance is the sum of individual variances

Vab = Va + Vb

Since V is the square of SD this we can substitute

SDab2 = SDa2 + SDb2

and

SDab = (SDa2 + SDb2)1/2

As Jochen indicates this only applies if the assumption that the variations in a and b are uncorrelated (in both cases). Is there is correlation (covariance), this means some variation in a and b is shared but should only be counted once, so must be corrected for.

I read somewhere that there is a bias in the standard deviation but the variance is unbiased. It has to do with taking the square root of a curved relationship (Jensen's inequality).

You may find this attachment useful for these questions and other similar things. Best wishes, David Booth