Is it mathematically possible to calculate the total standard deviation of a daily list of numbers from their daily rolling standard deviations

Question

Say I have the following list of numbers, each corresponding to a single day:

list = [1,   6,    7,    12,   5]

I can calculate the daily, rolling standard deviations between each as:

stds = [NaN, 3.54, 0.71, 3.54, 4.95]

Is it now possible to calculate the standard deviation across all values of list, using only the values of stds?

Answer 1

The rolling pairwise standard deviations don't give you enough information to get the total standard deviation. The two lists $$[0,1,0,1]\quad \&\quad [0,1,2,3]$$ both have the pairwise standard deviations $$\Big[\frac 1{\sqrt 2},\,\frac 1{\sqrt 2},\,\frac 1{\sqrt 2}\Big]$$

But the total standard deviation of the first is $.577$ and the standard deviation of the second is $1.29$.

Note: in all cases we are using the sample standard deviation (i.e. we divide by $n-1$ not $n$).