How to calculate cumulative S.D I performed an experiment in which I have 3 plastic columns (C1, C2, C3) filled with soil. I then poured specific amount of water in all the three columns and collected the filtered water into a container and labeled it as W1C1, W1C2, W1C3.
I again poured the same amount of water and collected and labeled as W2C1, W2C2, W2C3. I performed this experiment 7 times in total till I got W7C1, W7C2, W7C3. (21 total containers)
I then calculated the concentration of an Lead in EACH container and then found the mean concentration in each water pouring and Standard deviation (S.D) of the concentration as follows
WC1  = mean(W1C1, W1C2, W1C3); S.D1 (W1C1, W1C2, W1C3)
WC2  = mean(W2C1, W2C2, W2C3); S.D2 (W2C1, W2C2, W2C3)
WC3  = mean(W3C1, W3C2, W3C3); S.D3 (W3C1, W3C2, W3C3)
WC4  = mean(W4C1, W4C2, W4C3); S.D4 (W4C1, W4C2, W4C3)
WC5  = mean(W5C1, W5C2, W5C3); S.D5 (W5C1, W5C2, W5C3)
WC6  = mean(W6C1, W6C2, W6C3); S.D6 (W6C1, W6C2, W6C3)
WC7  = mean(W7C1, W7C2, W7C3); S.D7 (W7C1, W7C2, W7C3)
I got the results from MS Excel super easy which I wrote as WC1 +/- S.D1, and so on for each water pouring number.
Now I want to write in my findings that the mean total Lead concentration collected in the water = WC1+WC2+WC4+WC4+WC5+WC6+WC7 +/- S.D
The problem is I dont know how to find the cumulative S.D. Is is S.D1 + S.D2 + ........+ S.D7?
I do not know much statistics and I guess maybe I am even calculating the mean total Lead concentration wrong.
Anyone who can correct me in this?
Thanks
 A: In general, the sample standard deviation of the total set of data is not equal to the sum (or average) of the sample standard deviations of the subsamples.  Since you are the one that gathered the original data, the easiest thing to do here is to pool all the data and calculate the sample mean and sample standard deviation directly from this.
If you really want to calculate the sample standard deviation recursively from the sample means and sample variances of the subsamples then you can do this using the appropriate iterated variance formula (see e.g., O'Neill 2014, p. 283, Result 1).  For two subsamples, the pooled sample variance is:
$$s_{\text{pooled}}^2 = \frac{n_1-1}{n_1+n_2-1} \cdot s_1^2 + \frac{n_2-1}{n_1+n_2-1} \cdot s_2^2 + \frac{n_1 n_2}{n_1 + n_2} \cdot (\bar{x}_1 - \bar{x}_2)^2.$$
If you apply this rule iteratively to your subsamples you will be able to obtain the pooled sample variance (and thereby obtain the pooled sample standard deviation).  This is much more complicated than just using the original data directly.
