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This is perhaps a very basic question, but I am a bit confused. I have a time series of day some time varying data, say the amount of sunlight as the day passes.

I bucket the data in say 10 bins say according to just time, so later bins occur later in time and calculate the variance of each of the bins. Is there a relationship between the calculated variances of the 10 bins and the overall population variance (variance of the whole data set)?

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My favourite way to deal with variances of reals is using Steiner's Theorem (geometry Theorem, used in physics for calculating the moment of interia). If you plot the points on the line, the variance is just their moment of interia.

Suppose the i-th bin's variance is $V_i$, and the mean temperature in the i-th bin is $M_i$. Let $M$ be the mean temperature, and $\sigma$ the total variance. Then: $$ \sigma = \sum\limits_{i=1}^{n} V_i + \sum\limits_{i=1}^{n}M_i (M-M_i)^2 $$

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