# How to combine subsets consisting of mean, variance, confidence, and number of sampled points used?

I have a data set that has been divided into $n$ data subsets.

I am sampling from each of these subsets and getting a tuple consisting of mean, variance, confidence and number of sampled points used.

How can I combine these results?

I do not know how other than a simple function of the number of points and their averages. This wont take into account either the variance or the confidence of the score.

-

Let $n_i, m_i, v_i$ be the number of samples, observed mean, and variance in sample $i$. Let $n, m, v$ be similar figures for the combined data (sorry I redefined $n$ here). $m = \frac{1}{n}\sum_i n_i m_i$. Now for the variance: $v = \frac{1}{n-1}\sum_{i,j} (x_{i,j} - m)^2$, with $x_{i,j}$ the $j^{th}$ observation of sample $i$ and $j=1,2,\ldots, n_i$. Play around a little: $(x_{i,j} -m)^2 = (x_{i,j} - m_i + m_i - m)^2 = (x_{i,j} -m_i)^2 + (m_i-m)^2 +2(x_{i,j}-m_i)(m_i-m)$. Terms $(m_i-m)$ can be factored out of the summation over $j$: $v = \frac{1}{n-1}\left[\sum_i n_i(m_i-m)^2 + 2\sum_i(m_i-m)\sum_j(x_{i,j}-m_i) + \sum_{i,j} (x_{i,j} - m_i)^2\right]$. Since $\sum_j (x_{i,j}-m_i)=0$, the middle term cancels out. So you're left with $v=\frac{1}{n-1}\left[\sum_i n_i(m_i-m)^2 + \sum_i(n_i-1)v_i\right]$. Confidence intervals are obtained with $m$ and $v$. Is that what you were looking for ?
 It is close but not exactly. The mean is obtained in your answer by just taking the combined sum and dividing by the combined total. I wanted to take into consideration the variance and/or confidence of the individual means while combining them. I am going to continuously draw more and more samples from the different distributed datasets and want to take into consideration the variance in a dataset so as to weight it lesser – user12005 Jun 15 '12 at 2:52 Ok, well I'm not sure exactly what the objective is here but I guess you could assign a weight such as $w_i=1/v_i$ and compute the weighted average $m = \sum_i{w_i m_i}/\sum_i w_i$. – Chap Jun 15 '12 at 3:06 That would be heuristic. I was looking for some mathematical bounds on the error – user12005 Jun 15 '12 at 3:09 A bit more detailed context would help: what do the observations represent ? What is the confidence of the score ? Where does the error come from ? – Chap Jun 15 '12 at 3:23 I have say a dataset. I then divided into 5 datasets. I am going to continuously draw samples from these and get the mean and variance of each of the sample. At every few hundred milliseconds I want to combine the means that I have using the variance of each of the subsets. I want the answer with 95% confidence which would give me a certain confidence interval. So at every merge step I want to be able to say that the mean lies within this interval with this confidence of this error. – user12005 Jun 15 '12 at 3:28