How do you find the mean and variance with new observations?

Question

When given a mean and variance of a sample, without knowing the observations, how would you then find the new mean and variance given more observations?

Any help with this would be much appreciated.

Thank you in advance

Answer 1

Let us assume that $\left( x_1, \ldots, x_n \right)$ is the sample and let us call $\mu_n$ the empirical mean of size $n$. Then the updating rule is given by $\mu_n = \frac{n - 1}{n} \mu_{n - 1} + \frac{1}{n} x_n$. This is how to derive the result \begin{eqnarray*} \mu_n & = & \frac{1}{n} \sum_{i = 1}^n x_i\\ & = & \frac{n - 1}{n} \times \frac{1}{n - 1} \sum_{i = 1}^{n - 1} x_i + \frac{1}{n} x_n\\ & = & \frac{n - 1}{n} \mu_{n - 1} + \frac{1}{n} x_n \end{eqnarray*} The rule for the variance is a bit more complicated but follows from the same reasoning.