Updating mean and variance estimates
$$ Thus, $(n,\bar x_n,x_)$ yield $\bar x_$ and $(n,\bar\sigma^2_n,\bar x_n,\bar x_,x_)$ yield $\bar\sigma^2_$.
There are two problems in the preceding answer, the first being the formula for the variance is incorrect(see the formula below for the correct version) and the second is that the formula for the recursion ends up subtracting large, nearly equal, numbers.
These tradeoffs are explored using the example reliability modeling of a simple parallel-series system with three components.
Rather than dealing with abstract measures of total of uncertainty for a particular distribution or set of distributions, we explore the relationships between variance-based sensitivity analysis of the prior and posterior estimates of the mean and variance over all possible results of a particular test.
The goal is to gain insight into the many tradeoffs that occur when comparing different information collection actions, especially when the exact outcome of the action is uncertain.
Updating Mean and Variance Estimates: An improved method. A Fast Algorithm for Approximate Quantiles in High Speed Data Streams.
Survey error can be classified into sampling error and nonsampling error.These transaction prices are used to construct the broad range of PPIs.PPI variance estimates measure the sampling error in the estimate of the percent change in an index.Here are the iterative formulas (with derivations) for the population (N normalized) and sample (N-1 normalized) standard deviations, which express the $\sigma_$ ($s_$ for sample) for the $n 1$ value set in terms of $\sigma_$ ($s_$ for sample), $\bar x_$ of the $n$ value set plus the new value $x_$ added to the set.Essentially we need to find: $$\bar x_ = f(n, \bar x_n, x_)$$ and $$\sigma_ = g(n, \sigma_n, \bar x_n, x_)$$ Derivation for the Average For both cases, the average for $n\geqslant1$ is, for n values: $$ \bar x_n=\frac1n\sum_^nx_k $$ for n 1 values: $$ \bar x_=\frac1\sum_^x_k = \frac1(n\bar x_n x_) \leftarrow f(n, \bar x_n, x_) $$ Derivation for the Standard Deviation The standard deviation formulas for population and sample are: \begin \sigma_ &= \sqrt && \textit \textbf \textit\ \ s_ &= \sqrt && \textit \textbf \textit \ \end To consolidate the derivations for both population and sample formulas we'll write the standard deviation using a generic factor $\alpha_$ and replace it at the end to get the population and sample formulas.