The variance of a random variable is the average value of the squared deviation from the mean. Its square root is the standard deviation. In a sample of size n , the corresponding sample measure usually has a divisor n − 1 rather than n .David Clark-CarterStaffordshire University, Stoke-on-Trent, UKJohn Wiley & Sons, Ltd
The variance of a random variable or distribution is the expectation, or mean, of the squared deviation of that variable from its expected value or mean. Thus the variance is a measure of the amount of variation of the values of that variable, taking account of all possible values and their...
Formulae for the sample variance Until now, we have discussed how to calculate the variance of a random variable. However, there is another concept, that of sample variance, that applies when we need to assess the dispersion of some observations around their sample mean. If you are interested...
The variance is one of the measures of dispersion which computes the spreadness among the data values from the mean value. The variance of a random variable is computed by taking expectation of the squared random variable minus the square of expected value of the...
Thesample varianceof a random variable demonstrates two aspects of estimator bias: firstly, the naive estimator is biased, which can be corrected by a scale factor; second, the unbiased estimator is not optimal in terms ofmean squared error(MSE), which can be minimized by using a different sc...
Variance:The variance of a random variable is the averaged squared deviations of data from the mean. The variance is often denoted by {eq}\sigma^2 {/eq}, where {eq}\sigma {/eq} is the standard deviation. We will use these steps and definitions to calculate the variance of t...
The proof above uses the probability density function of the distribution. An alternative, simpler proof exploits the representation (demonstrated below) of as a sum of squared normal variables. Proof Variance Thevarianceof a Chi-square random variable ...
Taking expectations of both sides of the preceding yields, upon using the fact that for any random variable W,E[W2]=Var(W)+(E[W])2, (n−1)E[S2]=E[∑i=1nXi2]−nE[X‾2]=nE[X12]−nE[X‾2]=nVar(X1)+n(E[X1])2−nVar(X‾)−n(E[X‾])2=nσ2+nμ2−n...
The variance of a random variable is just an expected value of a function of . Specifically, . Let’s substitute into Markov’s inequality and see what happens. For convenience and without loss of generality, I will replace the constant with another constant, . Now, let’s substitute with ...
It provides a function that yields the mean value of a random variable under the condition that one (in bivariate regression) or more (in multivariate regression) independent variables have specified values. linear relationship Response or output is directly proportional to the input. moment Random ...