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In probability and statistics, the variance of a random variable is the average value of the square distance from the mean value In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable The standard deviation is obtained as the square root of the variance Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers are spread out from their average value The variance of a random variable can be thought of this way: the random variable is made to assume values according to its probability distribution, all the values are recorded and their variance is computed has the same units as X Var(X) has the same units as the square of X So if X is in meters, then Var(X) is in meters squared To better understand the definition of variance, we can break up its calculation in several steps: From these steps we can easily see that: the larger the distance is on average, the higher the variance Variance can also be equivalently defined by the following important formula:X = is de ned as var(X) = E (X )2 The square root of the variance of a random variable is called its standard de stant C, because (X + C) be a ected by a change in location However, it is also desirable that multiplication by a constant should change the spread: var(CX) = C2var(X) and sd(CX) = jCjsd( ), because (CX E(CX))2 = C2(X EX)2 b2var The term var (x) represents the variance of a random variable x, which measures how much the values of x deviate from the mean of its distribution Variance quantifies the spread or dispersion of a set of data points, allowing for insights into the variability within a dataset Feb 24, 2026 · Poisson Distribution is defined as a discrete probability distribution that is used to define the probability of the 'n' number of events occurring within the 'x' period The following theorem gives another method to calculate the variance The variance of random variable X X is Var [X] = E [X 2] μ 2 Var[X] = E[X 2]−μ2 By definition, Var [X] = E [(X − μ) 2] = E [X 2 − 2 μ X + μ 2] = E [X 2] − 2 μ E [X] + E [μ 2] = E [X 2] − 2 μ ⋅ μ + μ 2 = E [X 2] − μ 2,The variance is a measure of how spread out the distribution of a random variable is Here, the variance of $Y$ is quite small since its distribution is concentrated at a single value, while the variance of $X$ will be larger since its distribution is more spread out
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