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Mean squared deviation
- The variance is the mean squared deviation of a random variable from its own mean. If X has high variance, we can observe values of X a long way from the mean. If X has low variance, the values of X tend to be clustered tightly around the mean value.
www.stat.auckland.ac.nz/~fewster/325/notes/ch3.pdf
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Apr 24, 2022 · In both cases, \( f \) is the probability density function of \( X \). Variance is always nonnegative, since it's the expected value of a nonnegative random variable. Moreover, any random variable that really is random (not a constant) will have strictly positive variance.
- Understanding The Definition
- An Equivalent Definition
- Example
- How to Cite
To better understand the definition of variance, we can break up its calculation in several steps: 1. compute the expected value of , denoted by 2. construct a new random variable equal to the deviation of from its expected value; 3. take the square which is a measure of distance of from its expected value (the further is from , the larger ); 4. fi...
Variance can also be equivalently defined by the following important formula: This formula also makes clear that variance exists and is well-defined only as long as and exist and are well-defined. We will use this formula very often and we will refer to it, for brevity's sake, as variance formula.
The following example shows how to compute the variance of a discrete random variable using both the definition and the variance formula above. The exercisesat the bottom of this page provide more examples of how variance is computed.
Please cite as: Taboga, Marco (2021). "Variance", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-probability/variance.
Variance. Remember that the variance of any random variable is defined as Var(X) = E [(X − μX)2] = EX2 − (EX)2. So for a continuous random variable, we can write. Also remember that for a, b ∈ R, we always have Var(aX + b) = a2Var(X). (4.4) Example.
The variance is the probability weighted average of the square of these variances. The square of the error treats positive and negative variations alike, and it weights large variations more heavily than smaller ones.
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance.
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.
The variance measures how far the values of X are from their mean, on average. Definition: Let X be any random variable. The variance of X is Var(X) = E (X − µ X) 2 = E(X )− E(X) . The variance is the mean squared deviation of a random variable from its own mean. If X has high variance, we can observe values of X a long way from the mean.
