Variance Standard Deviation Variance Standard Deviation Here we will discuss Variance Standard Deviation, first we will discuss variance, and variance is shown below, Variance is a part of statistics and probability theory. Variance is basically used for measuring how far a set of number is spread out. Variance describes the farness of numbers from mean. Variance also describes the probability of numbers. Now let us take a look at the mathematical definition of variance, it is shown belowWe have ‘Y’ as a random variable ‘Y’ whose expected value or mean is ‘μ’ and μ = E [Y], therefore variance is given by, Var (Y) = E [(Y – μ)2], So we can say that variance is equals to the expected value of square of difference between the mean of variable and variable. This mathematical definition includes all types of random variables like discrete, continuous or mixed.
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Let us now discuss some important properties of variance, the properties of variance are shown below1.
Variance is always non – negative or Var (Y).
2.
Variance of a random variable which is constant is equals to zero, this can be written as
P (Y = b) = 1 <-> Var (Y) = 0. 3.
Variance of sum of two random variables and it can be written as shown below,
Var (X + Y) = Var (X) + Var (Y) + 2 cov (X, Y). Let us take an example to understand the concept of variance in detail, the example is shown belowExample: Calculate the variance for rolling of a perfect die? Solution) First we will calculate the mean, Mean = 1 / 6 (1 + 2 + 3 + 4 + 5 + 6) = 3.5 Now we will calculate the variance using the above mentioned formula, we get, Variance = 1 / 6 [(2.5)2 + (1.5)2 + (0.5)2 + (0.5)2 + (1.5)2 + (2.5)2] = 17.5 / 6 = 2.9. So variance is equals to 2.9. This is all about variance. Now we will discuss standard deviation,
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Standard deviation is used for measurement of variability and it is used in statistics and probability theory. It is used to show the variation between values and mean. Here high deviation means that data points are spread across large range of values. Standard deviation of a statistical population, random variable, data set or probability distribution is equals to the square root of variance. Standard deviation can be expressed in same units in which data values have been expressed. Let us now take a look at mathematical definition of standard deviation, it is given belowSuppose we have a random variable ‘Y’ having mean value ‘μ’, which is given by μ = E [Y]. The standard deviation can be calculated using the formula σ = √ E [(X – μ)2]. Where ‘E’ denotes the mean value. Standard deviation is denoted by ‘σ’ and it is equals to square root of variance of variable ‘Y’. Now let us take an example understand the concept of standard deviation in detail. Example: Here we have eight values 2, 4, 4, 4, 5, 5, 7, 9. Calculate the standard deviation? Solution) First we will calculate Mean, Mean = (2 + 4 + 4 + 4 + 5 + 5 +7 + 9) / 6 = 5 Standard Deviation = √[(2 – 5)2 + (4 – 5)2 + (4 – 5)2 + (4 – 5)2 + (5 – 5)2 + (5 – 5)2 + (7 – 5)2 + (9 – 5)2] / 8 = 2.
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