Variance and Standard Deviation

Variance

The measurement of the distance between the mean or average value of a data set and how far a data point has dispersed is called the variance. It is represented as “σ2”.

Properties of Variance

• Since the sum of variance is positive because each term of the variance is always going to be squared, the result of the variance is always going to be either a positive number or it is going to equal to zero. 

• The unit of variance is always going to be squared. For example, the variance of the volume of different vessels that are estimated in litres will be given as litre square. 

Standard Deviation

The standard deviation measures the spread of the statistical data. The deviation of data from the average position or its mean value is measured by distribution. With the help of the method that is used for the deviation of data points, the degree of dispersion is computed. The symbol “σ” represented the standard deviation.

Properties of Standard Deviation

• The square root of the means of all the squares of all values in a data set is described by the standard deviation. In other terms, the standard deviation is also called the root mean square deviation. 

• The smallest value of the standard deviation can only be number zero. This is because the value of the standard deviation cannot be a negative number. 

• The standard deviation’s value is low or close to zero when the data sets are similar to each other. But if the value of these data sets has a huge difference or have a high differential value, then the value of the standard deviation is going to be high.

Variance and Standard Deviation Formula

As explained earlier, The measurement of the distance between the mean or average value of a data set and how far a data point has dispersed is called the variance. The standard deviation measures the spread of the statistical data. The deviation of data from the average position or its mean value is measured by distribution. With the help of the method that is used for the deviation of data points, the degree of dispersion is computed. Let’s consider an example. Let’s say a set has ‘n’  number of elements and the elements in the set are { x, x2, x3, x4, . . . . . . . xn }. Then, the formulas for standard deviation variance formula goes by: 

Standard deviation = σ2 = \[\sqrt{\frac{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}{n}}\]

Variance, σ = \[\frac{\sum_{i=1}^{n}(x_{i} -\overline{x})^{2}}{n}\]

Here, 

xi = data set values

\[\overline{x}\] = mean of the data set

Solved Examples

Question 1: What are the variance and standard deviation values of rolling dice? 

Solution: The only possible outcomes of the dice are: { 6, 5, 4, 3, 2, 1 }. 

In order to find the variance, we should calculate the mean.

Mean of the six possible outcomes = sum of possible outcomes / total number of outcomes. 

Mean of the six possible outcomes = ( 6 + 5 + 4 + 3 + 2 + 1 ) / 6

Mean of the six possible outcomes = 21 / 6

Mean of the six possible outcomes = 3.5

To get the value of standard deviation, then: 

σ2 = \[\frac{\sum(x_{i}-\overline{x})^{2}}{n}\]

σ2 = \[\frac{1}{\text{6 ( 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 )}}\]

σ = \[\sqrt{\frac{1}{\text{6 ( 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 )}}}\]

σ = \[\sqrt{2917}\]

σ = 1708