Mean, Variance and Standard Deviation

Mean is average of a given set of data. Let us consider below example

$2,\ 4,\ 4,\ 4,\ 5,\ 5,\ 7,\ 9$

These eight data points have the mean (average) of 5:

$\frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = 5.$

$Formula : \mu=\frac{\sum_{i=1}^{N} x_{i}}{N}$

Where μ is mean and x₁, x₂, x₃…., x_i are elements.Also note that mean is sometimes denoted by $\bar{x}$

Variance is the sum of squares of differences between all numbers and means.
Deviation for above example. First, calculate the deviations of each data point from the mean, and square the result of each:

$\begin{array}{lll} (2-5)^2 = (-3)^2 = 9 && (5-5)^2 = 0^2 = 0 \\ (4-5)^2 = (-1)^2 = 1 && (5-5)^2 = 0^2 = 0 \\ (4-5)^2 = (-1)^2 = 1 && (7-5)^2 = 2^2 = 4 \\ (4-5)^2 = (-1)^2 = 1 && (9-5)^2 = 4^2 = 16. \\ \end{array}$

variance = $\frac{9 + 1 + 1 + 1 + 0 + 0 + 4 + 16}{8}$ = 4.

$Formula: \sigma^{2}= \frac { \sum_{i=1}^{N} (x_{i}-\mu)^{2}}{N}$

Where μ is Mean, N is the total number of elements or frequency of distribution.

Standard Deviation is square root of variance. It is a measure of the extent to which data varies from the mean.

Standard Deviation (for above data) = $\sqrt{ 4 }$ = 2

Why did mathematicians choose a square and then square root to find deviation, why not simply take the difference of values?
One reason is the sum of differences becomes 0 according to the definition of mean. Sum of absolute differences could be an option, but with absolute differences, it was difficult to prove many nice theorems. [Source: MIT Video Lecture at 1:19]

$\textup{Coefficient of variation } =\frac{ \textup{Standard deviation}}{Mean}*100$

Value of standard deviation is 0 if all entries in input are same.
If we add (or subtract) a number say 7 to all values in the input set, the mean is increased (or decreased) by 7, but the standard deviation doesn’t change.
If we multiply all values in the input set by a number 7, both mean and the standard deviation is multiplied by 7. But if we multiply all input values with a negative number say -7, the mean is multiplied by -7, but the standard deviation is multiplied by 7.
Standard deviation and variance is a measure that tells how spread out the numbers is. While variance gives you a rough idea of spread, the standard deviation is more concrete, giving you exact distances from the mean.
Mean, median and mode are the measure of central tendency of data (either grouped or ungrouped).

Below questions have been asked in previous year GATE exams
https://www.geeksforgeeks.org/gate-gate-cs-2012-question-64/

References:
https://en.wikipedia.org/wiki/Standard_deviation
http://staff.argyll.epsb.ca/jreed/math30p/statistics/standardDeviation.htm

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Last Updated : 14 Sep, 2023