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

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

Where μ is mean and x_{1}, x_{2}, x_{3}…., x_{i} are elements.Also note that mean is sometimes denoted by

**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:

variance = = 4.

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) = = 2

Why did mathematicians chose 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]

- 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, mean is increased (or decreased) by 7, but standard deviation doesn’t change.
- If we multiply all values in the input set by a number 7, both mean and standard deviation is multiplied by 7. But if we multiply all input values with a negative number say -7, mean is multiplied by -7, but the standard deviation is multiplied by 7.
- Standard deviation and varience is a measure which tells how spread out 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).

**Some Interesting Facts:**

Below questions have been asked in previous year GATE exams

http://quiz.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

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

Attention reader! Don’t stop learning now. Get hold of all the important CS Theory concepts for SDE interviews with the CS Theory Course at a student-friendly price and become industry ready.

## Recommended Posts:

- Variance and standard-deviation of a matrix
- Find Harmonic mean using Arithmetic mean and Geometric mean
- Find combined mean and variance of two series
- Given N and Standard Deviation, find N elements
- Program to implement standard deviation of grouped data
- Program for Mean Absolute Deviation
- Program to implement standard error of mean
- Mathematics | Rolle's Mean Value Theorem
- Mathematics | Lagrange's Mean Value Theorem
- Canonical and Standard Form
- Standard Normal Distribution (SND) - Java Program
- Data encryption standard (DES) | Set 1
- Modifications to standard Turing Machine
- Calculate the IST : Indian Standard Time
- Strength of Data encryption standard (DES)
- Set in C++ Standard Template Library (STL)
- Mathematics | Predicates and Quantifiers | Set 1
- Mathematics | Sum of squares of even and odd natural numbers
- Mathematics | Eigen Values and Eigen Vectors
- Mathematics | Introduction and types of Relations