**Standard Deviation**

If is the frequency distribution, then standard deviation, denoted by Greek letter σ (sigma) is given by

where is the arithmetic mean of the distribution and ∑f =N .

Mean Deviation | Standard Deviation |
---|---|

Uses absolute value to assure that final result is positive | Squares to make all values positive, then takes positive square root. |

Using absolute value makes it less suitable for further mathematical treatment. | Most suited measure of dispersion for further mathematical treatment. |

Makes use of all values in the distribution. | Makes use of all values in the distribution. |

Can be calculated about any measure of central tendency | Calculated using arithmetic mean. |

Standard Deviation is the most used measure of dispersion – and most suitable for further mathematical treatment.

## Variance

The square of standard deviation is called variance, denoted by

**Example 1**: Find the Standard Deviation and Variance

x: | 1 | 2 | 3 | 4 |

f: | 2 | 1 | 4 | 1 |

**Solution:**

**STEP 1**: To find Mean

x | f | fx |

1 | 2 | 2 |

2 | 1 | 2 |

3 | 4 | 12 |

4 | 1 | 4 |

Total | ∑f = 8 | ∑fx = 20 |

**STEP 2**: To find Standard Deviation

x | ||||

1 | 2 | -1,5 | 2.25 | 4.50 |

2 | 1 | -0.5 | 0.25 | 0.25 |

3 | 4 | 0.5 | 0.25 | 1 |

4 | 1 | 1.5 | 2.25 | 2.25 |

**Example 2**: Find the Standard Deviation and Variation

x: | 8 | 11 | 12 | 14 | 16 |

f: | 2 | 2 | 5 | 2 | 2 |

(Try it yourself!)

In fact, if the numbers are big, you can simplify the calculation!

- Choose any A (preferably median of observations or the observation with highest frequency) and
*h* - Put
- Calculate 𝜎
_{d} - Then 𝜎
_{𝑥}=*h*𝜎_{d}

**Example 3**

x: | 6 | 10 | 12 | 14 | 16 |

f: | 2 | 2 | 5 | 2 | 2 |

(Try it yourself with A = 12, h = 2)

x | f | fd | |

∑f= | ∑fd= |

d | f | |||

-3 | 1 | |||

-1 | 2 | |||

0 | 3 | |||

1 | 2 | |||

3 | 2 | |||

𝜎_{𝑑}^{2} =

𝜎_{𝑑} =

## Coefficient of Variation

Coefficient of Variation =

**Example 1**: Calculate the coefficient of variation

x: | 1 | 2 | 3 | 4 | 5 |

f: | 1 | 2 | 2 | 2 | 1 |

(Try it yourself)

## Moments

The *r ^{th }*moment of a distribution centred at any number A is given by

About 0,

- 𝜇
_{0}′ = 1 - 𝜇
_{1}′ is the arithmetic mean.

The *r ^{th }*moment about (i.e., ) is called

*r*central moment

^{th}- 𝜇
_{0}= 1 - 𝜇
_{1}= 0 - 𝜇
_{2}= 𝜎^{2}

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