Standard deviation is the measure of how distinct the average is from the given data set. The Relative standard deviation is the value of coefficient of variation that is denoted in percent(%).

Relative standard deviation is calculated by the formula,

SD = Standard deviation,

$\bar{x}$ = Mean.

Relative standard deviation is calculated by the formula,

Relative standard deviation = $\frac{SD}{\bar{x}}$ $\times 100$.

Where,SD = Standard deviation,

$\bar{x}$ = Mean.

**Step 1 :**Put down the set of data given in the question. First determine the mean of the given observation by using the formula,

$\bar{x}$ = $\frac{\sum x}{N}$.

Where, $\sum x$ = Sum of the observation and N = number of data.

**Step 2 :**Determine the deviation of each data from the observations with the mean and square the deviation by using the formula $(x-\bar{x})^{2}$.

**Step 3 :**To find the standard deviation, first find the variance by using the formula,

$\sigma$ = $\frac{\sum (x-\bar{x})^{2}}{N-1}$.

Standard deviation = $\sqrt{variance}$

**Step 4 :**Now, determine the relative standard deviation from the formula given.

Relative standard deviation = $\frac{SD}{\bar{x}}$ $\times 100$ Some of the problems are solved below based on Relative standard deviation.

### Solved Examples

**Question 1:**Determine the relative standard deviation of amit who obtained 69, 72, 91, 88, 67 marks in 12th grade examination.

**Solution:**

Given observations are : 69, 72, 91, 88, 67

Mean of the given observations,

$\bar{x}$ = $\frac{\sum x}{N}$

$\bar{x}$ = $\frac{69 + 72 + 91 + 88 + 67}{5}$ = $\frac{387}{5}$ = 77.4

Deviation of the given observations :

x |
$(x-\bar{x})$ |
$(x-\bar{x})^{2}$ |

69 |
-8.4 | 70.56 |

72 |
-5.4 | 29.16 |

91 |
13.6 | 184.96 |

88 |
10.6 | 112.36 |

67 |
-10.4 | 108.16 |

Variance, $\sigma$ = $\frac{\sum (x-\bar{x})^{2}}{N-1}$

$\sigma$ = $\frac{505.2}{4}$ = 126.3

Standard deviation = $\sqrt{126.3}$ = 11.238

Relative standard deviation is given by,

Relative standard deviation = $\frac{SD}{\bar{x}}$ $\times 100$

= $\frac{11.238}{77.4}$ $\times 100$

= 14.519%

**Question 2:**Suppose a dice is rolled 5 times to win a game and the results are recorded. 4, 6, 1, 5 and 1 are the results, calculate its relative standard deviation.

**Solution:**

Given observations are : 4, 6, 1, 5, 1

Mean of the given observations,

$\bar{x}$ = $\frac{\sum x}{N}$

$\bar{x}$ = $\frac{4 + 6 + 1 + 5 + 1}{5}$ = $\frac{17}{5}$ = 3.4

Deviation of the given observation :

Variance, $\sigma$ = $\frac{\sum (x-\bar{x})^{2}}{N-1}$

$\sigma$ = $\frac{21.2}{4}$ = 5.3

Standard deviation = $\sqrt{5.3}$ = 2.302

Relative standard deviation is given by,

Relative standard deviation = $\frac{SD}{\bar{x}}$ $\times 100$

= $\frac{2.302}{3.4}$ $\times 100$

= 67.706%

Mean of the given observations,

$\bar{x}$ = $\frac{\sum x}{N}$

$\bar{x}$ = $\frac{4 + 6 + 1 + 5 + 1}{5}$ = $\frac{17}{5}$ = 3.4

Deviation of the given observation :

x | $(x-\bar{x})$ |
$(x-\bar{x})^{2}$ |

4 | 0.6 | 0.36 |

6 |
2.6 | 6.76 |

1 |
-2.4 | 5.76 |

5 |
1.6 | 2.56 |

1 |
-2.4 |
5.76 |

Variance, $\sigma$ = $\frac{\sum (x-\bar{x})^{2}}{N-1}$

$\sigma$ = $\frac{21.2}{4}$ = 5.3

Standard deviation = $\sqrt{5.3}$ = 2.302

Relative standard deviation is given by,

Relative standard deviation = $\frac{SD}{\bar{x}}$ $\times 100$

= $\frac{2.302}{3.4}$ $\times 100$

= 67.706%