In statistics, the standard deviation is a measure that reflects the amount of variation of the set of data values. It also stipulates how individual data differs from the average of a dataset. When the standard deviation is low, it means that the data points are closer to the mean, while a high standard deviation denotes that the data points fall within a wide range of values.

There are two general formulas for standard deviation: one for a population and one for a sample. We will go through each,

Population Standard Deviation Formula:

Sample Standard Deviation Formula:

notations for Standard Deviation

σ = Standard Deviation

xi = Terms Given in the Data

x̄ = Mean

n = Total number of Terms

Standard Deviation Formula Based on discrete frequency Distribution

For discrete frequency distribution of the form:

x: x1, x2, x3, … xn and

f : f1, f2, f3, … fn

The formula for standard deviation becomes.

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Here, N is given as:

N = n∑i=1 fi

The other standard deviation formula which is derived from the variance. The formula for the variance is,

Question: In a survey, 6 students were asked how many hours per day they study on an average? Their answers were given as : 2, 6, 5, 3, 2, 3. Calculate the standard deviation.

Solution:

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Step A: Find the mean of data:

=3.5

x1 |
x1 − x̄ |
(x1 − x̄)2 |

2 |
-1.5 |
2.25 |

6 |
2.5 |
6.25 |

5 |
1.5 |
2.25 |

3 |
-0.5 |
0.25 |

2 |
-1.5 |
2.25 |

3 |
-0.5 |
0.25 |

= 13.5 |

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Step 3: Now use the formula for finding the Standard Deviation

Sample Standard Deviation =

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=√(13.5/[6-1])

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= √[2.7]

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= 1.643

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