**What is Standard
Deviation (SD)?**

The definition of Standard Deviation: “**Standard Deviation** is the positive square root of the arithmetic
mean of the squares of the deviations of the given observation from their
arithmetic mean.”

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Standard deviation is also known as ‘**Root Mean-Square Deviation**’ because it is the square root of the
arithmetic mean of the squares of the deviations.

It is denoted by the symbol σ (sigma).

**Significance of
Standard Deviation**

Among all the methods of finding dispersion of data, **Standard Deviation** is regarded as the
best. Because, it is free from the defects with which range, quartile deviation
and mean deviation suffer. Its value is based upon each and every item of the
series and it also takes into account algebraic signs.

**Calculation
of Standard Deviation**

**By Direct Method:**

We calculate the standard deviation for normal data by applying
the direct method. In this method, we find directly the arithmetic mean and then
deviations from each data.

__For Individual
Series__**:**

__For Individual Series__

The standard deviation of the set of n observations is given by

Where x̄ is the arithmetic mean of the given observations.

__For Discrete
or Continuous Series__**:**

__For Discrete or Continuous Series__

For a discrete and continuous frequency distribution, the standard deviation is given by

Where x̄ is the arithmetic mean and N is the total number of observations of
the given data. In the case of a continuous distribution, x is taken as the mid-value
of the corresponding class.

**By Deviation Method:**

If the values of the observations and their corresponding
frequencies are large, then the calculation of standard deviation is tedious
and time-consuming. In such cases, we can calculate the standard deviation by
taking deviation as follows:

__For Individual
Series__**:**

__For Individual Series__

When the values of the observations are very large in the individual series, the standard deviation can be calculated by using the following formula,

Where d = x – A and A is
assumed mean.

__For Discrete
or Continuous Series__**:**

__For Discrete or Continuous Series__

When the values of the observations and the frequencies are large in case of discrete or continuous series, the calculation of standard deviation can be made easy by using the following formula.

Where d = x – A and A is
assumed mean.

When the value of variables or mid-values have some common factor, the calculation of the standard deviation can be made still easier by taking the deviation as follows:

Where d = (x – A)/h

**Coefficient of Standard
Deviation**

The standard deviation is the absolute measure of dispersion. The relative measure of dispersion based on the standard deviation is known as the coefficient of standard deviation. It is given by the formula,

**Variance**

The measure of the dispersion of a random variable or a sample
is called **Variance**. It is given by
the square of the standard deviation. So, it is usually denoted by **σ ^{2}** or

**Var(X)**for random variable x.

For individual data,

For discrete data,

**Coefficient of Variance
(C.V.)**

The Coefficient of Variance (C.V.) is a measure of dispersion
equal to the standard deviation of a sample divided by the mean. It is dimensionless and not dependent on the units or scale in which the
observations are made. It is often expressed as a percentage.

∴ The coefficient of variance (C.V.) is given by the formula,

**Worked Out Examples**

**Example 1:** The weekly expenditure of 5 families in rupees are given below:

Calculate the standard deviation and its coefficient by:

a. Direct method

b. Deviation method

**Solution:** Here,

Calculation of Standard
Deviation:

**Direct Method**,

Table,

From the
Table,

No. of data (N) = 5

∑x = 1775

∑x^{2} = 639375

**Deviation method**,

Table,

From the
Table,

No. of data (N) = 60

∑d = 25

∑d^{2} = 9375

**Example 2:** Find the standard deviation and its coefficient from the following
frequency distribution.

**Solution:** Here,

Calculation of Standard
Deviation:

Table,

From the
table,

∑fx = 1396

∑fx^{2} = 44396

**Example 3:** Following are the marks obtained by students in a test exam.

Calculate the standard deviation
and it's coefficient by

a.
Direct method

b.
Deviation method

**Solution:** Here,

Calculation of Standard Deviation:

**Direct method**

Table,

From the
Table,

N = 60

∑fx = 3340

∑fx^{2} = 200300

**Deviation method**

Table,

From the
Table,

N = 60

∑fd = 4

∑fd^{2} = 144

**Example 4:** Find the variance and its coefficient from the following data.

**Solution:** Here,

Calculation of Variance:

Table,

From the
Table,

∑fx = 1660

∑fx^{2} = 42500

If you have any questions or problems regarding the **Standard Deviation**, you can ask here, in the comment section below.

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Find standard deviation coffeicient of standard deviation and coefficient of variation.

ReplyDeleteClass F

0≤x<10 12

10≤x<20 33

20≤x<30 30

30≤x<40 15

40≤x<50 10

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100-125

125-150

150–175

175-200

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75

57

81

19

12

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Cl : 50 - 54 55 - 59 60 - 64 65 - 69 70 - 74 75 - 79 80 - 84 85 - 89

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