Standard Deviation

Standard Deviation


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:


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


Formula of standard deviation by direct method for individual series.

Where is the arithmetic mean of the given observations.

 

 

For Discrete or Continuous Series:

 

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


Formula of standard deviation by direct method for discrete or continuous series.

Where 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:

 

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


Formula of standard deviation by deviation method for individual series.

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

 

 

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.


Formula of standard deviation by deviation method for individual series.

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:


Standard deviation formula by deviation method using common factor.

Where d = (x – A)/h , A = assumed mean and 'h' is the common factor from all x – A.

 

 

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,


Formula for coefficient of standard deviation


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 σ2 or Var(X) for random variable x.

 

For individual data,

Formula of variance for individual data.

For discrete data,

Formula of variance 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,

Formula for coefficient of variance


Worked Out Examples

 

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


Example 1: Table of data

Calculate the standard deviation and its coefficient by:

a.     Direct method

b.     Deviation method

 

Solution: Here,

Calculation of Standard Deviation:

 

Direct Method,

Table,

Example 1: Table for Standard Deviation by Direct Method

From the Table,

No. of data (N) = 5

∑x = 1775

∑x2 = 639375

Example 1: calculation of standard deviation by direct method


Deviation method,

Table,

Example 1 Table for Standard Deviation by Deviation Method

From the Table,

No. of data (N) = 60

∑d = 25

∑d2 = 9375

Example 1 calculation of standard deviation by deviation method

 

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


Example 2: Table of data

Solution: Here,

Calculation of Standard Deviation:

Table,

Example 2: Table for Standard Deviation

From the table,

∑fx = 1396

∑fx2 = 44396

Example 2: calculation of standard deviation



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


Example 3: Table of data

Calculate the standard deviation and it's coefficient by

a.     Direct method

b.     Deviation method

 

Solution: Here,

Calculation of Standard Deviation: 

Direct method

Table,

Example 3 Table for Standard Deviation by Direct Method

From the Table,

N = 60

∑fx = 3340

∑fx2 = 200300

Example 3: calculation of standard deviation by direct method


Deviation method

Table,

Example 3 Table for Standard Deviation by Deviation Method

From the Table,

N = 60

∑fd = 4

∑fd2 = 144

Example 3: calculation of standard deviation by deviation method



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


Example 4: Table of data

Solution: Here,

Calculation of Variance:

Table,

Example 4 Table for Variance

From the Table,

∑fx = 1660

∑fx2 = 42500

Example 4: calculation of variance by using formula



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

 

 

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8 comments:

  1. Find standard deviation coffeicient of standard deviation and coefficient of variation.
    Class F
    0≤x<10 12
    10≤x<20 33
    20≤x<30 30
    30≤x<40 15
    40≤x<50 10

    ReplyDelete
    Replies
    1. Here is the solution to your problems,
      [image src="https://1.bp.blogspot.com/-xZEI6a3dgc8/YOV2CUSnwLI/AAAAAAAAJbs/d44spVDCEHkippEn9zl1wHoIlZamg053gCLcBGAsYHQ/s16000/comment%2Bsolution%2B1.png"/]
      Comment here if you have more problems regarding the Standard Deviation.

      Delete
  2. A sample of 2 marks p and q is taken find mean of both and coffient variance of both
    X. P. Q
    0-2. 8. 10
    2-4. 12. 14
    4-6. 22. 19
    6-8. 14. 12
    8-10. 4. 5

    ReplyDelete
    Replies
    1. Here is the solution to your problems,
      [image src="https://1.bp.blogspot.com/-hNp8qhlBrdM/YOb8xViSC5I/AAAAAAAAJds/ZWqXwIV1iKQMa4vMyAcaytlt3gq4rnMogCLcBGAsYHQ/s16000/comment%2Bsolution%2B2.png"/]
      Comment here if you have more problems regarding the Standard Deviation.

      Delete
  3. Find standard deviation and coefficient of standard deviation
    (c) Daily wages (in Rs.
    100-125
    125-150
    150–175
    175-200
    200-225

    No. of workers
    75
    57
    81
    19
    12

    ReplyDelete
    Replies
    1. Here is the solution to your problems,
      [image src=" https://1.bp.blogspot.com/-JCJXX1_CjqM/YOb9bK0agnI/AAAAAAAAJd0/GwBFzoKK9eIg_XAHEW_hqwHxMJF_eJrNACLcBGAsYHQ/s16000/comment%2Bsolution%2B3.png"/]
      Comment here if you have more problems regarding the Standard Deviation.

      Delete
  4. Calculate standard deviation from the frequency distribution scores given
    below :
    Cl : 50 - 54 55 - 59 60 - 64 65 - 69 70 - 74 75 - 79 80 - 84 85 - 89
    F : 2 11 10 12 21 6 9 4

    ReplyDelete
    Replies
    1. Here is the solution to your problems,
      [image src="https://1.bp.blogspot.com/-v4KQYpfnqn0/YQl11brIjZI/AAAAAAAAJwY/JVCPnJ4GcJM8sndNRaKbuYErkVAp-SxyACLcBGAsYHQ/s16000/comment%2Bsolution%2B4.png "/]
      Comment here if you have more problems regarding the Standard Deviation.

      Delete