Standard Deviation in Discrete Series

Last Updated : 05 Jun, 2024

A scientific measure of dispersion that is widely used in statistical analysis of a given set of data is known as Standard Deviation. Another name for standard deviation is Root Mean Square Deviation. Standard Deviation is denoted by a Greek Symbol σ (sigma). Under this method, the deviation of values is taken from the arithmetic mean of the given set of data. Standard Deviation can be calculated in three different series; viz., Individual, Discrete, and Frequency Distribution or Continuous Series.

Methods of Calculating Standard Deviation in Discrete Series

Actual Mean Method
Direct Method
Short-cut Method or Assumed Mean Method
Step Deviation Method

Table of Content

Example

2. Direct Method

Example

3. Short-Cut Method or Assumed Mean Method

Example

4. Step Deviation Method

Example

1. Actual Mean Method

In actual mean method, the standard deviation is calculated by taking deviations from the actual mean. The steps taken to determine standard deviation through the actual mean method are as follows:

Step 1: Determine the actual mean (\bar{X}) of the given observation, using the formula, \frac{\sum{fX}}{N}

Step 2: Now, calculate the deviation of each item of the given series from the mean calculated in the first step; i.e., calculate (X-\bar{X}). Denote the deviations with x.

Step 3: After finding out the deviation, square these deviations and multiply them with their respective frequencies (f), and determine its total; i.e., ∑fx²

Step 4: Apply the following formula:

\sigma=\sqrt{\frac{\sum{fx^2}}{N}}

Where,

σ = Standard Deviation

∑fx² = Sum total of the squared deviations multiplied by frequency

N = Number of pairs of observations

Example:

Calculate the Standard Deviation for the following data by using the Actual Mean Method.

Solution:

Arithmetic Mean (\bar{X})=\frac{\sum{fX}}{N}=\frac{240}{20}=12

Standard Deviation (σ) = \sqrt{\frac{\sum{fx^2}}{N}}=\sqrt{\frac{970}{20}}=\sqrt{48.5}=6.9

Standard Deviation = 6.9 or 7

2. Direct Method

In Direct Method, there is no need to determine the deviation from the actual mean to calculate standard deviation. The steps taken to determine standard deviation through the direct method are as follows:

Step 1: First of all, calculate the actual mean (\bar{X}) of the given observations, using the formula, \frac{\sum{fX}}{f}

Step 2: Now square the observations and determine their total; i.e., ∑X².

Step 3: Multiply frequency (f) by their respective X² and determine the total; i.e., ∑fX².

Step 4: Now, apply the following formula:

\sigma=\sqrt{\frac{\sum{fX^2}}{N}-(\bar{X})^2}

=\sqrt{\frac{\sum{fX^2}}{N}-(\frac{\sum{fX}}{N})^2}

Where,

σ = Standard Deviation

∑fx² = Sum total of the squared deviations multiplied by frequency

\bar{X} = Actual Mean

N = Number of Observations

Example:

Calculate the Standard Deviation for the following data by using the Direct Method.

Solution:

Arithmetic Mean (\bar{X})=\frac{\sum{fX}}{N}=\frac{240}{20}=12

Standard Deviation (\sigma)=\sqrt{\frac{\sum{fX^2}}{N}-(\bar{X})^2}=\sqrt{\frac{3,850}{20}-(12)^2}

=\sqrt{192.5-144}=\sqrt{48.5}=6.9

Standard Deviation = 6.9 or 7

3. Short-Cut Method or Assumed Mean Method

Actual Mean can sometimes come in fractions, which can make the calculation of standard deviation complicated and difficult. In those cases, it is suggested to use Short-Cut Method to simplify the calculations. The steps taken to determine standard deviation through the assumed mean method are as follows:

Step 1: First of all, take any value of X in the series as Assumed Mean (A).

Step 2: Now determine the deviations of the items from an assumed mean and denote the deviations by d; i.e., d = X - A.

Step 3: Multiply the deviations calculated in the previous step by their respective frequencies and calculate their sum; i.e., ∑fd.

Step 4: Determine the square of the deviations, multiply it by their respective frequencies, and obtain the total; i.e., ∑fd².

Step 5: Now, apply the following formula:

\sigma=\sqrt{\frac{\sum{fd^2}}{N}-(\frac{\sum{fd}}{N})^2}

=\sqrt{\frac{\sum{fd^2}}{N}-(\frac{\sum{fd}}{N})^2}

Where,

σ = Standard Deviation

∑fd = Sum total of deviations multiplied by frequencies

∑d² = Sum total of the squared deviations multiplied by frequencies

N = Number of pairs of observations

Example:

Calculate the Standard Deviation for the following data by using the Assumed Mean Method.

Solution:

Arithmetic Mean (\bar{X})=\frac{\sum{fX}}{N}=\frac{240}{20}=12

Standard Deviation (\sigma)=\sqrt{\frac{\sum{fd^2}}{N}-(\frac{\sum{fd}}{N})^2}=\sqrt{\frac{\sum{1,150}}{20}-(\frac{-60}{20})^2}

=\sqrt{57.5-9}=\sqrt{48.5}=6.9

Standard Deviation = 6.9 or 7

Also Read:
Standard Deviation: Meaning, Coefficient of Standard Deviation, Merits, and Demerits

4. Step Deviation Method

This method is almost like the assumed mean method. The only difference in the formulas of both methods is that in the step deviation method, the deviations are divided by a common factor (C), and then the standard deviation is determined. The steps taken to determine standard deviation through the step deviation method are as follows:

Step 1: First of all take any value of X in the series as Assumed Mean (A).

Step 2: Now determine the deviations of the items from an assumed mean and denote the deviations by d; i.e., d = X - A.

Step 3: Divide these deviations by common factor (C) and obtain step deviations; i.e., d^\prime=\frac{d}{C}

Step 4: Multiply the step deviations determined in the previous step by their respective frequencies, and obtain their total; i.e., \sum{fd^\prime}

Step 5: Now determine the square of step deviations; i.e., d^\prime{^2}

Step 6: Multiply the squared step deviations by their respective frequencies and determine the total to get \sum{fd^\prime{^2}}

Step 7: Now, apply the following formula:

\sigma=\sqrt{\frac{\sum{fd^\prime{^2}}}{N}-(\frac{\sum{fd^\prime}}{N})^2}\times{C}

Where,

σ = Standard Deviation

\sum{fd^\prime{^2}} = Sum total of the squared step deviations multiplied by frequencies

\sum{fd^\prime} = Sum total of step deviations multiplied by frequencies

N = Number of pairs of observations

Example:

Calculate the Standard Deviation for the following data by using the Step-Deviation Method.

Solution:

Arithmetic Mean (\bar{X})=\frac{\sum{fX}}{N}=\frac{240}{20}=12

Standard Deviation (\sigma)=\sqrt{\frac{\sum{fd^\prime{^2}}}{N}-(\frac{\sum{fd^\prime}}{N})^2}\times{C}

=\sqrt{\frac{46}{20}-(\frac{-12}{20})^2}\times{5}

=\sqrt{2.3-0.36}\times{5}=\sqrt{1.94}\times{5}=1.39\times{5}=6.9

Standard Deviation = 6.9 or 7

Also Read:
Standard Deviation in Individual Series

Standard Deviation in Frequency Distribution Series

Combined Standard Deviation: Meaning, Formula, and Example

Standard Deviation in Frequency Distribution Series

nupurjain3

Improve

Article Tags :

Standard Deviation in Discrete Series

Methods of Calculating Standard Deviation in Discrete Series

1. Actual Mean Method

Example:

Solution:

2. Direct Method

Example:

Solution:

3. Short-Cut Method or Assumed Mean Method

Example:

Solution:

Also Read:

4. Step Deviation Method

Example:

Solution:

Also Read:

Similar Reads

Chapter 1: Concept of Economics and Significance of Statistics in Economics

Chapter 2: Collection of Data

Chapter 3: Organisation of Data

Chapter 4: Presentation of Data: Textual and Tabular

Chapter 5: Diagrammatic Presentation of Data

Chapter 6: Measures of Central Tendency: Arithmetic Mean

Chapter 7: Measures of Central Tendency: Median and Mode

Chapter 8: Measures of Dispersion

Chapter 9: Correlation