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., [Tex]d^\prime=\frac{d}{C}[/Tex]
Step 4: Multiply the step deviations determined in the previous step by their respective frequencies, and obtain their total; i.e., [Tex]\sum{fd^\prime}[/Tex]
Step 5: Now determine the square of step deviations; i.e., [Tex]d^\prime{^2}[/Tex]
Step 6: Multiply the squared step deviations by their respective frequencies and determine the total to get [Tex]\sum{fd^\prime{^2}}[/Tex]
Step 7: Now, apply the following formula:
[Tex]\sigma=\sqrt{\frac{\sum{fd^\prime{^2}}}{N}-(\frac{\sum{fd^\prime}}{N})^2}\times{C}[/Tex]
Where,
σ = Standard Deviation
[Tex]\sum{fd^\prime{^2}} [/Tex] = Sum total of the squared step deviations multiplied by frequencies
[Tex]\sum{fd^\prime} [/Tex] = 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 [Tex](\bar{X})=\frac{\sum{fX}}{N}=\frac{240}{20}=12 [/Tex]
Standard Deviation [Tex](\sigma)=\sqrt{\frac{\sum{fd^\prime{^2}}}{N}-(\frac{\sum{fd^\prime}}{N})^2}\times{C}[/Tex]
[Tex]=\sqrt{\frac{46}{20}-(\frac{-12}{20})^2}\times{5}[/Tex]
[Tex]=\sqrt{2.3-0.36}\times{5}=\sqrt{1.94}\times{5}=1.39\times{5}=6.9[/Tex]
Standard Deviation = 6.9 or 7
Also Read:
Standard Deviation in Individual Series
Standard Deviation in Discrete Series
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.