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:
[Tex]\sigma=\sqrt{\frac{\sum{fd^2}}{N}-(\frac{\sum{fd}}{N})^2}[/Tex]
Or
[Tex]=\sqrt{\frac{\sum{fd^2}}{N}-(\frac{\sum{fd}}{N})^2}[/Tex]
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 [Tex](\bar{X})=\frac{\sum{fX}}{N}=\frac{240}{20}=12 [/Tex]
Standard Deviation (\sigma)=\sqrt{\frac{\sum{fd^2}}{N}-(\frac{\sum{fd}}{N})^2}=\sqrt{\frac{\sum{1,150}}{20}-(\frac{-60}{20})^2}
[Tex]=\sqrt{57.5-9}=\sqrt{48.5}=6.9[/Tex]
Standard Deviation = 6.9 or 7
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.