If is the frequency distribution, then standard deviation, denoted by Greek letter σ (sigma) is given by
where is the arithmetic mean of the distribution and ∑f =N .
|Mean Deviation||Standard Deviation|
|Uses absolute value to assure that final result is positive||Squares to make all values positive, then takes positive square root.|
|Using absolute value makes it less suitable for further mathematical treatment.||Most suited measure of dispersion for further mathematical treatment.|
|Makes use of all values in the distribution.||Makes use of all values in the distribution.|
|Can be calculated about any measure of central tendency||Calculated using arithmetic mean.|
Standard Deviation is the most used measure of dispersion – and most suitable for further mathematical treatment.
The square of standard deviation is called variance, denoted by
Example 1: Find the Standard Deviation and Variance
STEP 1: To find Mean
|Total||∑f = 8||∑fx = 20|
STEP 2: To find Standard Deviation
Example 2: Find the Standard Deviation and Variation
(Try it yourself!)
In fact, if the numbers are big, you can simplify the calculation!
- Choose any A (preferably median of observations or the observation with highest frequency) and h
- Calculate 𝜎d
- Then 𝜎𝑥 = h 𝜎d
(Try it yourself with A = 12, h = 2)
Coefficient of Variation
Coefficient of Variation =
Example 1: Calculate the coefficient of variation
(Try it yourself)
The rth moment of a distribution centred at any number A is given by
- 𝜇0′ = 1
- 𝜇1′ is the arithmetic mean.
The rth moment about (i.e., ) is called rth central moment
- 𝜇0 = 1
- 𝜇1 = 0
- 𝜇2 = 𝜎2