Measures of dispersion

Measures of Dispersion:                                                                                      

Range & Semi inter quartile range, Mean deviation, Variance,        

Standard deviation, Coefficient of variation.                                                 

Range : is the difference between the highest and the lowest values Example: Range of normal Hb is 10 → 16 gm                                                  

Range = 16 – 10 =  6                                                                                              

Semi inter quartile range (S.I.Q.R) = (Q₃ – Q₁ ) / 2                                         

To calculate S.I.Q.R: Arrange the data from lowest to highest then order of Q₁ will be n  x ( 1 / 4 ) and order of Q₃ will be n x ( 3 / 4 )    

Mean deviation: is the deviation of observations from the mean.       

Example: data (1, 2, 3, 4, 5)                                                                                 

Mean = Ʃ x / n                                                                                                       

X = (1 + 2 + 3 + 4 + 5) / 5 = 15 / 5 = 3

Mean deviation is │X – X│                                                                                    

1 – 3 = – 2                        2 – 3 = -1                       3 – 3 = 0                           4 – 3= +1     5 – 3 = + 2                    Total = zero                                                             

Variance (S²) : is the square of the mean deviation.                                      

It has 2 formulas:                                                                                                 

 S² = Ʃ (X –  X)² / (n – 1)

S² = [Ʃ (X)² – (Ʃ X)² /(n )] / (n – 1)

Going back to (1,2,3,4,5 data) example:                                                          

(X): 1, 2, 3, 4, 5                                                                                                       

(X²): 1, 4, 9, 16, 25                                                                                                 

 Ʃ(X)² = 55                                                                                                                 (

X) = 3                                                                                                                       

(X – X): – 2, – 1, 0 , + 1, + 2                                                                                      

(X – X)²: 4, 1 , 0, 4 ,                                                                                                                                 

 Ʃ (X – X)²  = 10                                                                                                         

Ʃ X = 15                                                                                                                    

(Ʃ X)²  = 225                                                                                                             

n = 5

Using formula 1 for data 1, 2, 3, 4, 5:                                                              

Variance = 10 / (5 – 1) = 10 / 4                                                                           

Using formula 2 for data 1, 2, 3, 4, 5:                                                                 

Variance = 55 – (225 /5) / (5 – 1) = (55 – 45) / 4 = 10 / 4                      

Standard deviation (SD):                                                                                     

Is the square root of the variance.                                                                     

The SD will show the extent of variation (scattering or dispersion) of observations around the mean,                                                                  

The smaller the SD the less the scatter and vice versa.

SD =       Ʃ (X – X)²/ n -1

 

SD =     [Ʃ (X)² – (Ʃ X)² / n ] / n – 1

Coefficient of variation (CV): equals (SD/ Mean) x 100