Business Statistics - Measures of Dispersion (Formulas)

 

Measures of Dispersion

–      Formulas

 

A measure of dispersion is designed to state numerically the extent to which individual observations vary around the average. There are several measures of dispersion under the two broad categories as follows:

A.       Absolute Measures:

1.       Range,

2.       Quartile Deviation

        (Also known as 'Semi-Interquartile Range')

3.       Mean Deviation About Mean,

4.       Mean Deviation About Median, and

5.       Standard Deviation.

 

B.       Relative Measures:

1.       Coefficient of Range

2.       Coefficient of Quartile Deviation,

3.       Coefficient of M.D. About Mean,

4.       Coefficient of M.D. About Median, and

5.       Coefficient of Variation.

 

Note: Mean Deviation (M.D.) is usually calculated about arithmetic mean, and hence if it’s only ‘Mean Deviation’, it refers to M.D. About Mean only.

 

 

Formulas

 

1.           Range

= Largest Value (L) – Smallest Value (S)

2.           Coefficient of Range

= [(L – S) ÷ (L + S)] × 100

3.           Quartile Deviation = ½ (Q3 – Q1)

4.          Coefficient of Quartile Deviation (1^st Formula)

= [(Q3 – Q1) ÷ (Q3 + Q1)] × 100

5.          Coefficient of Quartile Deviation (2^nd Formula)

= [Quartile Deviation ÷ Median) × 100

6.          Mean Deviation About Mean (For Simple Distribution)

= [∑Mod (x_i – Mean)] ÷ n

7.           Mean Deviation About Mean (For Frequency Distribution)

= [∑f_i {Mod (x_i – Mean)}] ÷ ∑f_i

8.     Mean Deviation About Median (For Simple Distribution)

= [∑Mod (x_i – Median)] ÷ n

9.   Mean Deviation About Median (For Frequency Distribution)

= [∑f_i {Mod (x_i – Median)}] ÷ ∑f_i

10.      Coefficient of M.D. About Mean

= [M.D. About Mean ÷ Mean] × 100

11.      Coefficient of M.D. About Median

= [M.D. About Median ÷ Median] × 100

 

Standard Deviation

 

12.      For Simple Distribution (1^st Formula)

= √ [∑ {(x_i – Mean) ^2} ÷ n]

13.      For Simple Distribution (2^nd Formula)

= √ [{∑ (x_i^2) ÷ n} – {(∑x_i ÷ n) ^2}]

14. For Simple Distribution (3^rd Formula – Short-Cut Method)

= √ [{∑ (d_i^2) ÷ n} – {(∑d_i ÷ n) ^2}]

[Here, d_i = x_i – A; A = Assumed Mean (as near as possible to the true mean)]

15.     For Simple Distribution (4^th Formula – Step-Deviation Method)

= √ [{∑ (D_i^2) ÷ n} – {(∑D_i ÷ n) ^2}] × c

[Here, D_i = (x_i – A)/c; A = Assumed Mean (as near as possible to the true mean); c = Common factor]

16.      For Frequency Distribution (1^st Formula)

= √ [∑ {f_i (x_i – Mean) ^2} ÷ ∑f_i]

17.      For Frequency Distribution (2^nd Formula)

= √ [{∑f_i (x_i^2) ÷ ∑f_i} – {(∑f_ix_i ÷ ∑f_i) ^2}]

18.      For Frequency Distribution (3^rd Formula – Short-Cut Method)

= √ [{∑f_i (d_i^2) ÷ ∑f_i} – {(∑f_id_i ÷ ∑f_i) ^2}]

[Here, d_i = x_i – A; A = Assumed Mean (value of that observation or that mid-value which has the highest frequency)]

19.      For Frequency Distribution (4^th Formula – Step-Deviation Method)

= √ [{∑f_i (D_i^2) ÷ ∑f_i} – {(∑f_iD_i ÷ ∑f_i) ^2}] × c

[Here, D_i = (x_i – A)/c; A = Assumed Mean (value of that observation or that mid-value which has the highest frequency); c = Common factor or common width]

20.       Coefficient of Variation (i.e. Coefficient of SD)

        = (SD ÷ AM) × 100

21.       Variance = SD^2

 

Important note:

Formula of SD for simple frequency distribution and grouped frequency distribution under all the above four methods are same. Only thing to be remembered is that in case of grouped frequency distribution x_i will be the mid-values of the class intervals.

Important Properties of Standard Deviation

 

1.       If y = x ± c where 'c' is a constant,

SD of y = SD of x

i.e. σ_y = σ_x

2.       If x = c + dy where 'c' and 'd' are constants,

σ_x = (Mod 'd') × σ_y

3.       √ [1/n∑ (x_i – Mean of x) ^2] ≤ √ [1/n∑ (x_i – A) ^2] whatever be the value of A.

 

SD of Composite Group

Composite SD

= √ [(N₁σ₁^2 + N₂σ₂^2 + N₁d₁^2 + N₂d₂^2) ÷ (N₁ + N₂)]

 

[Here,

x_i = Observations of Group 1

y_i = Observations of Group 2

d₁ = Mean of Group 1 – Composite Mean

d₂ = Mean of Group 2 – Composite Mean

σ₁ = SD of Group 1

σ₂ = SD of Group 2

Composite mean = [(∑fx) + (∑fy)]/ (N1 + N2)

= [(N1*Mean of x) + (N2*Mean of y)]/ (N1 + N2)

n1 = Total number of observations of Group- 1

n2 = Total number of observations of Group- 2

N1 = Total frequency of Group- 1

N2 = Total frequency of Group- 2]