Moments - Measures of Dispersion, Business Mathematics & Statistics - Business Mathematics and Statistics - B Com

• Moments

A moment designates the power to which deviation are raised before averaging them.

Methods of Standard Deviation
1. Moments about Mean or Central Moments
2. Moments about Origin or Zero
3. Moments about Provisional Mean or Arbitrary Value (Non Central Moment)

1. Moments about Mean or Central Moments

For Ungrouped Data​

For Grouped Data​

2. Moments about Origin or Zero

For Grouped Data

3. Moments about Provisional Mean or Arbitrary Value (Non Central Moment)
4. Methods of Standard Deviation
i. Direct Method
ii. Short Cut Method
iii. Coding Method or Step-Deviation Method

i. Direct Method For Ungrouped Data

For Ungrouped Data

Where A is constant

For Grouped Data

          Where A is constant

ii. Short Cut Method
For Ungrouped Data
 

                Where D= X - A

                                                                    Where D= X - A

iii. Coding Method or Step Deviation Method
For Ungrouped Data

For Grouped Data

• Relation Between Central moments in Terms of Non Central Moments
  
• Moments – Ration

• Sheppard’s Correction for Moments of Group Data​

• Charliers Check

• Symmetry
  In a symmetrical distribution a deviation below the mean exactly equals the corresponding deviation above the mean. It is called symmetry. For symmetrical distribution the following relations hold.
  
  Mean = Median = Mode
  Q₃ - Median = Median - Q₁
  u₃ = m₃ = 0 
  β₁ = b₁ = 0
   
• Skewness
  Skewness is the lack of symmetry in a distribution around some central value i.e. means Median or Mode. It is the degree of asymmetry.
  
  Mean ≠ Median ≠ Mode
  Q₃ - Median ≠ Median - Q₁
  u₃  = m₃ ≠ 0
  β₁ = b₁ ≠ 0 There are two types of Skewness
   

1. Positive Skewness
If the frequency curve has a longer tail to right, the distribution is said to be positively skewed.

2. Negative Skewness
If the frequency curve has a longer tail to left, the distribution is said to negatively skewed.

Coefficient of Skewness (SK)

Karl Pearson’s Coefficient of Skewness

Bowly’s Quartile Coefficient of Skewness

Moment Coefficient of Skewness

Kurtosis

Moment coefficient β₂ is an important measure of kurtosis. These measures define as:

The moment coefficient β₂ is a pure numbers and independent of the origin and unit of measurement.

If β₂ > 3 distribution is Leptokurtic
If β₂ = 3 distribution is Normal or Mesokurtic
If β₂ < 3 distribution is Platy Kurtic

Or

  For Normal distribution, K = 0.263