Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Business Mathematics and Statistics

Created by: Arshit Thakur

B Com : Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

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  • 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​

 

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

 

For Grouped Data​

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

 

2. Moments about Origin or Zero

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

For Grouped Data

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

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

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRevWhere A is constant
Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

For Grouped Data

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev          Where A is constant

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev  

ii. Short Cut Method
For Ungrouped Data

 

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev                Where D= X - A

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev                                                                    Where D= X - A

                            Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

 

iii. Coding Method or Step Deviation Method
For Ungrouped Data

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev                             

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

 

 

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

 

For Grouped Data

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev
Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev
Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

  • Relation Between Central moments in Terms of Non Central Moments
    Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev
  • Moments – Ration

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

• Sheppard’s Correction for Moments of Group Data​

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Charliers Check

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

  • 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
Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Bowly’s Quartile Coefficient of Skewness

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev
Moment Coefficient of Skewness

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Kurtosis

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

Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev
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 β< 3 distribution is Platy Kurtic

Or

  Moments - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev For Normal distribution, K = 0.263

 

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