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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**

**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

Q_{3 }- Median = Median - Q_{1 }

u_{3 }= m_{3}= 0

β_{1}= b_{1}= 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

Q_{3}- Median ≠ Median - Q_{1 }

u_{3 }= m_{3}≠ 0

β_{1}= b_{1}≠ 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 β_{2 }is an important measure of kurtosis. These measures define as:

The moment coefficient β_{2} is a pure numbers and independent of the origin and unit of measurement.

If β_{2} > 3 distribution is Leptokurtic

If β_{2} = 3 distribution is Normal or Mesokurtic

If β_{2 }< 3 distribution is Platy Kurtic

Or

For Normal distribution, K = 0.263

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