Moments about the value Examples for grouped data

1. Calculate Moment about the value 2 from the following grouped data
Class Frequency
2 - 4 3
4 - 6 4
6 - 8 2
8 - 10 1

Solution:
Moments :
`A=2`

Class `(1)`	Mid value (`x`) `(2)`	`f` `(3)`	`(x-A)` `(5)`	`f*(x-A)` `(6)=(3)xx(5)`	`f*(x-A)^2` `(7)=(5)xx(6)`	`f*(x-A)^3` `(8)=(5)xx(7)`	`f*(x-A)^4` `(9)=(5)xx(8)`
2 - 4	3	3	1	3	3	3	3
4 - 6	5	4	3	12	36	108	324
6 - 8	7	2	5	10	50	250	1250
8 - 10	9	1	7	7	49	343	2401
---	---	---	---	---	---	---	---
--	--	`n=10`	--	`=32`	`=138`	`=704`	`=3978`

Now, calculate Raw Moments

First Raw Moment
`M_1=(sum f*(x-A))/n`

`=(32)/(10)`

`=3.2`

Second Raw Moment
`M_2=(sum f*(x-A)^2)/n`

`=(138)/(10)`

`=13.8`

Third Raw Moment
`M_3=(sum f*(x-A)^3)/n`

`=(704)/(10)`

`=70.4`

Fourth Raw Moment
`M_4=(sum f*(x-A)^4)/n`

`=(3978)/(10)`

`=397.8`

Find Central moments using Moments about the value 2

First Central Moment
`m_1=0`

Second Central Moment
`m_2=M_2-M_1^2`

`=13.8-3.2^2`

`=13.8-10.24`

`=3.56`

Third Central Moment
`m_3=M_3-3M_2M_1+2M_1^3`

`=70.4-3*13.8*3.2+2*3.2^3`

`=70.4-132.48+65.536`

`=3.456`

Fourth Central Moment
`m_4=M_4-4M_3M_1+6M_2M_1^2-3M_1^4`

`=397.8-4*70.4*3.2+6*13.8*3.2^2-3*3.2^4`

`=397.8-901.12+847.872-314.5728`

`=29.9792`

Skewness `beta_1=(m_3)^2/(m_2)^3`

`=(3.456)^2/(3.56)^3`

`=(11.9439)/(45.118)`

`=0.2647`

Kurtosis `beta_2=(m_4)/(m_2)^2`

`=(29.9792)/(3.56)^2`

`=(29.9792)/(12.6736)`

`=2.3655`

Moment coefficient of skewness
`beta_1>0` : The distribution is positively skewed (a longer tail to the right).

Moment coefficient of kurtosis
`beta_2<3` : platykurtic (flatter with lighter tails)

2. Calculate Moment about the value 10 from the following grouped data
X Frequency
10 3
11 12
12 18
13 12
14 3

Solution:
Moments :
`A=10`

`x` `(2)`	`f` `(3)`	`(x-A)` `(5)`	`f*(x-A)` `(6)=(3)xx(5)`	`f*(x-A)^2` `(7)=(5)xx(6)`	`f*(x-A)^3` `(8)=(5)xx(7)`	`f*(x-A)^4` `(9)=(5)xx(8)`
10	3	0	0	0	0	0
11	12	1	12	12	12	12
12	18	2	36	72	144	288
13	12	3	36	108	324	972
14	3	4	12	48	192	768
---	---	---	---	---	---	---
--	`n=48`	--	`=96`	`=240`	`=672`	`=2040`

Now, calculate Raw Moments

First Raw Moment
`M_1=(sum f*(x-A))/n`

`=(96)/(48)`

`=2`

Second Raw Moment
`M_2=(sum f*(x-A)^2)/n`

`=(240)/(48)`

`=5`

Third Raw Moment
`M_3=(sum f*(x-A)^3)/n`

`=(672)/(48)`

`=14`

Fourth Raw Moment
`M_4=(sum f*(x-A)^4)/n`

`=(2040)/(48)`

`=42.5`

Find Central moments using Moments about the value 10

First Central Moment
`m_1=0`

Second Central Moment
`m_2=M_2-M_1^2`

`=5-2^2`

`=5-4`

`=1`

Third Central Moment
`m_3=M_3-3M_2M_1+2M_1^3`

`=14-3*5*2+2*2^3`

`=14-30+16`

`=0`

Fourth Central Moment
`m_4=M_4-4M_3M_1+6M_2M_1^2-3M_1^4`

`=42.5-4*14*2+6*5*2^2-3*2^4`

`=42.5-112+120-48`

`=2.5`

Skewness `beta_1=(m_3)^2/(m_2)^3`

`=(0)^2/(1)^3`

`=(0)/(1)`

`=0`

Kurtosis `beta_2=(m_4)/(m_2)^2`

`=(2.5)/(1)^2`

`=(2.5)/(1)`

`=2.5`

Moment coefficient of skewness
`beta_1=0` : The distribution is perfectly symmetrical (like a normal distribution).

Moment coefficient of kurtosis
`beta_2<3` : platykurtic (flatter with lighter tails)

3. Calculate Moment about the value 5 from the following grouped data
Class Frequency
2 - 4 3
4 - 6 4
6 - 8 2
8 - 10 1

Solution:
Moments :
`A=5`

Class `(1)`	Mid value (`x`) `(2)`	`f` `(3)`	`(x-A)` `(5)`	`f*(x-A)` `(6)=(3)xx(5)`	`f*(x-A)^2` `(7)=(5)xx(6)`	`f*(x-A)^3` `(8)=(5)xx(7)`	`f*(x-A)^4` `(9)=(5)xx(8)`
2 - 4	3	3	-2	-6	12	-24	48
4 - 6	5	4	0	0	0	0	0
6 - 8	7	2	2	4	8	16	32
8 - 10	9	1	4	4	16	64	256
---	---	---	---	---	---	---	---
--	--	`n=10`	--	`=2`	`=36`	`=56`	`=336`

Now, calculate Raw Moments

First Raw Moment
`M_1=(sum f*(x-A))/n`

`=(2)/(10)`

`=0.2`

Second Raw Moment
`M_2=(sum f*(x-A)^2)/n`

`=(36)/(10)`

`=3.6`

Third Raw Moment
`M_3=(sum f*(x-A)^3)/n`

`=(56)/(10)`

`=5.6`

Fourth Raw Moment
`M_4=(sum f*(x-A)^4)/n`

`=(336)/(10)`

`=33.6`

Find Central moments using Moments about the value 5

First Central Moment
`m_1=0`

Second Central Moment
`m_2=M_2-M_1^2`

`=3.6-0.2^2`

`=3.6-0.04`

`=3.56`

Third Central Moment
`m_3=M_3-3M_2M_1+2M_1^3`

`=5.6-3*3.6*0.2+2*0.2^3`

`=5.6-2.16+0.016`

`=3.456`

Fourth Central Moment
`m_4=M_4-4M_3M_1+6M_2M_1^2-3M_1^4`

`=33.6-4*5.6*0.2+6*3.6*0.2^2-3*0.2^4`

`=33.6-4.48+0.864-0.0048`

`=29.9792`

Skewness `beta_1=(m_3)^2/(m_2)^3`

`=(3.456)^2/(3.56)^3`

`=(11.9439)/(45.118)`

`=0.2647`

Kurtosis `beta_2=(m_4)/(m_2)^2`

`=(29.9792)/(3.56)^2`

`=(29.9792)/(12.6736)`

`=2.3655`

4. Calculate Moment about the value 5 from the following grouped data
Class Frequency
0 - 2 5
2 - 4 16
4 - 6 13
6 - 8 7
8 - 10 5
10 - 12 4

Solution:
Moments :
`A=5`

Class `(1)`	Mid value (`x`) `(2)`	`f` `(3)`	`(x-A)` `(5)`	`f*(x-A)` `(6)=(3)xx(5)`	`f*(x-A)^2` `(7)=(5)xx(6)`	`f*(x-A)^3` `(8)=(5)xx(7)`	`f*(x-A)^4` `(9)=(5)xx(8)`
0 - 2	1	5	-4	-20	80	-320	1280
2 - 4	3	16	-2	-32	64	-128	256
4 - 6	5	13	0	0	0	0	0
6 - 8	7	7	2	14	28	56	112
8 - 10	9	5	4	20	80	320	1280
10 - 12	11	4	6	24	144	864	5184
---	---	---	---	---	---	---	---
--	--	`n=50`	--	`=6`	`=396`	`=792`	`=8112`

Now, calculate Raw Moments

First Raw Moment
`M_1=(sum f*(x-A))/n`

`=(6)/(50)`

`=0.12`

Second Raw Moment
`M_2=(sum f*(x-A)^2)/n`

`=(396)/(50)`

`=7.92`

Third Raw Moment
`M_3=(sum f*(x-A)^3)/n`

`=(792)/(50)`

`=15.84`

Fourth Raw Moment
`M_4=(sum f*(x-A)^4)/n`

`=(8112)/(50)`

`=162.24`

Find Central moments using Moments about the value 5

First Central Moment
`m_1=0`

Second Central Moment
`m_2=M_2-M_1^2`

`=7.92-0.12^2`

`=7.92-0.0144`

`=7.9056`

Third Central Moment
`m_3=M_3-3M_2M_1+2M_1^3`

`=15.84-3*7.92*0.12+2*0.12^3`

`=15.84-2.8512+0.0035`

`=12.9923`

Fourth Central Moment
`m_4=M_4-4M_3M_1+6M_2M_1^2-3M_1^4`

`=162.24-4*15.84*0.12+6*7.92*0.12^2-3*0.12^4`

`=162.24-7.6032+0.6843`

`=155.3205`

Skewness `beta_1=(m_3)^2/(m_2)^3`

`=(12.9923)^2/(7.9056)^3`

`=(168.7987)/(494.0882)`

`=0.3416`

Kurtosis `beta_2=(m_4)/(m_2)^2`

`=(155.3205)/(7.9056)^2`

`=(155.3205)/(62.4985)`

`=2.4852`

This material is intended as a summary. Use your textbook for detail explanation.
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3. Moments about the value Examples