Moments about origin Examples for grouped data

1. Calculate Moment about origin from the following grouped data
X Frequency
0 1
1 5
2 10
3 6
4 3

Solution:
Moments :

`x` `(2)`	`f` `(3)`	`f*x` `(4)=(2)xx(3)`	`f*x^2` `(5)=(2)xx(4)`	`f*x^3` `(6)=(2)xx(5)`	`f*x^4` `(7)=(2)xx(6)`
0	1	0	0	0	0
1	5	5	5	5	5
2	10	20	40	80	160
3	6	18	54	162	486
4	3	12	48	192	768
---	---	---	---	---	---
--	`n=25`	*`sum fx=55`**	`=147`	`=439`	`=1419`

Now, calculate Raw Moments

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

`=(55)/(25)`

`=2.2`

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

`=(147)/(25)`

`=5.88`

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

`=(439)/(25)`

`=17.56`

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

`=(1419)/(25)`

`=56.76`

Find Central moments using Moments about origin

First Central Moment
`m_1=0`

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

`=5.88-2.2^2`

`=5.88-4.84`

`=1.04`

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

`=17.56-3*5.88*2.2+2*2.2^3`

`=17.56-38.808+21.296`

`=0.048`

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

`=56.76-4*17.56*2.2+6*5.88*2.2^2-3*2.2^4`

`=56.76-154.528+170.7552-70.2768`

`=2.7104`

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

`=(0.048)^2/(1.04)^3`

`=(0.0023)/(1.1249)`

`=0.002`

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

`=(2.7104)/(1.04)^2`

`=(2.7104)/(1.0816)`

`=2.5059`

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 origin from the following grouped data
X Frequency
10 3
11 12
12 18
13 12
14 3

Solution:
Moments :

`x` `(2)`	`f` `(3)`	`f*x` `(4)=(2)xx(3)`	`f*x^2` `(5)=(2)xx(4)`	`f*x^3` `(6)=(2)xx(5)`	`f*x^4` `(7)=(2)xx(6)`
10	3	30	300	3000	30000
11	12	132	1452	15972	175692
12	18	216	2592	31104	373248
13	12	156	2028	26364	342732
14	3	42	588	8232	115248
---	---	---	---	---	---
--	`n=48`	*`sum fx=576`**	`=6960`	`=84672`	`=1036920`

Now, calculate Raw Moments

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

`=(576)/(48)`

`=12`

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

`=(6960)/(48)`

`=145`

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

`=(84672)/(48)`

`=1764`

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

`=(1036920)/(48)`

`=21602.5`

Find Central moments using Moments about origin

First Central Moment
`m_1=0`

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

`=145-12^2`

`=145-144`

`=1`

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

`=1764-3*145*12+2*12^3`

`=1764-5220+3456`

`=0`

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

`=21602.5-4*1764*12+6*145*12^2-3*12^4`

`=21602.5-84672+125280-62208`

`=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 origin from the following grouped data
Class Frequency
2 - 4 3
4 - 6 4
6 - 8 2
8 - 10 1

Solution:
Moments :

Class `(1)`	Mid value (`x`) `(2)`	`f` `(3)`	`f*x` `(4)=(2)xx(3)`	`f*x^2` `(5)=(2)xx(4)`	`f*x^3` `(6)=(2)xx(5)`	`f*x^4` `(7)=(2)xx(6)`
2 - 4	3	3	9	27	81	243
4 - 6	5	4	20	100	500	2500
6 - 8	7	2	14	98	686	4802
8 - 10	9	1	9	81	729	6561
---	---	---	---	---	---	---
--	--	`n=10`	*`sum fx=52`**	`=306`	`=1996`	`=14106`

Now, calculate Raw Moments

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

`=(52)/(10)`

`=5.2`

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

`=(306)/(10)`

`=30.6`

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

`=(1996)/(10)`

`=199.6`

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

`=(14106)/(10)`

`=1410.6`

Find Central moments using Moments about origin

First Central Moment
`m_1=0`

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

`=30.6-5.2^2`

`=30.6-27.04`

`=3.56`

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

`=199.6-3*30.6*5.2+2*5.2^3`

`=199.6-477.36+281.216`

`=3.456`

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

`=1410.6-4*199.6*5.2+6*30.6*5.2^2-3*5.2^4`

`=1410.6-4151.68+4964.544-2193.4848`

`=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 origin 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 :

Class `(1)`	Mid value (`x`) `(2)`	`f` `(3)`	`f*x` `(4)=(2)xx(3)`	`f*x^2` `(5)=(2)xx(4)`	`f*x^3` `(6)=(2)xx(5)`	`f*x^4` `(7)=(2)xx(6)`
0 - 2	1	5	5	5	5	5
2 - 4	3	16	48	144	432	1296
4 - 6	5	13	65	325	1625	8125
6 - 8	7	7	49	343	2401	16807
8 - 10	9	5	45	405	3645	32805
10 - 12	11	4	44	484	5324	58564
---	---	---	---	---	---	---
--	--	`n=50`	*`sum fx=256`**	`=1706`	`=13432`	`=117602`

Now, calculate Raw Moments

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

`=(256)/(50)`

`=5.12`

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

`=(1706)/(50)`

`=34.12`

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

`=(13432)/(50)`

`=268.64`

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

`=(117602)/(50)`

`=2352.04`

Find Central moments using Moments about origin

First Central Moment
`m_1=0`

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

`=34.12-5.12^2`

`=34.12-26.2144`

`=7.9056`

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

`=268.64-3*34.12*5.12+2*5.12^3`

`=268.64-524.0832+268.4355`

`=12.9923`

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

`=2352.04-4*268.64*5.12+6*34.12*5.12^2-3*5.12^4`

`=2352.04-5501.7472+5366.612-2061.5843`

`=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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2. Moments about origin Examples