Exercise Dispersion

8/14/2019 Exercise Dispersion

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CHEPTER 4MEASURES OF DISPERSION

EXERCISE # 4BY SHAHID MEHMOOD sm_078@hotm !" $om

DISPERSION

While measures of central tendency are used to estimate "normal" values of a dataset,measures of dispersion are important for describing the spread of the data, or its variationaround a central value. Two distinct samples may have the same mean or median, butcompletely different levels of variability, or vice versa. A proper description of a set ofdata should include both of these characteristics. There are various methods that can beused to measure the dispersion of a dataset, each with its own set of advantages anddisadvantages.

MEASURES OF DISPERSION

The measures use to find the dispersion are called measure of dispersion or measures ofvariation.

TYPES OF MEASURES OF DISPERSION

1. Absolute Measures of Dispersion.2. Relative Measures of Dispersion.

ABSO%UTE MEASURES OF DISPERSION

Absolute measures of dispersion are one which gives the amount of dispersion in thesame unit as the unit of observations. For example if the observations are in !g theamount of dispersion is also in !g.

RE%ATI&E MEASURES OF DISPERSION

Relative measures of Dispersion are free from units and amount of dispersion is expressin percentage or ratios.

TYPES OF ABSO%UTE MEASURES OF DISPERSION

The most common absolute measures of dispersion are"i. The range.ii. The mean deviationiii. The standard deviation# andiv. The variance.

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TYPES OF RE%ATI&E MEASURES OF DISPERSION

The most common absolute measures of dispersion are"i. The coefficient of range.ii. The coefficient of mean deviationiii. The coefficient of standard deviation# andiv. The coefficient of variance.

THE RAN,E

Defined as the difference between the largest and smallest sample values.

One of the simplest measures of variability to calculate. Depends only on extreme values and provides no information about how the

remaining data is distributed.

. $%mbolicall% it is given b%

R & ' ( $)here ' & 'argest value

$ & $mallest *alue

RAN,E FOR ,ROUPED DATA

The range for grouped data ma% therefore be defined as the difference between theupper class boundar% of the highest class and the lower class boundar% of the lowestclass.

THE COEFFICIENT OF RAN,E

The relative measure of range is called coefficient of range and is given b% the formula"

+oefficient of range & ' ( $ ,1-- ' $

E- m."( # )Different prices/Rs0 are given below" Rs 1- 12 13 1

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Find range and coefficient of range.So"/t!oR '( 1 %2SR '( 1 )32)0 1 Rs 3

CR 1S LS L

+

,1--

+R &1-1

1-1

+

,1-- & 2 4

E- m."( #

Find arithmetic and harmonic mean from the following fre5uenc% distribution"X 6. 16. 26. 36.F 7 6 3

So"/t!oRange & '8$Range & 36. 86. & 3-

CR 1S LS L

+

,1--

+R &6.36.

6.836.

+,1-- & 9.:24

E- m."( # Distribution of marks obtained by 100 candidates in an examination is givenbelow:

Marks 10-24 25-3 40-54 55-! "0-#4 #5-frequency 10 1# 23 2 14 !

$om%ute&i' (ange&ii' $oefficient of range

)olutionR '( 1 5 3255 31 50

CR 1.::.:

.::.:

+

,1-- & 72. 94

THE MEAN DE&IATION

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The mean deviation is defined as the average of the deviations of the values from anaverage. /e.g. the mean median or the mode0# the deviations are ta!en withoutconsidering algebraic signs. ;t is denoted b% MD.

THE MEAN DE&IATION ABOUT MEAN

The mean deviation about mean of a set of n values < 1 < 2 =< n is given b%

MD &n

x x

THE MEAN DE&IATION ABOUT MEAN FROM ,ROUPED DATA

;f < 1 < 2 =< n are the classes mar!s with f 1 f 2 =.f n as the corresponding classfre5uencies the mean deviation is given b%

M.D &

f

x x f

COEFFICIENT OF MEAN DE&IATION ABOUT MEAN

The relative measure of mean deviation is called coefficient of mean deviation and it isdenoted b% +MD.

CMD 1 1--, x

MD

THE MEAN DE&IATION ABOUT MEDIAN

The mean deviation about mean of a set of n values < 1 < 2 =< n is given b%

MD &n

median x

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THE MEAN DE&IATION ABOUT MEDIAN FROM ,ROUPED DATA

;f < 1 < 2 =< n are the classes mar!s with f 1 f 2 =.f n as the corresponding classfre5uencies the mean deviation is given b%

M.D &

f

Median x f

COEFFICIENT OF MEAN DE&IATION ABOUT MEDIAN

The relative measure of mean deviation is called coefficient of mean deviation and it isdenoted b% +MD.

CMD 1 1--,median

MD

E- m."( # )Find mean deviation and coefficient of mean deviation about mean from the values"2 6 9 7 1-So"/t!o

X 6-2 x x x 2 86 66 82 29 - -7 2 2

1- 6 6

CMD 1 1--, x

MD

MD &n

x x &12

& 2.6

+MD & 1--, x

MD 1 1--,

9

6.2

+MD & 6-4

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E- m."( # +Fo th( *o""o9! ' * (:/( $;

C" ss(s 8 781- 11813 16819 1 81:F ( :/( $; 1 17 2 1- 9

So"/t!o

$" ss(s * - F- x x * x x 8 1 9 :-

781- 17 : 192 2 3911813 2 12 326 1 216819 1- 1 1 - 6 6-1 81: 9 17 1-7 62Total 9 736 22-

x *

f

fx *

9

736 * 11

M.D &

f

x x f &

9

22- * 2+

CMD 1 1--, x

MD

CMD 1 1--,11

:.2 1 29.394

E- m."( #

Find mean deviation and coefficient of mean deviation about median from the followingfre5uenc% distribution"C" ss(s 8 781- 11813 16819 1 81:F (:/( $; 1 17 2 1- 9

So"/t!o

CB * - CF median x * median x

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6. 8 . 1 9 1. 81-. 17 : 33 2 39

1-. 813. 2 12 9- 1 213. 819. 1- 1 - 6 6-19. 81:. 9 17 9 62

Total 9 22-

M(


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are the deviation among individual measurements from the population mean for theentire population. For the sample variance and standard deviation s 2 and s it is howmuch each individual measurement deviates from the sample mean. As is the case withthe mean the population variance and standard deviation are the expected or truedeviations.

*ariance & $ 2 & ( )2

n

x x for ungrouped data.

THE &ARIANCE FROM ,ROUPED DATA

;f < 1 < 2 =< n are the classes mar!s with f 1 f 2 =.f n as the corresponding classfre5uencies the variance is given b%"

*ariance & $ 2 &( ) 2

f

x x f for ungrouped data.

STANDARD DE&IATIONThe positive s5uare root of variance is called standard deviation and sample standarddeviation is denoted b% $ and is given b%

6! $ 1 ( ) 2

n

x x for ungrouped data

6!! $ 1( )

f

x x f 2

for grouped data.

E- m."( # )Find variance standard deviation and coefficient of variation from the following sampleobservations"2 6 9 7 1-.

So"/t!o

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- 6-2 x ( ) 2 x x 2 86 196 82 69 - -7 2 6

1- 6 19

$ 2 & ( ) 2

n

x x for ungrouped data.

$ 2 &6-

& 7

$ & 7 & 2.72

+* & 1--, x s

& 1--,9

72.2 & 6 4

E- m."( # +

+alculate mean variance standard deviation and coefficient of variation from thefollowing fre5uenc% distribution"

C" ss(s F2. 82.: 23.-83.63. 83.: 16.-86.6 26. 86.: 2-

.-8 .6 12

. 8 .: :9.-89.6 7

So"/t!oC" ss(s F - *-

( 2 x x * ( )

2

x x

2. 82.: 2 + 7 3 4 3 ? 703.-83.6 + ++ 4 ) 77 )+ 83. 83.: 1 7 ?+ 5 0 ?5 )) 7)6.-86.6 2 4 + )03 0 )) + 7+6. 86.: 2- 4 7 54 0 0 0 38

.-8 .6 12 3 + ?+ 4 0 43 3 5

. 8 .: : 3 7 3) ) 7 )+ +

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9.-89.6 7 ? + 45 ? + 75 ++ )1-- 43 74 ))

x *

f

fx *

1--

6 3 * 4+53

$ 2 &( )2

f

x x f &

1--

11.6 & -. 6

$ 1( )

f

x x f 2

1 6.- 1 -.79

+* & 4:7.171--,3.6

79.-=

SHORTCUT COMPUTATION OF &ARIANCE AND STANDARD DE&IATION

$ 2 &( )

2

22

n

x

n x for ungrouped data

$ 2 &( )( )

2

2

2

f

fx

f

fx for fre5uenc% distribution

E- m."( # Find variance standard deviation and coefficient of variation b% short cut method fromthe following sample observations"2 6 9 7 1-.$olution

- X +

2 66 199 397 96

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1- 1--0 ++0

$ 2 &( )

2

22

n

x

n

x for ungrouped data

$ 2 & 3922-

& 7

$ & 7 & 2.72

+* & 1--, x s & 1--,

9

72.2 & 6 4

E- m."( # ++alculate mean variance standard deviation and coefficient of variation b% short cutmethod from the following fre5uenc% distribution"

+lasses F2. 82.: 23.-83.63. 83.: 16.-86.6 26. 86.: 2-

.-8 .6 12

. 8 .: :9.-89.6 7

So"/t!oC" ss(s * - *- F- +

2. 82.: 2 2. .6 16. 7

3.-83.6 3.2 22.6 1.973. 83.: 1 3. 92.: 232. 36.-86.6 2 6.2 1- 6616. 86.: 2- 6. :6 661.7

.-8 .6 12 .2 92.6 326.67

. 8 .: : . 1.3 2:2.619.-89.6 7 9.2 6:.9 3- . 2

1-- 6 3 2129.2-

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x *

f

fx *

1--

6 3 * 4+53

$ 2 &( )( )

2

2

2

f

fx

f

fx &

( )( )2

2

1--

6 3

1--

2-.2129

$ 2 & -. 6S 1 -.79

+* & 4:7.171--,3.679.-

=

SYMMETRY

A distribution or data set is s%mmetric if it loo!s the same to the left and right ofthe center point.

A distribution or data whose mean median mode and median are e5ual is calleds%mmetrical distribution or data

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S E NESS

Departure from s%mmetr% is called s!ewness

TYPES OF S E NESS

There are two t%pes of s!ewness !nown as" >ositive s!ewness. ?egative s!ewness.

Pos!t! ( s (9 (ss;f the right tail is longer than the left tail the distribution is called positivel% s!ewed.;n positive s!ewness mean is greater than mode and median lies between mean andmedian.

N(' t! ( s (9 (ss;f the left tail is longer than the right tail the distribution is called negativel% s!ewed.;n negative s!ewness mode is greater than mean and median lies between mean andmedian.

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,ormal $urve &symmetrical'

(i g t-)kewed

.ean

.edian


/eft-)kewed

.ean .edian


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/(st!o # )

For the following measurements"

29 1 26 29 2: 21 33 31 2: 2+ompute"a0 Range.

b0 +oefficient of Range.c0 Mean Deviation.d0 +oefficient of Mean Deviation.

A s 6 )? 6= + 6$ 3 6< ) 0

/(st!o # +

For the following wages in Rs."

12- 122 12 12 13- 16-.+ompute"a0 Range.

b0 +oefficient of Range.c0 Mean Deviation.d0 +oefficient of Mean Deviation.e0 *ariancef0 $tandard deviation.g0 +oefficient of variation.

A s 6 Rs +0 6= 7 7 6$ Rs 4 58 6< 546( Rs 40 )5 6* Rs ? 4 6' 3 0+

/(st!o # A firm of +hartered Accountants on the basis of their audit reports grouped the assets of 7- @ointstoc! companies in million rupees as follows"Assets 689 987 781- 1-812 12816 16819 19817

?o. of firms 2 : 13 3- 1 7 3+alculate"

a0 Range. b0 +oefficient of Range.

A s 6 Rs )4 m!""!o s 6= ? ?

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/(st!o # 4

The fre5uenc% distribution of the ages of the emplo%ees of a business firm is given below"A'( 1 81: 2-826 2 82: 3-836 3 83: 6-866 6 86: -8 6 8 : 9-896* 2 1- 1: 2 19 1- 9 3 2+alculate variance

/(st!o # 3

A firm of +hartered Accountants on the basis of their audit reports grouped the assets of 7- @ointstoc! companies in million rupees as follows"Assets 689 987 781- 1-812 12816 16819 19817

?o. of firms 2 : 13 3- 1 7 3+alculate"

a0 *ariance b0 $tandard deviation.c0 +oefficient of variation.

/(st!o # ?The average grade of a group of candidates in conomics was 76 with a standard deviation of 1-.;n accountanc% the mean grade of the same group of candidates was 7 with a standard deviationof 7. +ompute the coefficient of variation of economics and accountanc%. ;ndicate for whichsub@ect the performance is more consistentB

A s )) 5 )0 ! $$o/ t $; .( *o m $( !s mo ( $o s!st( t

/(st!o # 7

+oefficient of variation of two series are 74 and 9:4 their standard deviations are21.2 and 1 .9 respectivel%. )hat are their arithmetic meansB

/(st!o # 8

;n a surprise chec!ing of passengers in a local bus 2- passengers without tic!ets werecaught. The sum of s5uares and the standard deviation of the amount found in their

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poc!ets were Rs.2--- and Rs.9 respectivel%. ;f the total fine imposed is e5ual to theamount recovered from them and fine imposed is uniform. )hat is the amount each ofthem will have to pa% as fineB

/(st!o #5

Talent 'td. a Coll%wood casting compan% is selecting a group of extras for a movie.The ages of the first 2- men to be interviewed are

- 9 6: 2 9 9 :

6 91 9- 1 : 92 2 6 6:

The director of the movie wants men whose ages are fairl% tightl% grouped around %ears. eing a statistics buff of sorts the director suggests that a standard deviation of 3%ears would be acceptable. Does this group of extras 5ualif%B

/(st!o # )0

The universit% has decided to test three new !inds of light bulbs. The have three identicalrooms to use in the experiment. ulb 1 has an average lifetime of 16 - hours and avariance of 1 9. ulb 2 has an average lifetime of 16-- ours and a variance of 71. ulb 3

has an average lifetime of 13 - hours and standard deviation of 9 hours. +alculate +* ineach of the above cases in terms of relative variabilit%. )hich was the best bulbB

/(st!o # ))

$tudents ages in the regular da%time M A program and the evening program of +entralEniversit% are described b% these two samples"

R('/" MBA E ( ! ' MBA

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21 323 2222 6-2 227 3-29 3723 6

Ese a measure of relative variabilit% to suggest which of the two groups will be easier toteach.

A s>Rs 8

/(st!o #)+

The following are the number of defective bolts produced b% two machines"

M $h! ( A M $h! ( B2 369 -

27 11

1- :13 19 7

)hich machine showing the more consistent performanceBA s M $h! ( A

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/(st!o # )

+onsider the following fre5uenc% distribution"

C" ss ! t( " F (:/( $;1-81: 1-

2-82: 163-83: 266-86: 1

-8 : :

Find the mean variance and standard deviation.6 ICAP S. ! ' +008 M s 03

/(st!o #)4

A research organi ation selected a sample of 3- visitors to a prestigious shopping mall. The dataabout the ages of the selected persons have been organi ed into the following table"

Age /in %ears0 17 to 23 23 to 27 27 to 33 33 to 37 37 to 63f 2 12 9 3Gou are re5uired to calculate the following"/a0 Range./b0 $ample variance and sample standard deviation./c0 +oefficient of variation. 6 ICAP A/t/m +007 M s 08

/(st!o # )3

The following table shows the distribution of monthl% salaries of a compan%Hsemplo%ees"S " ; S$ "( 3---83 -- 3 -186--- 6--186 -- 6 -18 --- --18 --

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/eft-)kewed

.ean .edian


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No o* (m."o;((s

3-- 2 - 21- 1 - 1--

Gou are re5uired to calculate"/a0 Mean deviation from Median/b0 +o8efficient of Mean deviation from Median. 6 ICAP S. ! ' +00? M s 07

/(st!o # )?

For the following distribution calculate the variance standard deviation and coefficientof variation"

- 2- 8 2: 3- ( 3: 6- ( 6: - ( : 9- ( 9: - ( : 7- 8 7:* 6 12 1: 2 2- 16 9

6 ICAP S. ! ' +004 M s 0?

/(st!o # )7

The mean temperature in Iarachi in the month of Januar% is 19 0 + with a standarddeviation of -. 0 +. Kn Januar% 1 the temperature is 6 0 + standard deviation above themean. )hat is the temperature on Januar% 1 B

6 ICAP S. ! ' +00 M s 0

Liven"CI -86 8: 1-816 1 81: 2-826 2 82:* 12 2- 67 1- 3

R(:/! (< " /i0 +alculate Mean Deviation from Median./ii0 +alculate coefficient of Mean Deviation from Median.

6 ICAP A/t/m +00+ M s 04G0+

/(st!o # )8

Find the sample variance sample standard deviation and co8efficient of variation for thefollowing data"$i e of orders /x0 ?o. of orders /f0888888888888888888 888888888888888882- 3- 33- 6- 7

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6- - 12- 9- 9

9- - 16 ICAP S. ! ' +00) M s 0?

/(st!o # )5

The length of 32 items are given below. Find the mean length and the standard deviationof the distribution"

%( 'th6! $h(s 2-822 2382 29827 2:831 32836* (:/( $; 3 9 12 : 26ICAP /t/m )555 M s 04

A s +7 )7

/(st!o # +0The mean of the numbers 3 9 a 16 is 7. find the standard deviation of the set of abovenumbers.

6ICAP S. ! ' )555 M s 04A s 8

Exercise Dispersion

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