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LESSON – 1  STATISTICS FOR MANAGEMENT

Meaning The t ermofstStatistics at i st i cs  m ean t hat t he num eri cal st at em ent as wel l as st at i st i cal m et et hodol ogy. When i t i s used i n t he sense of st at i st i cal dat a it re fe rs to qu an ti ta ti ve aspec aspects ts of thin things gs and is a num numeri erical cal descr descrip ipti tion. on. E x a m p l e

: I ncom e of f am i l y, product i on of aut om obi l e i ndust ry, sal es of cars etc.T h e r e q u a n t i t i e s a r e n u m e r i c a l . B u t t h e r e a r e s o m e q u a n t i t i e s w h i c h a r e n o t i n t he h e ms m s el el v ve e s n um u m er e r iic c al a l b ut u t c an a n b e m ad ad e s o by co un ti ng . Th e se x of a ba by is no t a numb number er,, but but by coun counti ting ng the the nu numb mber  er  of boys, we can associate a numerical descriptionto sex of all new born babies, for an example, when saying that 60% of all live-born babies are boy. This information then, comes within the realm of statistics.

Definition  The word st at i st i cs can be used i s t wo senses, vi z, si ngul ar and p l u r a l . I n narrow sense and plural sense, statistics statistics denotes denotes some numerical numerical data data (statistical data).In a wide and singular sense statistics refers to the st at is ti ca l me th od s. Th er ef or e, thes these e hav have e bee been ng gro roup uped ed unde underr two two head heads s –   –  „Statistics as a data” and “Statistics as amethods”.  amethods”.

Statistics as a Data  Some defi Some defini niti tion ons s of of s sta tati tist stic ics s as as a dat data a area)S area)S ta ti st ic s ar e nu me ri ca l st at em en t o f f a c ts in a n y d e pa r tm e n t o f en qu ir in g p la c ed in relation to each other.Powley b ) B y s t a t i s t i c s w e m e a n q u a n t i t i e s d a t a a f f e c t e d t o a m a r k e d e x t e n t b y multiplasticity of course.Yule and Kendallc ) B y s t a t i s t i c s w e m e a n a g g r e g a t e s o f f a c t s a f f e c t e d t o a m a r k e d e x t e n t b y multiplicity of causes, numerically ex expre presse ssed, d, enum enumera erate ted d or es esti tima mate ted d acc accord ordin ingto gto re a so na bl e st an da rd of  accuracy, collected in a systematic manner for pre-determinated purpose and placed in relation to each other.- H. SecristThis definition is more comprehensive and exhaustive. It shows more light oncharacteristics of statistics and covers different aspects.Some characteristics the statistics should possess by H. Secrist can be listed asfollows.1

Statistics are aggregate of facts Statistics are affected to a marked extent by multiplicity of causes. Statistics are numerically expressed Statistics should be enumerated / estimated Statistics should be collected with reasonable standard of accuracy Statistics should be placed is relation to each other.

Statistics as a methods    Definition  a)“St at i st i cs m ay be cal l ed t o sci ence of   count i ng”ng” - A.L. Bowley b)“St at i st i cs i s t he sci ence of est i m at es and probabi l i t i es”. - Boddington c)Dr. Croxt on and Cowden have gi ven a cl ear and conci se def i ni t i on.“ on. “ S t a t i s t i c s m a y b e d e f i n e d a s t h e c o l l e c t i o n , p r e s e n t a t i o n , a n a l y s i s a n d interpretation of numerical data”.  data”.

According to Croxton and Cowden there are 4 stages.   a )C o l l e c t i o n

o f

D a t a

A s t r u c t u r e o f s t a t i s t i c a l i n v e s t i g a t i o n i s b a s e d o n a s y s t e m a t i c c o l l e c t i o n of data. The data is classified into two groups) Internal data and External dataI n t e r n a l d a t a a r e o b t a i n e d f r o m i n t e r n a l r e c o r d s r e l a t e d t o o p e r a t i o n s o f b u s i n e s s o r g a n i s a t i o n s u c h a s p r o d u c t io n , s o u r c e o f i n c o m e a n d e x p e n d i t u r e , inventory, purchases and accounts. Th e e xt e rn a l dat a are col l ect ed and purchased by ext ernal agenci es. The e xt ernal dat a coul d be ei t her pri m ary dat a or secondary dat a. The pri m ary dat a are collected for first time and original, while secondary data are collected by published by some agencies. b ) O r g a n i z at i o n s

o f

d a t a

The collected data is a large mass of figures that needs to be organized. The col l ect ed dat a m ust be edi t ed t o rect i f y f or a ny om i ssi ons, i rrel evant a n s w e r s , a n d wrong computations computations.. The edited data must must be classified classified and and tabulated tabulated to suit further analysis.

c ) P r es e n t a t i o n

o f

d a t a

The large data that are collected cannot be understand and analysis easily and quickly. Therefore, collected data needs to be presented in tabular or graphic form. This systematic order and graphical presentation helps for further analysis. D ) A n a l y s i s

o f

d a t a

The analysis requires establishing the relationship between on e o r m o r e v a r i ab ab l e s . A na na l y s is is o f d a t a in in c l u de d e s c o n d en en s a t io io n , a b s tr tr a c t in in g , sum m ari zat i on, concl usi on et c. Wi t h t he hel p of st at i st i cal t ool s and te ch ni qu es li ke me as u re s of di dispe spersi rsion on centr central al tend tendenc ency, y, corre correla lati tion, on, vari varianc ance e analysis etc analysis can be done. E )

In t e r p r e t a t i o n

o f

d a t a

The interpretation requires deep insight of the subject. Interpretation involves drawing the valid conclusions on the bases of the analysis of data. This work requires good experience and skill. This process is very important as conclusions of results are done based on interpretation. We can define statistics as per Seligman as follows.

“Statistics is a science which deals with the method and of  collecting, classifying, presenting, comparing and interpretating the n u m e r i c a l d a t a collec ectted to throw li lig ght on enquiry”.  Importance of statistics  Bowleys Skewness Coefficient  An alternative measure of skewness has been proposed by the late professor pr ofessor Bowley. Bowley‟s Bowley ‟s quartiles are based on quartiles. In a symmetrical distribution first and third quartiles are equidistant from the median as can be seen from the following diagram. In an asymmetrical distribution the third quartile is the same distance over the median as the first quartile is below it i.e.

Q3  – –Med.

=

Med.

–

Q1 or

Q3 =

Q1  – –

2

Med.

=

If this distribution is positively skewed the top 25 per cent of the values will tend to be farther from median than the bottom 25 per cent. i.e. Q3 will be farther from median than Q1 is form median and the reverse for negative skewness. Hence a possible measure is

– Med.)  – (Med.  – Q1)/ (Q3 – Skb = (Q3 –  – Med.) + (Med.  – Q1) Q1 Med.  Skb =

Bowley’s

coefficient

of

–  or Q3 + Q1 – 2/ Q3 – skewness.

It must be with remembered that the results obtained by these two measures are nottotoone be compared one another especially. The numerical values are not related

another since the burley‟s measure, because of its computational basis, is limited to values between -1 and +1, while person‟s measure has no such limits.  limits.  Not only do the numerical values obtained from these two formulae bear no necessary relationship to one another but, on rare occasions, with unusually shaped distributions, it is possible for them to emerge with opposite sings.

Illustration:  find Bowley‟s Coefficient of Skewness for the following frequency distribution:

No. of children per family No. of families

0

1

2

3

4

5

6

7

10

16

25

18

11

8

Skewness  Solution: calculation of Bowley‟s Coefficient of Skewness

Number of children per family X

No. of families

c.f

0

7

7

1

10

17

2 3

16 25

33 58

4

18

76

5

11

87

6

8

95

SkB =Q3 +Q1 –  –2Med./Q3 –  –Q1  Q1 =Size

of N

Q3 =Size

=

of N

Size

+

+

2

=

+

1/2th item 3.

2

(3)/4

24th item,

hence Q1

3×96/4

=

hence Q3

4,

is

–

1 =

is

of 48th item

4

= 95

3(N+1)th item

of 72th item

Size

SkB =

¼th item

of

Size Med.

+

= 98/2 Hence

–

2

= =

0/2

72th item 4  48th item. = 3

median

=

=

=

0.

Karl Pearsons Skewness This method of measuring skewness, also known as Pearsonian coefficient of  skewness, was suggested by Karl Pearson, a great British biometrician and statistician. It is based upon the difference between mean and mode, this difference is divided by standard deviation to give a relative measure the formula thus becomes:

Skp = Skp =

median Karl

– Pearson‟s

mode/standard coefficient

deviation

of

skewness.   skewness.

There is no limit to this measure in theory and this is a slight drawback. But in practice the value given by this formula is rarely very high and usually lies between ± 1. When a distribution is symmetrical, the values of mean, median and mode coincide and, therefore the coefficient of skewness will be zero. When a distribution is positively skewed, the coefficient of skewness shall have plus sign and when it is negatively skid, the coefficient of skewness shall have minus sign. The degree of skewness shall be obtained by the numeral value. Say, 0.8 or 0.2 etc. thus this formula given both the direction as well as the extent of skinniness. The above method of measuring skewness cannot be used where mode is ill defined; however, in moderately skewed distribution the averages have the following relationship:

Mode

=

3

median

–

2

mean

And therefore, if this value of mode is substituted in the above formula we arrive at another formula for finding out skewness.

Skp = [X  – (3 med. - 2X)]/ σ = X  – 3 med.

/ σ = 2 X = 3 (X- med.)/ σ

Theoretically value of of thisskewness coefficientobtained varies between practice ±it 1 is. 3; however, rare that the the coefficient by the ± above methodinexceeds data;   Illustration: calculate Karl Pearson‟s coefficient of skewness from the following data;

Profits (\$ 0.1 million)

No. of Cos.

Profits (\$ 0.1 million)

No. of Cos.

70-80

12

110-120

50

80-90

18

120-130

45

90-100

35

130-140

30

100-110

42

140-150

8

method   Solution: calculation of Coeff. of skewness by Karl Pearson‟s method

Profits (\$ 0.1 million)

m.p.m

f

(m – 115)/10d

fd

fd

70-80

75

12

-4

-48

192

80-90

85

18

-3

-54

162

90-100

95

35

-2

-70

140

100-110

105

42

-1

-42

42

110-120 120-130

115 125

50 45

0 1

0 45

0 45

130-140

135

30

2

60

120

145

8

3

24

72

Coeff.

of

N = 240  =

skewness

Σ = fd = -85  Σ = fd = 77  –  Mode/σ

Mean

Mean: ‾X  ‾X

= A = Σfd/N × I = 115 – 85/240 × 10 =115  – 3.54 = 111.46

Mode:

by

Mode L

=

inspection

= 110,

Therefore,

mode

L ∆1 =

Mode

|50 = 110

Standard deviation σ

lies

+ 42|

+

8/8

in

∆1 =

8, +

5

/ ∆2 = ×

∆1 +

|50 10

=

–

120.

∆2 ×

class 110

the

-

45|

110

+

=

5,

I

6.15

+

=

10

116.15

= √Σfd2 /N  –  [Σfd/N]2 × I = √773/240 – [85/240]2 × 10

Coeff. of Sk = 111.46 – 116.15/17.795 = -4.69/17.595 = - 0.266

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