Formulas of Central Tendency & Dispersion

Formulas For Measures of central Tendency & Dispersion Prep Prepar ared ed by Ifti Iftikh khar ar Ali Ali Msc Msc Econ Econom omic ics, s, Re Rese sear arch ch Meth Method odol olo oy y & Econometrics E!pert Pun"ab #ollee of E!cellence in #ommerce $amra

Arithmetic Mean Method%s ame

ature of Data 'nrouped Data (rouped Data

Direct Method Indirect or )hort*#ut Method Method of  )tep*De+iation Where Indi Indica cate tes s valu values es of the the vari variab able le Indicates number of values of

. .

Indicates Indicat es frequency of dierent groups. Indicates assumed mean. Indicates deviation from

Step-deviation and

i.e,

Indicates common divisor

Indicates size of class or class interval in case of grouped data. Summation or addition.

Median Median from 'nrouped Data

Median  !alue of item ote Another simple method to calculate median for unrouped data is as follo-s ". #or #or odd values values $ust pic% the the central central most value value that &ill be the the median. '. #or #or even values values $ust ad up the the t&o central central most values values and divide divide it by ' the ans&er &ill be the median. Median from (rouped Data

 

Where  (o&er class boundary of the model class  #requency #requency of the median class  )umber of values or total frequency frequency  *umulative frequency frequency of the class preceding the median

class  *lass interval size of the model class

Mode "+age

Mode from 'nrouped Data Mode is calculated from ungrouped data by inspecting the given data. We pic% out that value &hich occur the greatest numbers of times in the data. Mode from (rouped Data When frequency distribution &ith equal class interval sizes, the class &hich has maimum frequency is called model class.

r

Where  (o&er class boundary of the model class  #requency of the model class /maimum frequency0  #requency preceding the model class frequency  #requency follo&ing the model class frequency  *lass interval size of the model class Mode from Discrete Data When the data follo&s discrete set of values, the mode may be found by inspection. Mode is the value of  X  corresponding to the maimum frequency.

(eometric Mean For 'nrouped Data

For (rouped Data

.armonic Mean For 'nrouped Data

For (rouped Data

/eihted Arithmetic Mean Where1 Stands for &eighted arithmetic mean. Stands for values of the items and Stands for &eight of the item

0uartiles 0uartile for Indi+idual 1bser+ations 2'nrouped Data3

'+age

0uartile for a Fre4uency Distribution 2Discrete Data3

0uartile for (rouped Fre4uency Distribution

Deciles Deciles for Indi+idual 1bser+ations 2'nrouped Data3

0uartile for a Fre4uency Distribution 2Discrete Data3

0uartile for (rouped Fre4uency Distribution

Percentiles

2+age

Measures of Dispersion ". The Rane For 'nrouped Data Rane 5 R 5  X m

− X o

Where  X m  the largest value.  X o  the smallest value. For (rouped Data 3ange  3  4pper class boundary of the highest class 5 lo&er class boundary of  the lo&est class or

3ange  3  *lass Mar%s /60 of the highest class 5 *lass Mar%s of the lo&est class

#oe6cient 1f Rane #oe6cient of Rane  

 X m − X o  X m + X o

78 )emi Inter 0uartile Rane or 0uartile De+iation )8I808R 5 08D 5

Q3 − Q1 2

Where Q1  #irst, (o&er quartile

Q3   7hird, 4pper quartile

#oe6cient 1f 0uartile De+iation #oe6cient of 08D 5

Q3 − Q1 Q3 + Q1

Where Q1  #irst, (o&er quartile

Q3   7hird, 4pper quartile

98Mean De+iation or A+erae De+iation :8 Mean De+iation From Mean For 'nrouped Data M8D 5

∑  X − X  n

r

M8D 5

∑  X − Mean n

For (rouped Data M8D 5

∑  f X − X  ∑  f   r

M8D 5

∑  f

X

− Mean

∑  f  

#oe6cient 1f Mean De+iation From Mean #oe6cient of M8D from Mean 5

 Mean Deviation From Mean  Mean

8+age

r #oe6cient of M8D from Mean 5

 M .D From X   X 

78 Mean De+iation From Median For 'nrouped Data M8D 5

∑  X − Median n

For (rouped Data M8D 5

∑  f

X

− Median

∑  f  

#oe6cient 1f Mean De+iation From Median #oe6cient of M8D from Median 5

 Mean Deviation From Median  Median

98 Mean De+iation From Mode For 'nrouped Data M8D 5 9 6-Mode n For (rouped Data

#oe6cient 1f Mean De+iation From Mode

;8)tandard De+iation 2)3 Methods of )tandard De+iation

I. II. III.

:irect Method Short *ut Method *oding Method or Step-:eviation Method

:8 Direct Method For 'nrouped Data )8D 5 ) 5

∑  X n

2

 X   −  ∑ ÷ ÷  n  

2

or )8D 5 ) 5

∑ (  X − X  )

 2

n

For (rouped Data

;+age

)8D 5 ) 5

∑  fX ∑  f

)8D 5 ) 5

∑  f ( X − X  ) ∑   f  

2

 fX   −  ∑ ÷ ÷  ∑ f   

2

 2

78 )hort #ut Method For 'nrouped Data

∑ D

)8D 5 ) 5

 D  −  ∑ ÷ ÷  n  

2

n

2

Where : 6 5

< For (rouped Data

∑  fD ∑  f

)8D 5 ) 5

2

 fD  −  ∑ ÷ ÷  ∑ f   

2

98 #odin Method or )tep*De+iation Method For 'nrouped Data

)8D 5 ) 5 h × u

=

 X

−A h

or 

∑u

 u  −  ∑ ÷ ÷  n  

2

n

2

Where

D h

For (rouped Data

)8D 5 ) 5 h ×

∑ ∑  f

 fu  −  ∑ ÷ ÷  ∑ f   

2

 fu

2

#oe6cient 1f )tandard De+iation S .D

#oe6cient of )8D 5

 X 

< =ariance 2 S  3 2

Methods of =ariance ". :irect Method '. Short *ut Method 2. *oding Method or Step-:eviation Method :8 Direct Method For 'nrouped Data =ar2>3 5 S 2  5

2

=ar2>3 5 S   5

∑ X n

2

 X   −  ∑ ÷ ÷  n  

∑ (  X − X  )

2

 2

n

For (rouped Data

=+age

=ar2>3 5 S 2  5

=ar2>3 5 S 2  5

∑ ∑  f

 fX   −  ∑ ÷ ÷  ∑ f   

∑

− X  )

 fX 2

 f ( X

2

 2

∑  f  

78 )hort #ut Method For 'nrouped Data

∑ D =ar2>3 5 S   5

 D  −  ∑ ÷÷  n  

2

2

n

2

Where :

65< For (rouped Data

∑  fD =ar2>3 5 S   5 ∑  f

2

2

 fD  −  ∑ ÷ ÷  ∑ f   

2

98 #odin Method or )tep*De+iation Method For 'nrouped Data

 u  u   ×  ∑ −  ∑ ÷ ÷  n  n    2

2

=ar2>3 5 S 2  5 h

u=

 X

−A h

or 

2

Where

D h

For (rouped Data

 ∑ fu  ∑ fu   × − ÷  ∑  f  ∑ f  ÷    2

2

2 =ar2>3 5 S   5 h

2

#oe6cient 1f =ariation 2#8=3 #oe6cient of =ariation 5 #8= 5

S .D  X 

×100

?8 #oe6cient of )ke-ness 2)$3 $arl Pearson%s #oe6cient of )ke-ness )$ 5

)$ 5

 Mean − Mode S .D 3

(  Mean − Median ) S .D

@o-ly%s 0uartile #oe6cient of )ke-ness )$ 5

Q3 + Q1 − 2Median Q3 − Q1

>+age