All the formulas related to statistical Topics Measures of central Tendency & Dispersion
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This is a lecture about central tendency of data of statistics i.e. mean, median, mode and the ways to calculate them
Central tendency
Central tendency, assessment
notes
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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 oy y & Econometrics E!pert Pun"ab #ollee of E!cellence in #ommerce $amra
Arithmetic Mean Method%s ame
ature of Data 'nrouped 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 dierent 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 'nrouped Data
Median !alue of item ote Another simple method to calculate median for unrouped 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 'nrouped 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 maimum frequency is called model class.
r
Where (o&er class boundary of the model class #requency of the model class /maimum 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 maimum frequency.
(eometric Mean For 'nrouped Data
For (rouped Data
.armonic Mean For 'nrouped Data
For (rouped Data
/eihted 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'nrouped Data3
'+age
0uartile for a Fre4uency Distribution 2Discrete Data3
0uartile for (rouped Fre4uency Distribution
Deciles Deciles for Indi+idual 1bser+ations 2'nrouped Data3
0uartile for a Fre4uency Distribution 2Discrete Data3
0uartile for (rouped Fre4uency Distribution
Percentiles
2+age
Measures of Dispersion ". The Rane For 'nrouped Data Rane 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 Rane #oe6cient of Rane
X m − X o X m + X o
78 )emi Inter 0uartile Rane 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+erae De+iation :8 Mean De+iation From Mean For 'nrouped 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 'nrouped 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 'nrouped 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 'nrouped 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 'nrouped 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 'nrouped 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 'nrouped 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 'nrouped 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 'nrouped 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