Measures of Central Tendency1

7/24/2019 Measures of Central Tendency1

1/24

NIFTMFM

Business Statistics

for Research


2/24

MEASURES

OFCENTRAL TENDENCY

SESSION 2


3/24

Summar StatisticsSummar Statistics describe characteristicsof the data set in two dierent ways:

Centra! Ten"enc is the middle point of afrequency distribution. Dis#ersionis spread of the data in a

distribution, that is, the extent to which theobservations are scattered.

Dispersion is contrasted with central tendency,and together they are the most usedproperties of distributions used for better"ecision ma$in%.


4/24

Summar

Statistics&Measures

Centra!Ten"enc

Mean

Me"ian

Mo"e

SummarStatistics

Dis#ersion

'ariance

Stan"ar"De(iation

Ran%e

Coe)cient of'ariation


5/24

Fre*uenc Distri+ution ,ra#hs-

Summar Measures Data sets with dierent

Mean but same Mode.

Data sets with same Meanand Dispersion but dierentMode/urtosis.

Data sets with same Meanbut dierent Dispersion.


6/24

Fre*uenc Distri+ution ,ra#hs-

S$e.ness S$e.e" Cur(es are not mirror images on both

sides of the vertical axis, as in Smmetrica!!ell "haped #Norma!$ curve.

%eft "&ewed 'ormal (ight "&ewed#'egative$ #"ymmetrical$ #)ositive$

*x. Mar&s in *asy *xam *x. "tores "ales *x. +ruits

"ale


7/24

Arithmetic Mean- Un%rou#e"

Data Sam#!e Arithmetic Mean is sum of the values divided

by the number of values.

x: observation values

n: number of observations /o#u!ation Mean

*xercise .- bservations: , 0, 0, 1, 2, 2,

"ample Mean 3 4/0 3 5.1 +or large no. of observations, data is 6rst grouped in

frequency distribution table and then Mean is

calculated.

x x

n=

1 2

1

/

N

N

i

i

X X XX N

N

=

+ + += =


8/24

Arithmetic Mean- ,rou#e" Data "ample Arithmetic Mean for %rou#e" "ata

f : frequency in each class

x : midpoint for each class in the sample

n : total number of observations in the sample "ample Arithmetic Mean for %rou#e" "atausin%

co"es

7ssign integer codes to each class midpoint

x8 : value of the midpoint assigned code 8

w : width of the class interval

u : code assigned to each class

f : frequency in each class

n : total number of observations in the sample

x f x

n

( )=

x x w u f

n

( )0

= +


9/24

9lass +requency 9lass +requency

-8.8;-8. - -5.8;-5. --

--.8;--. 2 -1.8;-1. t is highly in?uenced by extreme values. +orexample if values are near 2 and - value is58, then the mean


11/24

0ei%hte" Mean =he .ei%hte" mean enables us to calculate an average,

that ta&es into account the importance of each value to theoverall total.

w : weight of each observation

x : value of each observation *xample : 9alculation of "@)7 in '>+=

*xample: +ind 7verage "alary :

7rithmetic Mean 3 -888/43 0888 #>ncorrect *stimate$

Aeighted Mean 3 #-8888B-8C1888B8C5888B48$/18

3 408888/18 3 1-10 #9orrect *stimate$

x

w x

w

( )w =

Desi%nation

Sa!ar

Stren%th

Manager -8888 -8

*ngineer 1888 8"ta 5888 48

=otal -888 18


12/24

0ei%hte" Mean- E1ercise 23 7 9ompany with 4 branches is forecasting regional sales for

next year. !ranch -, with current yearly sales of (s. -4.n such cases, geometric mean is preferred

over arithmetic mean.

@eometric mean is used to show

multiplicative eects over time in compoundinterest and in?ation calculations. @eometric Mean 3 nth root of F#x-BxBx4BG.xn$ >t is li&e the area is the sameH


14/24

,eometric Mean *xample : +ind 7verage @rowth +actor over 5 years

Arithmetic Mean 3 #-.80C-.8


15/24

Me"ian =he median is a single value that is the 4mi""!e5 num+erin the data set.

9alculating Me"ian from Un%rou#e" "ata:

7rrange the data in ordered array.

>f n is odd, the median is the middle number.

>f n is even, the median is the average of the middle

numbers.

>n both cases, Median 3 #n C-$/th item in data array.

*xample-: 4, 2, 0, , -4, -, 4. Median is 2rth item : . *xample: 4, 2, 0, , -4, -, 4, 1.

Median is average of 2thand 5thitems : #C-4$/ 3 --.

A"(anta%e 6- Median is not as strongly in?uenced by

extreme values as Mean.


16/24

Me"ian for ,rou#e"

Data 9alculate the cumulative frequency +ind the median class J that contains the #nC-$/th

item =he median for grouped data is calculated as:

n : total number of items+ : 9umulative frequency of class preceding themedian class

fm:frequency of the median class

w : width of the class interval


17/24

*xercise .2

9lass +requency 9lass +requency

-8;-.5 < 18;1.5 5

8;.5 -5 08;0.5


18/24

*xercise .2c!ass Fre*uenc C

Fre*uencc!ass Fre*uenc C

Fre*uenc

-8J-.5 < < 18J1.5 5 676

8 J .5 -5 4 89:8;< 7= 2>2>>; =2

10;0-. 4

0;01. 8

00;


23/24

Com#arin% the Mean Me"ian an"

Mo"e

%eft "&ewed 'ormal (ight "&ewed#'egative$ #"ymmetrical$ #)ositive$

*x. Mar&s in *asy *xam *x. "tores "ales *x. +ruits "ale

"ymmetrical distribution have the same value for Mean,Median and Mode.

+or s&ewed distributions #left or right$, Median is oftenthe best choice for central location as Mode is in?uenced

by occurrence of a single value and Mean is pulled byextreme values.


24/24

Assi%nment 2

9lass -5;- 8;2 5; 48;42 45

+requency -8 4 2 2

=he ages of a sample of the students is:

#a$ 9ompute the Mean of the data.#b$ *stimate the Median value.#c$ 9alculate the Mode of the data.

Documents

Measures of Central Tendency1