MEASURES OF CENTRAL TENDENCY.ppt

8/19/2019 MEASURES OF CENTRAL TENDENCY.ppt

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MEASURES OF CENTRAL TENDENCY

By JEMIMAH NINSIIMA


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Introduction• One of the most important objectives ofstatistical analysis is to get one single value thatdescribes the characteristic of the entire mass of

data• Such a value is called the central value or anaverage or the expected value of the variable

• The word average is commonly used in day to dayconversation

• Average is defined as attempt to find a singlefigure to describe whole of figures


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Objectives of averaging• To get single value that describes

the characteristic of the entire

group• Measures of central value by

condensing the mass of data in one

single enable us to get a bird !s eyeview of the entire data


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MEASURES OF

CENTRAL TENDENCY• Measures of central tendency aremeasures of the location of the middle orthe center of a distribution"

• There are a number of measures ofcentral tendency and these include# meanthe median the mode


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Measures of $entral

tendencyA good Measure of $entral tendency should havethe following characteristics

It should be easy to calculate and understand

It should be uni%ue and exist at all times It should consider all observations It should not be affected by extreme values It should be suitable for further mathematical

manipulation


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Mean • This is the summation of allobservations divided by the number

of observations in the sample


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Mean For Ungrouped dataIt i gi!en "y # $ % #i

N

$#& '#( ')*'#n

• N


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MeanFor +rouped data

• The mean for grouped data is given

by& ' ( f&i

(f


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Mean

advantages)disadvantages• Advantages

* It summari+es the entire distribution

* It could be processed further into thestandard distribution

* It is unbiased)meaning it always gives usthe population mean ,


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Mean -isadvantages• It may be some distance from the majorityof observations

• $an be misleading

• It is approximated for grouped data

• Sometimes the figure obtained is notanywhere in the distribution"

• $an give fractional values even for ungroupeddata


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,ropertie o- t.e Mean."The product of the mean and numberof the values on which the mean is

based is e%ual to the sum of all givenvalue e"g" if the variables are /012and we substitute these by the mean

the total is 34


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,ropertie o- t.e Mean3"The algebras sum of the deviations of

the values from the arithmetic mean is

e%ual to +ero i"e" ( 5x6 x7 ' 8/" The sum of the s%uares of deviations

from the mean is the least


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Median• The median conveys the notion ofbeing the middle most value with in

the data distribution


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Median• 9or :n grouped data determined byfirst arranging the data in order of

magnitude and then selecting themiddle observation e"g" the medianfor the value ; .8 . / and 0"


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Median/+rouped dataMedia ' )3 * $fbm cm fm

?here

• is the total number of observations• $9bm is the cumulative fre%uency of the class

below the median class• $m is the class interval of the median class


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Advantages)disadvantage

s of the median• Advantages@* Simple to calculate#

* It is representative of entire distribution#

* It is uni%ue and representative of an actualfigure in the distribution#

Diad!antage@

* It cannot be subjected to further processing


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Advantages)disadvantage

s of the median• Advantages@* Simple to calculate#

* It is representative of entire distribution#

* It is uni%ue and representative of an actualfigure in the distribution#

Diad!antage@

* It cannot be subjected to further processing


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Mode • The Mode is the most common valuein a given range of data

:ngrouped data"In the following observations /0 43

42 42 0 18" the mode is 42


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Mode +rouped data•


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Advantages)disadvantages

• Advantages@• It is simple

• :seful for %ualitative data say the mosthandsome man#

• -isadvantages@* $annot be called unbiased

* $annot be used to reconstruct the distribution

* $an not be further processed

* Some distributions are bimodal


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E0er1ie$lass 9re%uency• /"8 * /1"1 • /1"; * /2"0 1

• /2" * 4."/ 34• 4."4 * 4/". 1• 4/"3 * 44"2 3• 40"8 * 4"1 4

Co2pute t.e 2ean3 2ode and 2edian -or t.ea"o!e ditri"ution o- ditan1e to and -ro24or5 in 6i7o2eter -or UMI e2p7oyee*

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MEASURES OF CENTRAL TENDENCY.ppt