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Measures of CentralTendency
Mean,Median and Mode
for Ungrouped Data
Basic Statistics
By: Noshielta Castillo
Rizza Maalac
Jaycon Malicdem
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Measures of CentralTendency
In laymans term, ameasure of central tendencyis an AVERAGE. It is a singlenumber of value which canbe considered typical in a set
of data as a whole.
or e!ample, in a class of"# students, the averageheight would be the typicalheight of the members of this
class as a whole.
Measures of central tendency are set of data by
identifying the central position within the set of
data and are numerical descriptive measures
which indicate or locate the center of a
distribution or data set.
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MEAN
Among the three measures of centraltendency, the mean is the most popular and
widely used. $he mean is e%ual to the sum of allthe values in the data set divided by the numberof values in the data set it is sometimes called thearithmetic mean.
If we compute the mean of the population, wecall it the parametric or population mean, denotedby (read mu).
If we get the mean of the sample, we call it thesample mean and it is denoted by (read x bar).
$he MEANof a set of values or measurements isthe sum of all the measurements divided by thenumber of measurements in the set.
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Mean for Ungrouped Data
Example 1:
&s. 'ulit collects the data on the ages of &athematicsteachers in 'anta Rosa 'chool, and her study yields thefollowing(
)* )+ * )- )+ ))"#
Solution:
= 35
Based on the computed mean, 38 is the average age of
Mathematics teachers in SRS.
For ungrouped or raw data, the mean has the following
formula.
where = mean
= sum of the measurements or values
n = number of measurements
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Your turn!
Mang John is a meat vendor. The following are his sales forthe past six days. Compute his daily mean sales.
Tuesday P !""#ednesday ! $""Thursday $ ""Friday % &""
'aturday () ""'unday (& %""
'olution*
= (, (""
The average daily sales of Mang John is P(,("".
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Example
-elow are maya/s sub0ets and the orresponding number
of units and grades she got for the previous grading
period. Compute her grade point average.
Subject Units rade
Filipino .1 !$
2nglish (. !
Mathematis (. !!
'iene (.! !3
'oial 'tudies .1 !$
T42 (.) !&
MP25 (.) !3
= !$.(
maya/s average grade is !$.(
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Your turn!
James obtained the following grades in his five sub0ets forthe seond grading period. Compute his grade point average.
'olution*
= !3.$3
James general average is !3.$3
Subject Units rade
Math (. 1"
2nglish (. !$
'iene (.! !!
Filipino ".1 !3
M6-78 (. !3
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ier"type #uestion
e!t are e!amples of /i0ert1type statements.
Respondents will choose the number which bestrepresents their feeling regarding the statements.ote that the statements are grouped accordingto a theme.
This is used if the researher wants to 9now the feelings oropinions of the respondents regarding any topi or issues of
interest.
!hoices
:'; 'trongly gree% :; gree
& :8; 8eutral
) :
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Students" persona# confidence in #earningStatistics
5 $ 3 % &
(. am sure that an learn 'tatistis
). thin9 an handle diffiult lessons in'tatistis.
&. an get good grades in 'tatistis.
Source: B.E. Blay, Elementary Statistics
-elow are the responses in the 4i9ert+type ofstatements above. The table below shows the meanresponses and their interpretation. >sing the formula foromputing the weighted mean, he9 the orretness of thegiven means on the table.
5 $ 3 % & Mean 'nterpretation
( &$ ( (! " ( %.(% gree
) (! %% &3 ! ( &.$ gree
& (! %! )! " ( &.!$ gree
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ie"type Mean$nterpretation
(." + (.31 + 'trongly
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Your turn!-elow is the result of the responses to the following 4i9ert+type statements . 'olve for the mean and give the
interpretation.
5 $ 3 % & Mean 'nterpretation
( && %1 )$ ( (
) & % &( " (
& &% ! )( " "
Students" perception on Statistics as asubject
5 $ 3 % &
(. thin9 'tatistis is a worthwhile, neessary
sub0et
). will use 'tatistis in many ways as aprofessional
&. /ll need a good understanding of 'tatistisfor my researh wor9
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%roperties of Mean
(. Mean an be alulated for any set of
numerial data, so it always exists.
). set of numerial data has one and only one
mean.
&. Mean is the most reliable measure of entraltendeny sine it ta9es into aount every item
in the set of data.
%. t is greatly affeted by extreme or deviant
values :outliers;
. t is used only if the data are interval or ratio.
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MED$AN
$he MED$AN& denoted &d, 2lace the value
order and 3nd the middle number or the mean of
the two middle numbers.
($ (3 (! (1 )" )( ))
Median
($ (3 (! (1 )" )( )) )&
Median
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Your turn!
Compute the median and interpret the result.(. n a survey of small businesses in Tondo, (" ba9eries
report the following numbers of employees*(, (%, (), (1, (&, (% (, (!, (&, (1.
). The random savings of )ndyear high shool studentsreveal the following urrent balanes in their ban9
aounts*
&. The following are the lifetimes of 1 lightbulbs in
thousands of hours.
'tudents - C < 2 F ? 5
Current -alanes P&%" &" %" "" &$" 3$" !"" 3%"
4ightbulb - C < 2 F ? 5
4ifetime (.( (.( (.) (.( (.% .1 .) (.) (.3
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%roperties of Median
(. Median is the sore or lass in the distributionwherein "@ of the sore fall below it and
another "@ lie.
). Median is not affeted by extreme or deviant
values.
&. Median is appropriate to use when there are
extreme or deviant values.
%. Median is used when the data are ordinal.
. Median exists in both Auantitative or Aualitative
data.
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M'DE
$he M'DE& denoted &o&4r modal value, place
the numbers in value order then count on how many
of each number.
2xamples*
Find the Mode.
(. The ages of five students are* (3, (!, )&, )", and (1
). The following are the desriptive evaluations of
teahers* B', ', B', B',
&. The grades of five students are * %.", &., %.", &., and
(."%. The weights of five boys in pounds are* ((3, )(!, )&&,
()", and ((3
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%roperties
(. t is used when you want to find the valuewhih ours most often.
). t is a Aui9 approximation of the average.
&. t is an inspetion average.
%. t is the most unreliable among the three
measures of entral tendeny beause its
value is undefined in some observations.
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Your turn!
(ind the mode and interpret it.
(. The following table shows the freAueny of errorsommitted by (" typists per minute.
). random sample of ! mango trees reveals thefollowing number of fruits they yield
&. The following are the sores of 1 students in aMathematis AuiD.* (), (, (), !, 3, (, (1, )%, (&
Mango Tree - C < 2 F ? 5
8o. of fruits !" 3" !" 1" !) !) 1" !)
Typists - C < 2 F ? 5 J
8o. of errors per min. & & 3 ) ! ! % 3 ("
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(eferrences)
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