TOPIC:MEASURES OF CENTRAL TENDENCY-MEAN,MEDIAN , MODE AND    COMPUTATION


                                                             

MEASURES OF CENTRAL TENDENCY
The three most commonly used measures of central tendency are the mean,the median and the mode.
THE MEAN(M)
The mean of a distribution is commonly understood as the arithmetic average.It is perhaps the most familiar,most frequently used and well understood average.
The mean of a set of observations or scores is obtained by dividing the sum of all the values by the total numbers of values.
MEAN FOR UNGROUPED DATA
The formula for finding the mean for ungrouped data is
M=∑X/N
in which
 M=Mean
∑=sum of
X=scores in distribution
N=total number of scores
Eg:calculate the mean from the following data 3,5,10,7,8,10
Mean=sum of obs ervations /no.of observations
3+5+10+7+8+12/6
45/6=7.6
DEFINITION FOR FREQUENCY DATA
For a frequency data if x1,x2,x3……..xn are ‘n’ observations or middle values of ‘n’ class with the corresponding frequencies f1,f2,f3……fn then AM is given by
AM=∑fx/∑f
Where N=∑f=Total frequency
Eg:calculate the AM of the following data
Class 0-4 4-8 8-12 12-16
Frequency 1 4 3 2
Solution:
Class F Midvalues(x) fx
0-4 1 2 2
4-8 4 6 24
8-12 3 10 30
12-16 2 14 28
Total 10 84
AM=∑fx/N
84/10=8.4
THE MEADIAN
Median is defined as the middlemost observation when the observations are arranged in ascending or descending order of magnitude.Median is denoted by M.
DEFINITION FOR A RAW DATA
For a raw data if three are odd number of observations,there will be only one middle value and it will be the median.That means, if there are n observations arranged in order of their magnitude,the size of n+2/2 th observations the average of two middle values will be the median.That means,median will be the average of n/2 th and (n+1/2) th observations.
Eg 1. :find the median height from the following heights(cms)of 9 soldiers.
160,164,171,175,178,179,179,180,181
Solution:
Step1:heights are arranged in ascending order
          160,164,171,175,178,179,179,180
Step2:position of median =n+1/2 is calculated.It 9+1/2=5
Step3:median is identified (5th value)
M=178cms
It is to be noted that n+1/2 may be a fraction,in which case,median is found as follows.
       Eg 2 :find the median weight from the following weights(in kgs)of soldiers.
75,71,73,70,74,80,85,86,79.
Step 1:weights are arranged in ascending order
                       70,71,73,75,79,80,85,86
Step 2:position of the median n+1/2
            10+1/2=5 ½ (5.5)is calculated.
Step 3:Median is found.It is the mean of values at    
              5th and 6th positions and so
              M=75+79/2 =77kgs.
Eg 3:find median
Hight(cms) 160 164 170 173 178 180 182
No.of soldiers 1 2 10 22 19 14 2
Solution :
Step 1:Heights are in ascending order.Cumulative            
Frequencies(cf)are found (They help to                                                                                                                          know the values at different positions).                          

Height(cms)
No.of soldires Cf(cumulative frequency)
160 1 1
164 2 3
170 10 13
173 22 35
178 19 54
180 14 68
182 2 70
Total 70
Step 2:position of median n+1/2
70+1/2=35 ½ is calculated.
Step 3:Median is identified as the average of the
             values at the positions 35 and 36.The values                                                                                       are 173 and 178 respectively.
             M=173+178/2=175.5cm.
MEDIAN FOR GROUPED DATA
In case of grouped data,we first prepare the cumulative frequency distribution and median is calculated using the formula m=l+(N/2 -F)/Fm )*i
In which ‘l’=exact lowerlimit of the medianclass.
N/2=One half the total number of scores
F=cumulative frequency up to the median class
Fm=frequency of median class
i=width of class interval in which the median lies
Eg:calculate median for the following data.
Class 0-5 5-10 10-15 15-20 20-25
F 5 10 15 12 8
solution
Class f Cf
0-5 5 5
5-10 10 15(5+10)
10-15 15 30(5+10+15)
15-20 12 42(15+12)
20-25 8 50(42+8)
Total 50

M=l+(N/2-F)/fm)*i
Median class is 10-15
Here l =10
N/2=50/2=25
i=5
F=15
fm=15
 M=10+(25-15)/15*5
= 10+10*5/15
=10+3.3333
=13.33                              
THE MODE
The mode is defind as the most frequently occuring score in a distribution.If there is only one value which occures a maximum number of times,then the distribution is said to have one mode or to be unimodal.
In some distributions there may be more than one mode.A two mode distribution is bimodal;more than two,multimodal.Mode is denoted by ‘Z’ or ‘Mo’.
MODE FOR UNGROUPED DATA
Eg:9,10,11,16,18,18,19
Mode=18
MODE FOR GROUPED DATA
MODE=3median-2mean
COMPUTATION-MEAN,MEDIAN AND MODE
Class interval Actual class Midpoint(x) F cf Fx
140-144 139.5-144.5 142 1 1 142
145-149 144.5-149.5 147 3 4 441
150-154 149.5-154.5 152 2 6 304
155-159 154.5-159.5 157 4 10 628
160-164 159.5-164.5 162 4 14 648
165-169 164.5-169.5 167 6 20(F) 1002
170-174 169.5-174.5 172 10(fm) 30 1720
175-179 174.5-179.5 177 8 38 1416
180-184 179.5-184.5 182 5 43 910
185-189 184.5-189.5 187 4 47 748
190-194 189.5-194.5 192 2 49 384
195-199 194.5-199.5 197 1 50 197
N=50 8540


1.MEAN

= MEAN=∑FX/N
        8540/50=170.8
Fx=8540  N=50
2.MEDIAN=l+(N/2-F/fm)*I                                  l=169.5
169.5+(25-20)/10*5                                            N/2=25
=169.5+5/10*5                                              F=20,i=5
=169.5+2.5=172                                                   fm=10  
3.MODE
=3median-2mean
=3*172-2*170.8
=516-341.6
=174.4
ADVANTAGES & DIS ADVANTAGES OF MEAN
ADVANTAGES DISADVANTAGES
1.It is easy to understand 1.It is unduly affected by extreme items.One greatest item may pull up the mean of the set to such an extent that its representative character is questioned
2.It is easy to calculate 2.theoratically,it cannot be calculated for open end data.
3.It utilizes the entire data in the group 3.Itcannot be found graphically
4.Aggregate can be calculated if the number of items and the mean are known. 4.It is not defind to deal with qualities
5.It is ameanable to algibraic manipulation 5.In the absence of actual data it can mislead
6.It provides a good comparison
7.It is most widely used and accurate measure of central tendency
8.It is very much useful in day to day activities
9.It is based on all the items are considered for its computation
10.It is most widely used and accurate measure of central tendency
WHEN TO USE MEAN
1. We compute mean for the given data when a reliable and accurate measure of central tendency is needed.
2. We use mean for the computation of various statistics like standard deviation ,co-efficient of correlation etc.
3. We calculate mean when the series have no extreme items and each score carries equal weight in determining the central tendency.
ADVANTAGES AND DIS ADVANTAGES OF MEDIAN
ADVANTAGES DIS ADVANTAGES
1.It is simple to understand and easy to calculate. 1.It is not based on the magnitudes of all the items.It is a positional measure.It is the value of the middle most item.
2.It is not unduly affected by extreme items 2.It cannot be algibraically manipulated.for eg.the median of the combined set cannot be found from the medians and the sizes of the individual sets alone.
3.It can be calculated for open end data. 3.It is difficult to calculate when there are large number of items which are to be arrangrd in order of magnitude.
4.It can be determined graphically. 4.It does not have sampling stability.It varies more markedly than AM from sample to sample although all the samples are from one and the same population.
5.It can be used to deal with qualitative data. 5.Its use is lesser than that of AM.
WHEN TO USE THE MEDIAN
1. Median is used when the exact midpoint of the distribution is desired.
2. We use median when a series contains extreme scores.
3. Median is most reliable measure in case of an open end distribution(incomplete distribution 80 and above or 20 and below etc.)
4. We compute median when we have suitable graphs like frequency curve,polygon etc.
ADVANTAGES AND DIS ADVANTAGES OF MODE
ADVANTAGES DIS ADVANTAGES
1.It is simple to understand and easy to calculate 1.It is not rigidly defined.It is obvious when different books on statistics are referred.
2.Mode is not unduly affected by extreme items. 2.It is not based on all the items.It is a positional value.It is the value which is repeated more number of times than any other value.The frequencies of other values are considered but not their magnitudes.
3.It is the most typical or representative value in the sense that it has the greatest frequency dencity. 3.It cannot be algibraically manipulated.The mode of the combined set cannot be determined as in the case of AM.
4.It can be calculated for many open end data. 4.It is less stable than the AM.
5.It can be determind graphically.It is the x-co ordinate of the frequency curve. 5.It has very limited use.Modal wage,modal size of shoe,modal size of family etc.are determined.Consumer preferences are also dealt with.
6.It can be found for qualities also.The quality which is observed more often than any otherquality is the modal quality. 6.It is not capable of further mathamatical treatment.
WHEN TO USE MODE
1. Mode is used where quick and approximately measure of central tendency is desired.
2. Mode is used when we need to know the most often recurring score or value of the item in a series.
3. We compete mode when we have a graphical representation of the distribution.