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Measures of Central Tendency

               

 A measure of central tendency is a value that can represent a set of data or group of values. It is used to describe a group of values since it is the most representative of all the values in a distribution. It is a convenient point to use as a reference in looking at individual scores or values. When comparing two or more groups of values, their measures of central tendency are usually compared (Caintic & Cruz, 2008, p.56).

Commonly Used Measures of Central Tendency

1.       Mean: the average value of a distribution or an array

2.       Median: the value at the middle of a distribution or an array

3.       Mode: the most typical value of a distribution or an array

Mean

                The mean is the average of the values in the sample. The sum of all the values in a set of data is divided by the total number of values in the set. It is the most stable measure of central tendency although it can easily be affected by extreme values. Its value need not be among the values in the set.

                There are several methods of finding the mean. The mean of an ungrouped data is given by the formula   

X = (X1+X2+...+Xn)
/ n

Where:

X = sample mean

Xi= values in the set

n = sample size, total number of values in the set

Example 1: The following numbers are the number of minutes for a customer to be served by the cashier in JDV convenience store (in an ungrouped data set): 2, 2, 4, 5, 4, 6, 5, 6, 5, 3, 3, and 3.

 

Calculate the mean of the sample.

Solution:

(2+2+4+5+4+6+5+6+5+3+3+3)/12

= 48/12
= 4

Interpretation: JDV convenience store’s cashier serves her customers for 4 minutes / transaction

UNGROUPED FREQUENCY DISTRIBUTION

                When the data are in the form of an ungrouped frequency distribution where each different value of X has a frequency f, then the samples has to be calculated using the following equation:

X= (f1X1+ f2X2+ ....+fnXn)/(f1+ f2...+fn) or
X=
ΣfiXi/n

where X1,X2 …. and Xn are values of X in the set of values, n is the number of values, and f1, f2 … and fn are their respective frequencies.

Time (mins.) Xi

Frequency fi

fiXi

32

2

 

31

2

 

30

0

 

29

1

 

28

5

 

27

4

 

26

7

 

25

10

 

24

8

 

23

1

 

22

1

 

21

2

 

22

0

 

 

Σ fi =

Σ fiXi =

Example 2: The time spent by IV-Maxwell students in answering the 1st periodical exam in English IV-B (Find the mean of the ungrouped frequency distribution).

 

 
Solution: X= ΣfiXi/n

X = ------------ =

Interpretation:

 

How do we calculate the mean of the scores when the set of scores is organized in a grouped frequency distribution? To get the mean of a set of values arranged in a grouped frequency distribution, use the following formulas:
X= Σfi(Xmid)i/N

   where:  fi                       = frequency of a class interval

                                                                Xmid             = midpoint of a class interval

                                                                N             = total number of scores

                                                                n              = number of class intervals

X = A.M. + {fidi)/N}i   where: A.M.         = assumed mean; midpoint of the class interval assumed to be containing

    the mean

fi                = frequency of a class interval

di                       = number of deviations of the class intervals from the assumed mean

i                 = size of the class intervals

N               =total number of scores

Example 3: Ages of persons with friendster account in Barangay Dalandanan (grouped frequency distribution).

Class Intervals

Frequency fi

Class Deviations di

fidi

55-59

1

 

 

50-54

3

 

 

45-49

8

 

 

40-44

11

 

 

35-39

16

 

 

30-34

17

 

 

25-29

16

 

 

20-24

10

 

 

15-19

9

 

 

10-14

4

 

 

5-9

1

 

 

 

Σ fi =

 

Σ fidi=

 

 

X = A.M. + {fidi)/N}i

 
 



X  =

 

Interpretation:

 

Median

                It is not the advisable to use the mean as the measure of central tendency when the data are skewed or most of the scores at one end of the distribution and very few are at the other end. The mean is easily influenced by extreme scores. In such cases, the median is preferred.

                The median is the value that falls in the middle position when the measurements are ranked from the lowest to the highest. It divides the ranked scores into two equal parts. To obtain the value of the median, rank the values from the smallest to the largest value. If the total number of scores is odd, locate the middle value. If the total number of scores is even, locate the two middle scores and take their average, that is,

 

Median = (highest valeu + lowest value)/2

Example 4: Calculate the median of the following ranked scores: 2, 3, 5, 6, 7, 9, 11, 13, 15, and 16

Solution: the middle scores are 7 and 9. The median therefore is (7+9)/2 = 8.

 

For a grouped frequency distribution, the median can be determined using the formula

Median = L + [{(n/2)-CFb}/Fi]i

where:   L    = lower real limit of the median class or the class interval that contains the median

                N   = total number of scores in the distribution

CFb= cumulative frequency of the class interval before reaching the median class starting from the lowest 

         class interval

Fi     = frequency in the class interval that contains the median

i     = size of the class interval

First, locate the class interval containing the median. Once it is located, the information needed to plug into the formula will be easily found. Then, the median can be calculated.

 

Example 5: Calculate the median of the following grouped frequency distribution.

Class interval

Frequency f

Cumulative Frequency

25-29

6

 

20-24

5

 

15-19

7

 

10-14

6

 

5-9

1

 

0-4

1

 

     

Solution:

First, locate the median class. It is the class interval containing the middle score. In this case, it is the 13th and 14th scores. They are included in the class interval 15 to 19.

                The median class is 15-19 and its lower real limit, L is 14.5, N = 26, CFb = 8, Fi = 7 and i = 5.

Median = ?

Mode

                The mode is the most frequently occurring score in a set of measurements. It is the easiest to determine measure of central tendency. In an ungrouped frequency distribution, the score with the highest frequency is the mode.

Example 6: Find the mode in the following distribution.

Score

Frequency

10

1

9

3

8

2

6

1

1

1

The most frequently occurring score in this set is ______?

 

In a grouped frequency distribution, the mode can be determined by locating the class interval with the highest frequency, the midpoint of this class interval is the mode of the distribution.

 

 Example 7: Consider the following grouped frequency distribution.

Class Interval

Frequency

60-69

5

50-59

6

40-49

7

30-39

2

20-29

4

10-19

3

0-9

2

 

 

The class interval with the highest frequency is 40-49. The midpoint of this class interval is 44.5. Therefore, the mode for this distribution is 44.5.

 

How can we tell the best measure of central tendency to be used in a particular case? The mean is the most commonly used measure of central tendency because it is the most stable. It is easily calculated without having to arrange the data in either ascending or descending order. However, it is easily affected by extreme scores like in skewed distributions. The term skew refers to the general shape of a graphed distribution. A distribution is skewed when most of the scores are at one end and very few are at the other end of the graph. Symmetrical graphs are not skewed. For skewed distributions, the median is the best measure of central tendency to represent the set of scores. On the other hand, since the mode is the most easily determined measure of central tendency, it is most appropriate to use when a quick and rough estimate of the most typical case is desired. However, it is rarely used since it is the most unstable measure of central tendency aside from the fact that there can be more than one mode in asset of scores (Caintic & Cruz, 2008, p.63).

 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


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