MEASURE OF CENTRAL TENDENCY, MODE, MEDIAN, ARITHMETIC MEAN, ITS TYPES WITH EXAMPLES, ADVANTAGES AND DISADVANTAGES

 

INTRODUCTION:

·         “A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data”.

·         The term central tendency dates from the late 1920s.

·         A central tendency can be calculated for either a finite set of values or for a theoretical distribution, such as the normal distribution.

 MEASURES OF CENTRAL TENDENCY:

There are three measures of central tendency which are as follows:

·       Arithmetic mean

·       Median

·       Mode

·       Geometric mean 

v ARITHMETIC MEAN:

ü  “ In statistics, the arithmetic mean is the sum of a collection of numbers divided by the   number of numbers in the collection”.

ü  The collection is often a set of results of an experiment, or a set of results from a survey.

 PROPERTIES:

§  Arithmetic mean is rigidly defined.

§  It is based on all observations of series.

§  It is easy to calculate and simple to understand.

§  It is also capable of further algebraic treatments.

§  It is free from extreme values and can not be affected by extreme values.

FORMULA:


ADVANTAGES

ü  It is easy to understand.

ü  It is easy to calculate.

ü  It is use to calculate standard deviation.

ü  It is based on all observations.

ü  Stable in the sampling sense.

ü  It is use to calculate mean deviation.

ü  It has definite value.

 DISADVANTAGES:

ü  If the of items in a series is very small , the extreme items effect the arithmetic mean. 

ü  The average can not be evaluated even if one item in a given series is not known.

ü  In an organization , the production rate of four of its member per hour is A – 30 , B – 30 ,     C – 30 and D – 46 . The average comes out of 34 ; hence , the average 34 can not be fixed as the production standard because many of employes can not achieve this.

ü  It can not be located by mere inspection . It  requires computation. 

·       MEDIAN:

“ The median is the size of middle – most item when the items from an array”.

 FORMULA:



EXAMPLE 1:

Find the  median for the following list of values:

13, 18, 13, 14, 13, 16, 14, 21, 13

SOLUTION:

The median is the middle value, so I'll have to rewrite the list in order:

13, 13, 13, 13, 14, 14, 16, 18, 21

There are nine numbers in the list, so the middle one will be the (9 + 1) ÷ 2 = 10 ÷ 2 = 5th number:

13, 13, 13, 13, 14, 14, 16, 18, 21

So the median is 14. 

EXAMPLE 2:

Find the  median for the following list of values:

           8, 9, 10, 10, 10, 11, 11, 11, 12, 13

SOLUTION:

The median is the middle value. In a list of ten values, that will be the (10 + 1) ÷ 2 = 5.5th value; that is, I'll need to average the fifth and sixth numbers to find the median:

(10 + 11) ÷ 2 = 21 ÷ 2 = 10.5 

EXAMPLE 3:

The Doran family has 5 children, aged 9, 12, 7, 16 and 13. What is the age of the middle child?

SOLUTION:

Ordering the childrens' ages from least to greatest, we get:

                      7,  9,  12,  13,  16

The age of the middle child is the middlemost number in the data set, which is 12.

 EXAMPLE 4:

The Jameson family drove through 7 states on their summer vacation. Gasoline prices varied from state to state. What is the median gasoline price?

   $1.79,  $1.61,  $1.96,  $2.09,  $1.84,  $1,75,  $2.11

SOLUTION:

Ordering the data from least to greatest, we get:

$1.61,  $1.75,  $1.79,  $1.84,  $1.96,  $2.09,  $2.11

The median gasoline price is $1.84.

 EXAMPLE 5:

During the first marking period, Nicole's math quiz scores were 90, 92, 93, 88, 95, 88, 97, 87, and 98. What was the median quiz score?

SOLUTION:

Ordering the data from least to greatest, we get:

The median quiz score was 92.87,  88,  88,  90,  92,  93,  95,  96,  98

The median quiz score is 92.

 ADVANTAGES: 

ü The median is easily calculated and can be understood easily.

ü The median can be evaluated even if data are incomplete.

ü The median may be located when items in a series can not be measured quantitatively. 

DISADVANTAGES: 

ü   If there is a high degree of variation among the data set , median can not be viewed as representative.

ü  It can not be considered as representative when there are few items.

ü   It is not capable of further algebraic treatment.

·       MODE:

“ The value of variables that commonly occurs within a given area is called mode”.

TYPES:

There are three types of mode which are as follows:

Ø  Unimodal

Ø  Bimodal

Ø  Multimodal 

Ø UNIMODAL:

If the set of information contains only one mode , then it is said to be unimodal.

Ø  BIMODAL:

If the set of information contains two modes , then it is said to be bimodal.

Ø MULTIMODAL:

If the set of information contains more than two modes  , then it is said to be multimodal.

EXAMPLE 1:

The number of points scored in a series of football games is listed below. Which score occurred most often?

 7,  13,  18,  24,  9,  3,  18

 SOLUTION:

 Ordering the scores from least to greatest, we get:

           3,  7,  9,  13,  18,  18,  24

 The score which occurs most often is 18.

 EXAMPLE 2:

 The following is the number of problems that Ms. Matty assigned for homework on 10 different days. What is the mode?

  8,  11,  9,  14,  9,  15,  18,  6,  9,  10

SOLUTION:

 Ordering the data from least to greatest, we get:

           6,  8,  9,  9,  9,  10,  11  14,  15,  18

           The mode is 9.

 EXAMPLE 3:

      In a crash test, 11 cars were tested to determine what impact speed was required to obtain

        minimal bumper damage. Find the mode of the speeds given in miles per hour below:

        24,  15,  18,  20,  18,  22,  24,  26,  18,  26,  24

        SOLUTION:

        Ordering the data from least to greatest, we get:

         15,  18,  18,  18,  20,  22,  24,  24,  24,  26,  26

         Since both 18 and 24 occur three times, the modes are 18 and 24 miles per hour.

 EXAMPLE 4:

        A marathon race was completed by 5 participants. What is the mode of these times given in

        hours?

         2.7 hr,  8.3 hr,  3.5 hr,  5.1 hr,  4.9 hr

     SOLUTION:

         Ordering the data from least to greatest, we get:

          2.7,  3.5,  4.9,  5.1,  8.3

         Since each value occurs only once in the data set, there is no mode for this set of data.

 ADVANTAGES:

§    It is simple.

§     It is useful in the study of popular sizes.

§     It is not affected by extreme values and can be calculated even if extreme values are

   unknown . 

DISADVANTAGES:

§   It is not suitable for mathmetical treatment.

§   A distribution may be bimodal or multimodal.

§   It is not well defined.

§   Sometimes it is not possible to locate it properly.

·        GEOMETRIC MEAN:

“ The geometric mean is a type of mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values”.

 FORMULA:

  Geometric Mean = ((X1)(X2)(X3)........(XN))1/N

     where

       X = Individual score N = Sample size (Number of scores)

    EXAMPLE 1:

     Find the geometric mean of 10,12 ,18 ,20.

SOLUTION:

Geometric Mean = ((X1)(X2)(X3)........(XN))1/N

       By putting values , we get:

       (10 * 12 * 18 * 20) 1/4  = 10800

         EXAMPLE 2:

      Find geometric mean of  6,9,7,2,3.

        SOLUTION:

   Geometric Mean = ((X1)(X2)(X3)........(XN))1/N

          By putting values , we get:

         (6 * 9 * 7 * 2 * 3) 1/5 

                               = 453.6

   EXAMPLE 3:

         Find the geometric mean of 1 , 4 , 6 , 9 , 0, 8.

            SOLUTION:

           Geometric Mean = ((X1)(X2)(X3)........(XN))1/N

           By putting values , we get:

           (1 * 4 * 6 * 9 * 4 * 8 ) 1/6

                              =  1152

 EXAMPLE 4:

         Find the geometric mean of 1, 2 , 3.

             SOLUTION:  

              Geometric Mean = ((X1)(X2)(X3)........(XN))1/N

                     By putting values , we get:

              (1 * 2 * 3) 1/3

                    = 1.99

      ADVANTAGES:

§      It is use to calculate the rate of change.

§      It is use in certain cases of averaging and percentages.

§      It is use in evaluation of certain index.


Comments

Post a Comment

Popular posts from this blog

life truth (zindagi ka such)

Image
  Life is so unpredictable. It is full of happiness and sad moments. Sometimes we are so hurt but we can not tell anyone else about our worries. We have lots of relatives, friends, siblings but still alone and you think what is purpose of your creation. You spent your whole life in making happy others peoples but your inner heart is still empty. No one tries to understand you, your feelings, your emotions nothing. You know everything, others behaviours, attitude everything but trying to show everybody that you are happy which is not true. You are lying to yourself also along with others by ignoring your so much little happiness, memories. No one bother about your emotions memories feeling.   No one have time to talk with you. You think everytime to make smile your loved ones but they never bother. We want to hear that we are important in their lives and they love us but nothing in response. A time come when we are preparing ourselves no to bother or expect these things but our...
Read more

PHENYLKETONEURIA, SYMPTOMS, DIAGNOSTIC TESTS, PHARMACOGENOMICS, TREATMENTS, MEDICATIONS

Image
  DEFINITION : “ Phenylketonuria (PKU)   is an   inborn error of metabolism   involving impaired metabolism of   phenylalanine , one of the   amino acids . Phenylketonuria is caused by absent or virtually absent   phenylalanine hydroxylase (PAH)   enzyme activity . ”   EXPLAINATION : ·          Protein-rich foods or the sweetener   aspartame   can act as poisons for people with phenylketonuria. ·          The role of PAH is to break down excess phenylalanine from food. ·           Phenylalanine is a necessary part of the human diet and is naturally present in all kinds of dietary protein. ·          It is also used to make aspartame , known by the trade name Nutrasweet , which is used to sweeten low-calorie and sugar free soft drinks, yogurts, and desserts. ...
Read more

how to impress people with your speaking skills

Image
  POWER OF VOICE The human voice: It's the instrument we all play. It's the most powerful sound in the world, probably. It's the only one that can start a war or say " I love you ." And yet many people have the experience that when they speak, people don't listen to them. And why is that? How can we speak powerfully to make change in the world? What I'd like to suggest, there are a number of habits that we need to move away from. I've assembled for your pleasure here seven deadly sins of speaking. I'm not pretending this is an exhaustive list, but these seven, I think, are pretty large habits that we can all fall into. SEVEN SINS OF SPEAKING First, gossip. Speaking ill of somebody who's not present. Not a nice habit, and we know perfectly well the person gossiping, five minutes later, will be gossiping about us.   Second, judging . We know people who are like this in conversation, and it's very hard to listen to somebody if you kno...
Read more