Central Tendency

Central Tendency:

Data can be classified in various forms. One way to distinguish between data is in terms of grouped and ungrouped data.

What is ungrouped data?

When the data has not been placed in any categories and no aggregation/summarization has taken placed on the data then it is known as ungrouped data. Ungrouped data is also known as raw data.

Height of students:

 (171,161,155,155,183,191,185,170,172,177,183,190,139,149,150,

150,152,158,159,174,178,179,190,170,143,165,167,187,169,182,

163,149,174,174,177,181,170,182,170,145,143):

This is raw/ungrouped data.

When raw data have been grouped in different classes then it is said to be grouped data.

Height of students:

(139, 143, 143, 145, 149, 149,150,150,152, 155,155, 158,159, 161, 163, 165,167, 169, 170, 170, 170 170, 171, 172, 174, 174,174, 177, 177, 178,179, 181, 182, 182, 183, 183, 185, 187, 190, 190, 191)

Before we study more about grouped and ungrouped data it is important to understand what do we mean by “Central Tendencies”?

 

Measures of central tendency

These are statistical constants which give us an idea about the concentration of the values in the central part of the distribution. The various measures of central tendency are:

1. Arithmetic Mean
2. Median
3. Mode
4. Gerometric Mean
5. Harmonic Mean

• Arithmetic Mean

    Arithmetic mean of a set of observations is their sum divided by

 the number of observations, E.g., the Arithmetic mean $\bar{x}$ of $n$ observations $x_1, x_2, x_3,........, x_n$ is given by

                  $\bar{x}=\frac{x_1+x_2+x_3+....+x_n}{n}=\frac{1}{n}\sum_{i=1}^n x_i$ (Ungrouped data) 

        In case of frequency distribution $\ x_i, f_i, i=1,2,3,......, n$ where $\ f_i$ is the frequency of the variable $\ x_i$

$\bar{x}$=$\frac{f_1x_1+f_2x_2+f_3x_3+.......+f_nx_n}{f_1+f_2+f_3+.......+f_nx_n}$=$\frac{\sum_{i=1}^n f_ix_i}{\sum_{i=1}^n f_i}$ (Grouped Data with frequency Distribution)  

       In case of continuous frequency distribution, $x$ is taken as the mid value of the corresponding class.

 

Example: Find the arithmetic mean of the following distribution

        (a) x: 150,200,300,650,250,180,400,500,550,220

Solution (a): Here the data is group data. So the arithmetic mean is

           $\bar{x}$ = $\frac{150+200+300+650+250+180+400+500+550+220}{10}$=340

Solution (b) :  Here the given data are grouped data with discrete frequency distribution. So the arithmetic mean is

                  $\bar{x}$=$\frac{1\times5+2\times9+3\times12+4\times17+5\times14+6\times10+7\times6}{5+9+12+17+14+10+6}$=4.09

Example: Find the arithmetic mean of the following distribution

                            If the values of $x$ or $f$ are large, the calculation of mean is quite time consuming and tedious. The arithmetic is required to a great extent by taken deviations of the given values from any arbitrary point "A".

                      

                      $d_i$ = $x_i - A$ ⇒$\sum_{i=1}^n d_i$ = $\sum_{i=1}^n x_i - A_n$

 

            Now , 

                         $\frac{\sum_{i=1}^n d_i}{n}$ = $\frac{\sum_{i=1}^n x_i}{n}$ - $A$

                     

                         $\bar{x}$ = $A$ + $\frac{\sum_{i=1}^n d_i}{n}$         (for ungrouped data)  

 

         

                            $\bar{x}$ = $A$+$\frac{1}{N}{\sum f_id_i}$     where $N$=$\sum f_i$

 

 

                            $d_i$ = $\frac{x_i-A}{h}$

                        $\bar{x}$= $A$+$\frac{h}{N}{\sum f_id_i}$      where, $N$= $\sum f_i$

 

 Example : Calculate the mean for the following frequency distribution

 

Solution :

  

                            

Example : The average salary of male employees in a firm was Rs. 5,200 and that of female was Rs. 4,200. The mean salary of all the employees was Rs. 5000. Find the percentage of male and female employees.