# 12.1.2 Measures of Central Tendencies

   
     
      
      
12.1.2 Mode, Median and Mean
      
1.   The mode of a set of data is the value of item which occurs most frequently.

      
Example:
      
3, 7, 6, 9, 7, 1, 5, 7, 2
      
Mode = 7
  
   
      
2.   When a set of data is given in a frequency table, the value or item which has the highest frequency is the mode.
 
 
 
         
      
      
      
3.   The median of a set of data is the value located in the middle of the set when the data is arranged in numerical order.
      
- If the total number of data is odd, then the median is the value in the middle of the set.
      
- If the total number of data is even, then the median is the average of the two middle values of the set
      

        
Example 1:
Find the medians of the following sets of data:
      
(a) 10, 9, 11, 6, 5, 8, 7
      
(b)   10, 9, 11, 6, 5, 8, 7,12
      

      
Solution:
      
(a) Number of data values = 7 ← (Odd number)
      
Rearranging the data in order of magnitude:
      
5, 6, 7, 8, 9, 10, 11
      
Therefore, median = 8

   
      
(b)   Number of data values = 8 ← (Even number)
      
Rearranging the data in order of magnitude:
      
5, 6, 7, 8, 9, 10, 11, 12
   
   
         $\begin{array}{l}\therefore \text{Median}=\frac{8+9}{2}\\ \text{}=8.5\end{array}$       
 
   
      
4.   When a set of data is given in a frequency table, the value situated in the middle position of the data is the median.
   
   
      
 
 
 
    
    
      
      
      
5.   The mean of a set of data is obtained by using formula below.
  
  
      
      
       $\text{Mean =}\frac{\text{sum of all values of data}}{\text{total number of data}}$       
      
      

        
Example:
      
Find the mean of the following sets of data items:
      
-5, -2, -1, 7, 4, 9

   
      
Solution:
      
       $\begin{array}{l}\text{Mean}=\frac{\left(-5\right)+\left(-2\right)+\left(-1\right)+7+4+9}{6}\\ \text{}=\frac{12}{6}\\ \text{}=2\end{array}$       
   
 
   
 
      
6.   When data is given in a frequency table, the mean can be found by using the formula below.
      

      
       $\text{Mean =}\frac{\text{sum of}\left(\text{value}×\text{frequency}\right)}{\text{total frequency}}$       
      

        
Example:
The table below shows the scores obtained by a group of players in a game.

      
      
        Score          1          2          3          4          5          Frequency          5          12          8          15          10  
      
      
Find the mean of the scores.
 
        
Solution:
         $\begin{array}{l}\text{Mean}\\ =\text{}\frac{\text{sum of}\left(\text{score}×\text{frequency}\right)}{\text{total frequency}}\\ =\frac{\left(1×5\right)+\left(2×12\right)+\left(3×8\right)+\left(4×15\right)+\left(5×10\right)}{5+12+8+15+10}\\ =\frac{163}{50}\\ =3.26\end{array}$