Types of Measures of Central Tendencies 






(i) 

Mode 






(ii) 

Median 






(iii) 

Mean 






Mode, mean and median for a set of ungrouped data: 






Mode 



Definition 

The mode of a set of data is the highest value of its frequency.





 Sometimes there are two modes in a set of data where the highest frequency is equal.
 When the frequency of a set of data is the same, then the set of data is to be said no mode.







Median 







Definition 





 The median for a set of data with an odd number of items is the value in the middle.
 The median for a set of data with an even number of items is the average value of two numbers in the middle arranged in ascending or descending order.










Formula 



For even data: 



Average data at position
\(\bigg[\bigg(\dfrac{n}{2}\bigg)^\text{th}\text{ and }\bigg(\dfrac{n}{2} + 1\bigg)^\text{th}\bigg]\)




For odd data: 



Data at the position \(\bigg(\dfrac{n +1}{2} \bigg)^\text{th}\) 



Mean 



Definition 

Mean for a set of data is the value obtained when the sum of the data values is divided by the number of data.





Formula 



\(\text{Mean}= \dfrac{\text{Total value of data}}{\text{Number of data}}\) 



Mean for the data in the frequency table: 



\(\text{Mean}= \dfrac{\text{Sum (data}\times\text{frequency)}}{\text{Number of frequencies}}\) 



Extreme Value 



Definition 

 A value that is too small or too large in a set of data.
 It means the value is too far from the value of the other data in the set.





The effect of changing a set of data to the mode, median and mean: 



 A uniform change in data will result in a uniform change in values for mean, median and mode.
 However, if the data is changed in a nonuniform manner, the values of mean, median and mode will also change in a nonuniform manner.




Organise data in frequency tables for grouped data: 



 To ensure the data is classified with a uniform class interval.
 To prevent the data from overlapping.
 To categories those data into appropriate groups and help to make a conclusion.




Example 







Modal class and mean of a set of grouped data: 



Mean for grouped data: 



\(\text{Mean} \\\\= \dfrac{ \text{The sum (frequency}\times\text{midpoint)}}{\text{Number of frequencies}}\)




The most appropriate measure for central of tendencies: 



Mean 


 Selected to represent data when it involves the whole data when the extreme value does not exist.



Median 


 Selected to represent the data when extreme values exist.
 It is not influenced by extreme values.



Mode 


 Selected to represent data when we intend to determine the item with the highest frequency.
 Involves category data.


