Mean, Median and Mode - Definitions and Solved examples

MEAN  or Arithmetic Mean:  Mean of 'n' numbers is defined as the sum of all 'n' numbers  divided by 'n', number of observations.

Important notes regarding Mean:

1. If the each observation is increased by 'a' then the mean is also increased by 'a'.

2. If the each observation is decreased by 'a' then the mean is also decreased by 'a'.

3. If a number greater than the mean value is added to the observations then the mean is increases.

4. If a number less than the mean value is added to the observations then the mean is decreases.

5. We can replace all the observations by the mean value keeping the total sum of the observations the same.

Example:

Find the mean of the numbers 44,65,65,32,76,9,10,4,5

Solution:

Total number of observations = 9

 

Sum of the numbers is   44+65+65+32+76+9+10+4+5 = 310

Therefore there arithmetic mean = 310/9 = 34.44

 

MODE of an observations is a number that is most repeated among a set of observations.

Example:

Find the mode of the numbers 44,65,65,32,76,9,10,4,5

Solution:

Only the number 65 is repeated 2 times and all the other numbers have occured once. Therefore the mode of this set of observations is 65.

 

MEDIAN: If we arrange all the observations in either ascending or descending order then the median of the 'n' observation is

1.  If 'n' is odd, then the median is located at (n+1)/2 th location.

2.  If 'n' is even then the median is the arithmetic mean of the numbers located at (n/2) th and {(n+2)/2} th location.

Example for odd number of observations:

Find the median of the numbers 44,65,65,32,76,9,10,4,5

Solution:

Arrange the above observations in increasing order. we get

                                   4,5,9,10,32,44,65,65,76

Total number of observations = 9 which is odd. Therefore the median is (9+1)/2 = 10/2 = 5

Therefore the median is located at the 5th location which in above series is 32.

Example for even number of observations:

 Find the median of the numbers 44,65,65,32,76,9,10,4,5,29

Solution:

Arrange the above observations in increasing order. we get

                                   4,5,9,10,29,32,44,65,65,76

Total number of observations = 10 which is even.

Therefore the median is the mean of  (10)/2 = 10/2 = 5 th and (10+2)/2= 12/2=6th numbers.

5th number is 29 and 6th number is 32 thus the median is (29+32)/2 = 61/2 = 30.5