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Middle/High School Algebra, Geometry, and Statistics (AGS)
9.22 Median of Discrete Values

Median

When the observation in the data are arranged in the order (ascending or descending), the median is the middle-value. That is 50% of the observations are below the median and 50% are above it.

Following are the steps to find the median of an ungrouped frequency distribution

Step 1. Arrange the variates in the ascending order of their magnitude.
Step 2. Find the total number of variates, say n.
Step 3. If n is odd then the median is the value of [(n+1)/2]th item. If n is even then the median is the A.M. of the values of (n/2)th and [(n/2) +1]th items.

Example
  • Find the median of the following observations
    i)15,40,25,16,28,32,36,42,16,19,28
    ii)72,68,42,33,35,39,40,41,65,69

  • Solution

    i)Arranging the given data in the ascending order we get
    15,16,16,19,25,28,28,32,36,40,42
    Here n = 11(odd)
    Median = Value of the [(11+1)/2] th item = Value of the 6th item.
    So Median = 28

    ii)Arranging the given data in the ascending order we get
    33,35,39,40,41,42,65,68,69,72.
    Here n = 10(even)
    Median = Value of the A.M. of (n/2)th and [(n/2)+1]th item = A.M. of the values of the 5th and the 6th items.
    So Median = (41+42)/2 = 41.5

Merits of the median
1.) It can be calculated graphically.
2.) It is not affected by extreme values.
3.) It can be determined by inspection in many cases.
Demerits of the median
1.) It is not directly based on all the values.
2.) It is not as stable as the arithmetic mean.


Q 1: {1, 2, 3, 6, 8} is a set of 5 positive integers whose mean is 4 and median is 3. Write down another set of 5 positive integers, each having the same mean and median as this set.
{1, 2, 3, 6, 10}
{1, 2, 4, 5, 9}
{1, 2, 3, 4, 10} or {1, 2, 3, 5, 9}
{1, 2, 3, 4, 11}

Q 2: Numbers 50,42,41,35,2x+10,2x-8,12,11,8,6 are written in descending order and if their median is 25, find x.
12
13
15
18

Q 3: The median of the following obervation arranged in ascending order is 24. Find x.
11,12,14,18,x+2,x+4,30,32,35,41
30
21
15
18

Q 4: The following data have been arranged in the ascending order :
24, 27, 28, 31, 34, x, 37, 40, 42, 45
In the above data, if 45 is changed to 33, find the new median.
32.5
33.5
36
30

Q 5: The following data have been arranged in an ascending order :
12, 14, 17, 20, 22, x, 26, 28, 32, 36.
If the median of the data is 23, find x.
24
26
32
22

Q 6: Find the median of the prime numbers between 51 and 80
67
73
61
59

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