Median
Median is the middle value of a distribution. It divides the data in two equal parts.
Middle number of an ungrouped data : The ungrouped data are x
_{1}, x
_{2},x
_{3},…,x
_{n} then the middle value after arranging the data either ascending or descending order is the middle number of the data.
Steps to find the mid point of the ungrouped data :
1) Arrange the data either ascending or descending order of their values.
2) Determine the total number of observations, say n.
3) If n is odd then the middle number will be the median. And if n is even then mean of middle two numbers will be the median.
Merits and Demerits of Median
Example 1: Find the median of 12,15,10,18 ,8.
Solution : Data in ascending order : 8,10,12,15,18
From the above middle number 12 is the median.
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Example 2 : Find the median of 23,46,18,32,65,20.
Solution : Data in ascending order : 18,20,23,32,46,65.
From the above we can say that there are two middle numbers 23 and 32.
So, the median = (23 + 32 ) / 2 = 55 / 2 = 27.5
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Example 3 : Numbers 50,42,2x + 10, 2x  8 ,12,11, 8, 6 are written in descending order and their median is 25, find x.
Solution : 50,42,2x + 10, 2x, 8 ,12,11, 8, 6 from this the median is mean of 2x + 10 and 2x  8.
Median = ( 2x + 10 + 2x 8) / 2 = 25
⇒ ( 4x + 2 ) / 2 = 25
⇒ 4x + 2 = 50
⇒ 4x = 48
∴ x = 12
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Cumulative Frequency Distribution :
The total of a frequency and all frequencies below it in a frequency distribution.
It is the 'running total' of frequencies.
Example : Scores : 4,4, 5,5,1,1,2,3,3,3,2
Scores 
Frequency 
Cumulative frequency (cf) 
1 
2 
2 
2 
2 
2 + 2 = 4 
3 
3 
2 + 2 + 3 = 7 
4 
2 
2 + 2 + 3 + 2 = 9 
5 
2 
2 + 2 + 3 + 2 + 2 = 11 

11 

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There are two types of cumulative frequency distribution : 1) Less than
2) Greater than.
For Less than cumulative frequency distribution, we
add up the frequencies from the above and in Greater than cumulative frequency distribution we
add up the frequencies from below.
Example :
Class intervals 
Frequency 
Less than type Cumulative frequency 
Greater than type Cumulative frequency 
0  10 
9 
9 
50 
10  20 
14 
9 + 14 = 23 
50  9 = 41 
20  30 
8 
9 + 14 + 8 = 31 
50  9 14 = 27 
30  40 
10 
9 + 14 + 8 + 10 = 41 
50  9 14  8 = 19 
40  50 
9 
9 + 14 + 8 + 10 + 9 = 50 
50  9 14  8  10 = 9 


50 

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Median of a Grouped or Continuous Frequency Distribution :
To find the median of a grouped data, use the following steps :
1) Obtain the frequency distribution and obtain N = Σ fi.
2) Prepare cumulative frequency distribution.
3) Find N / 2.
4) Find the cumulative frequency just greater than N / 2 and determine the corresponding class. This class is known as the Median Class.
5) Use the following formula :
Where, l = Lower limit median class
f = frequency of the median class
h = width of the median class
F = Cumulative frequency of the class preceding the median class.
N = Σ fi.
Some solved examples :
1) Find the median for the following data:
Classes 
10  20 
20  30 
30  40 
40  50 
50  60 
60  70 
70  80 
Frequency 
4 
8 
10 
12 
10 
4 
2 
Solution :
Classes 
Frequency (fi) 
Cumulative frequency 
10  20 
4 
4 
20  30 
8 
12 
30  40 
10 
22 
40  50 
12 
34 
50  60 
10 
44 
60  70 
4 
48 
70  80 
2 
50 
Total 
N = Σ fi = 50 

N / 2 = 50 / 2 = 25,
So, 40 – 50 is the median class and its lower limit = l = 40
Cumulative frequency preceding the median class = f = 22
Frequency of the median class = F = 12
Class width h = 50 40 = 10
⇒ Median = 40 +[ ( 25 – 22) / 12 ] x 10
⇒ = 40 + ( 3 / 12 ) x 10
⇒ = 40 + 0.25 x 10
⇒ = 40 + 2.5
∴ Median = 42.5
Statistics
• Statistics
• Pictograph
• Pie chart
• Bar Graph
• Double Bar Graph
• Histogram
• Frequency polygon
• Frequency distribution (Discrete )
• Frequency distribution continuous (or grouped)
• Measures of central tendency (Meanmedian and Mode
• Ogive or Frequency curve.
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