Median for Grouped Data

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The median is a measure of central tendency which gives the value of the middle-most observation in the data when arranged in an increasing or decreasing order.
↪ It divides the data in two group with equal number of observations.
↪ For odd (n) number of data,
Median = ^n +1/₂ th term
↪ For even (n) number of data,
Median = mean of the (ⁿ/₂)th term & (ⁿ/₂+1)th term

Finding Median from Cumulative frequency distribution

Finding median for large data by arranging them in ascending or descending order can be quite time consuming. Therefore, for large data we use cumulative frequency distribution to find the median.
↪ Arrange the data in cumulative frequency table, then
median = ^n +1/₂th term, if number of data (n) is odd, or
median = mean of ⁿ/₂th and ( ⁿ/₂ +1 )th term, if number of data (n) is even
↪ Ex- find the median of the following data, which gives the marks, out of 50, obtained by 100 students in a test :

Solution:- We arrange the marks in ascending order and prepare a cumulative frequency table as follows:

Here n = 100, which is even
∴ The median will be the mean of the ⁿ/₂th and the ( ⁿ/₂ +1 )th observations, i.e., the 50th and 51st observations.
From the table above, we find that, 50th observation is 28 & 51st observation is 29
So, Median = ²⁸⁺²⁹/₂ = 28.5

Median for Grouped Data

In a grouped data, it is not possible to find the median by looking at the cumulative frequencies as the middle observation will be some value in a class interval.
↪ The class which divides the whole distribution into two halves is called the median class. It can be located as the class whose cumulative frequency is greater than (and nearest to) ⁿ/₂.
↪ After locating the median class in cumulative frequency distribution of the less than type, we use the following formula to find the median,

Median

=

,

where, l = lower limit of median class,
n = number of observations,
cf = cumulative frequency of class preceding the median class,
f = frequency of median class,
h = class size (assuming class size to be equal).

Ex - The distribution below gives the weights of 30 students of a class. Find the median weight of the students.

Solution:

here, n = 30 ⇒ ⁿ/₂ = 15, which lies in class interval 55 – 60
So, Median Class = 55 – 60
l = 55, cf = 13, f = 6, h = 5
∴ Median = 55 + (^15˗13/₆)✕5
= 55 + ^2✕5/₆
= 55 + 1.67
= 56.67 kg

Finding median from cumulative frequency curves (Ogives)

Method I
On either more than or less than Ogive ↴
↪ Locate n/2 on the y-axis
↪ From this point, draw a line parallel to the x-axis cutting the curve at a point.
↪ From this point, draw a perpendicular to the x-axis.
↪ The point of intersection of this perpendicular with the x-axis gives the median of the data.
↪ Ogive for previous example:

Method II
↪ Draw both ogives (i.e., of the less than type and of the more than type) on the same coordinate plane.
↪ From the point of intersection of the Ogives, draw a perpendicular on the x-axis.
↪ The point at which it cuts the x-axis gives us the median.

Frequencies and Ogive
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