Median
2026-02-28 13:08 Diff
https://brightchamps.com/en-us/math/data/median

Median works for both ungrouped lists and class-interval data. Ungrouped data uses the middle number, but grouped data requires a frequency-based formula.


Grouped Data: In grouped data, the information is arranged in class intervals with their frequencies and cumulative frequencies. The median is then found using this Formula:
 

Median = \(l + \frac{\left(\frac{n}{2} - \text{cf}\right)}{f} \times h\)

Where:

l = lower limit of the median class

n = total number of observations

f = frequency of the median class

h = class size (class width)

cf = cumulative frequency of the class just before the median class

For example, find the median of the following data:

Step 2: Total observations 
            n = 50

Step 3: Find \(\frac{n}{2} = \frac{50}{2} = 25\)

Locate the class where CF ≥ 25 → Median Class = 20–30

Step 4: Apply the Formula

l = 20
f = 14
cf = 14 (CF before median class)
h = 10

Median = \(20 + \left( \frac{25 - 14}{14} \right) \times 10\)
 =\(20 + \left( \frac{11}{14} \right) \times 10\)

\(= 20 + 7.86 = 27.86\)

Median ≈ 27.86.

Ungrouped Data: In ungrouped data, the information is listed as individual values rather than the class intervals. To find the median, the data is first arranged in ascending order, and then the median formula is applied depending on whether the number of observations (n) is odd or even.

Median of Ungrouped Data: Ungrouped data means that the numbers are listed individually, without grouping them into intervals. The way we find the median depends on whether n is odd or even.

Median Formula When n Is Odd

If the number of values is odd:

Median = \(\left( \frac{n + 1}{2} \right)^{\text{th}} \text{ value}\)

Example (Odd Number of Values)

Find the median of
12, 18, 25, 30, 45

Arrange in order → already sorted.

Number of values = 5 (odd)

Median = \(\frac{5 + 1}{2}\) = 3rd value

The 3rd value is 25

 Median = 25

Median Formula When n Is Even

If the number of values is even:

Median =\(\left( \frac{n + 1}{2} \right)^{\text{th}} \text{ value}\)

Example (Even Number of Values)

Find the median of:
\(10, 15, 20, 30, 40, 55\)

Arrange in order is already sorted.

Number of values = 6 (even)

Middle values = 3rd (20) and 4th (30)

Median = \(\frac{20 + 30}{2}\)

Median = 25