Mean, Median, and Mode
2026-02-28 21:32 Diff
https://brightchamps.com/en-us/math/data/mean-median-mode

Median of ungrouped data: To find the median of ungrouped data, follow these steps.

Step 1: Sort the data in ascending or descending order.

Step 2: Consider n to be the total number of observations. If the n is an odd number, then
the median \(=\frac{n+1}{2}\)th observation in the sorted list. 
If n is even, then the median is the average of the \(\frac{n}{2}\)th and the \(\frac{n}{2}+1\)th observation. 

Example 1: 
Find the median of the data 56, 67, 54, 34, 78, 43, 23. 

Solution: 

Sort the data : 23, 34, 43, 54, 56, 67, 78.
Here, the number of observations n = 7.

Then, the median = \(\frac{n+1}{2}\)
\(=\frac{7+1}{2}\)
\(=\frac{8}{2}=4\)

Therefor, the 4th observation is the median.
That is 54.

Example 2: 
Find the median of the data 50, 67, 24, 34, 78, 43.

Solution: 

By sorting the data: 24, 34, 43, 50, 67, 78.
Here, the number of observations n = 6
\(\frac{n}{2}=\frac{6}{2}=\) 3rd term.

\(\frac{n}{2}+1=\frac{6}{2}+1=4\)th term.

Median \(=\frac{43+50}{2}\)

\(=46.5\)


Median of grouped data: When data are grouped into class intervals (with frequencies), use these steps:

Step 1: Compute the cumulative frequency of each class, to determine where \(\frac{n}{2}\) lies. Here, \(n=∑fi (n = \sum f_i) \) (total number of observations). 

Step 2: Identify the median class: the class interval in which \(\frac{n}{2}\) falls.

Step 3: Use the formula,
Median \(= l+\frac{(\frac{n​}{2}−c​)}{f}×h\)

Where, 
l = lower limit of the median class.
c = cumulative frequency of the class just before the median class.
f = frequency of the median class.
h = class width (size of the class interval).
 

Example: 
Classes: 0–10, 10–20, 20–30, 30–40, 40–50
frequencies : 2, 12, 22, 8, 6
Total n = 50
So, \(\frac{n}{2}\) = 25.

The cumulative frequencies are: 
0–10 → 2
10–20 → 14
20–30 → 36, ……

Since 25 lies in the class 20–30, that’s the median class.
Here, l = 20, c = 14, f = 22, h = 10. 

By applying the formula: 

Median \( = 20 + \left(\frac{25 - 14}{22}\right) \times 10 = 20 + \left(\frac{11}{22}\right) \cdot 10 = 20 + 5 = 25 \)