This document provides instructions for calculating the median of both ungrouped and grouped data. For ungrouped data, it describes how to order the values from lowest to highest and find the middle value. For grouped data, it explains how to determine the median by first calculating the cumulative frequency, then using a formula to find the class containing the median value. An example is provided to demonstrate finding the median of grouped data with class boundaries, frequencies, cumulative frequencies, and the formula to calculate the median.

1. Presented by: Mahrukh Shehzadi

2. middle number
order from lowest to highest
or highest to lowest.

3. Methods Formula Formula
Ungrouped Data Case- 1 :
When its n is odd
Case- 2
When its n is even
𝑋 = 𝑠ⅈ𝑧𝑒 𝑜𝑓
𝑛 + 1
2
𝑡ℎ 𝑋 =
1
2
𝑠𝑖𝑧𝑒 𝑜𝑓
𝑛
2
+
𝑛 + 2
2
obs.
Grouped Data Continuous frequency Non-continuous frequency
Make cumulative frequency
column.
Determine the median by
using cumulative frequency.
By using formula:
𝑙 +
ℎ
𝑓
𝑛
2
− 𝑐 .

4. 𝑿 = 𝒔𝒊𝒛𝒆 𝒐𝒇
𝒏 + 𝟏
𝟐
𝒕𝒉 𝒐𝒃𝒔 ⋅

5. For even
When n is even:
Example:
Find the mean of marks 2.3, 2.7, 2.5, 2.9, 3.1, 1.9 ?
1.9, 2.3, 2.5, 2.7, 2.9, 3.1,
As n= 6 so,
𝐗 =
𝟏
𝟐
𝒔𝒊𝒛𝒆 𝒐𝒇
𝟔
𝟐
𝐭𝐡 +
𝟔+𝟐
𝟐
𝐭𝒉 𝐨𝐛𝐬.
𝑿 =
𝟏
𝟐
𝒔𝒊𝒛𝒆 𝒐𝒇 𝟑𝐫𝐝 + 𝟒𝐭𝐡 𝒐𝒃𝒔.
𝑿 =
𝟏
𝟐
𝟐. 𝟓 + 𝟐. 𝟕
𝑿 =
𝟐⋅𝟓+𝟐⋅𝟕
𝟐
𝑿 = 𝟐 ⋅ 𝟔
𝑋 =
1
2
𝑠𝑖𝑧𝑒 𝑜𝑓
𝑛
2
+
𝑛 + 2
2
obs.

7. 𝐶𝑙𝑎𝑠𝑠 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑖𝑒𝑠 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
1 − 2 3
2 − 3 8
3 − 4 5
4 − 5 3
5 − 6 1
To find median, we will follow some points:
1. We will find the cumulative frequency.
2. Then find the median by cumulative frequency distribution table, i.e.
𝑛
2
𝑡ℎ
𝑜𝑏𝑠.
3. Lets find the median.

8. 𝐶𝑙𝑎𝑠𝑠 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑖𝑒𝑠 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝐶𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑣𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
1 3 3
2 8 8 + 3 = 11
3 5 11 + 5 = 16
4 3 16 + 3 = 19
5 1 19 + 1 = 20
𝑀𝑒𝑑𝑖𝑎𝑛 = c𝑙𝑎𝑠𝑠 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑔
𝑛
2
th
𝑜𝑏𝑠.
𝑀𝑒𝑑𝑖𝑎𝑛 = 𝑐𝑙𝑎𝑠𝑠 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑖𝑛𝑔
20
2
𝑡ℎ
𝑜𝑏𝑠 ⋅
Medⅈan = class contaⅈnⅈng 10 th obs.
Medⅈan = 2

9. No of Leaves 1 2 3 4 5 6 7
No of
Branches
2 11 15 20 25 18 10
𝑵𝒐 𝒐𝒇 𝑳𝒆𝒂𝒗𝒆𝒔
𝑿
𝑵𝒐 𝒐𝒇 𝑩𝒓𝒂𝒏𝒄𝒉𝒆𝒔
𝒇
𝑪𝒖𝒎𝒖𝒍𝒂𝒕𝒊𝒗𝒆
𝑭𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 𝑪. 𝑭
𝟏 𝟐 𝟐
𝟐 𝟏𝟏 𝟏𝟑
𝟑 𝟏𝟓 𝟐𝟖
𝟒 𝟐𝟎 𝟒𝟖
𝟓 𝟐𝟓 𝟕𝟑
𝟔 𝟏𝟖 𝟗𝟏
𝟕 𝟏𝟎 𝟏𝟎𝟏
𝑻𝒐𝒕𝒂𝒍 𝟏𝟎𝟏
Solution:
𝑀𝑒𝑑𝑖𝑎𝑛 = 𝑐𝑙𝑎𝑠𝑠 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑖𝑛𝑔
101
2
𝑡ℎ
𝑜𝑏𝑠 ⋅
Medⅈan = class contaⅈnⅈng 50.5 th obs.
Medⅈan = 5

10. 𝑮𝒓𝒐𝒖𝒑 𝟔𝟎 – 𝟔𝟒 𝟔𝟓 – 𝟔𝟗 𝟕𝟎 – 𝟕𝟒 𝟕𝟓 – 𝟕𝟗 𝟖𝟎 – 𝟖𝟒 𝟖𝟓 – 𝟖𝟗
𝑭𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 𝟏 𝟓 𝟗 𝟏𝟐 𝟕 𝟐
Solution:
We should follow these steps:
1. Determine the class boundaries.
2. Determine the median through cumulative frequency.
3. Use the formula:
𝑙 +
ℎ
𝑓
𝑛
2
+ 𝑐 .

11. Group f
Class
Boundary
Cumulative
Frequency
60 – 64 1 59.5 – 64.5 1
65 – 69 5 64.5 – 69.5 6
70 – 74 9 69.5 – 74.5 15
75 – 79 12 74.5 – 79.5 27
80 – 84 7 79.5 – 84.5 34
85 – 89 2 84.5 – 89.5 36

12. Q: find the median of the record:
Solution:
We should follow these steps:
1. Determine the class boundaries.
2. Determine the median through cumulative frequency.
3. Use the formula:
𝑙 +
𝒉
𝒇
𝒏
𝟐
− 𝒄 .

13. 𝑙 +
ℎ
𝑓
𝑛
2
− 𝑐 .
1𝟒𝟒. 𝟓 +
𝟗
12
𝟒𝟎
2
− 𝟏𝟕
1𝟒𝟒. 𝟓 +
𝟗
12
𝟑
Median= 𝟏𝟒𝟔. 𝟖
𝒍 = 𝒍𝒐𝒘𝒆𝒓 𝒄𝒍𝒂𝒔𝒔 𝒃𝒐𝒖𝒏𝒅𝒂𝒓𝒊𝒆𝒔 𝒐𝒇 𝒎𝒆𝒅𝒊𝒂𝒏 𝒄𝒍𝒂𝒔𝒔
= 𝟏𝟒𝟒. 𝟓
𝒉 = 𝟗 ,
𝒏 = 𝟒𝟎 ,
𝒄 = 𝟏𝟕,
𝒏 = 𝟒𝟎