Calculating 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