Mean of Grouped Data
If n observations in the raw data consists of n distinct values denoted by x_{1}, x_{2}, x_{3}, â€¦, x_{n} of the observed variable x occurring with frequencies f_{1}, f_{2}, â€¦, f_{n} respectively, then the arithmetic mean of the variate x is given by x, where=
where N =
Example
Find the mean of the following data
x: | 19 | 21 | 23 | 25 | 27 | 29 | 31 |
f: | 13 | 15 | 16 | 18 | 16 | 15 | 13 |
Solution
x_{i} | f_{i} | x_{i}f_{i} |
19 | 13 | 247 |
21 | 15 | 315 |
23 | 16 | 368 |
25 | 18 | 450 |
27 | 16 | 432 |
29 | 15 | 435 |
31 | 13 | 403 |
Total | N = Î£f_{i }= 106 | Î£x_{i}f_{i} = 2650 |
We know that x_{i}f_{i} = Ã— 2650 = 25.
Example
The marks of 41 students of Xth class are given in the table. Five the mean marks secured by one student.
Marks: | 20 | 15 | 17 | 30 | 22 | 25 |
Frequency: | 10 | 2 | 5 | 4 | 12 | 8 |
Solution
Marks (x_{i}) | Frequency (f_{i}) | x_{i}f_{i} |
20 | 10 | 200 |
15 | 2 | 30 |
17 | 5 | 85 |
30 | 4 | 120 |
22 | 12 | 264 |
25 | 8 | 200 |
Total |
N = Î£f_{i }= 41 |
Î£x_{i}f_{i }= 899 |
= = 21.93.
Example
In the first proof reading of a book containing 800 pages, the following distribution of misprints were obtained.
No. of Misprints per page |
No. of Pages |
0 |
654 |
1 | 95 |
2 | 36 |
3 | 9 |
4 | 5 |
5 | 1 |
Total |
800 |
Find the average number of misprints per page.
Solution
No. of misprints Per page (x_{i}) |
No. of pages |
f_{i} x_{i} |
0 | 654 | 00 |
1 | 95 | 95 |
2 | 36 | 72 |
3 | 9 | 27 |
4 | 5 | 20 |
5 | 1 | 5 |
N = Î£f_{i} = 800 | Î£f_{i}x_{i }= 219 |
= = 0.27375
Hence the average number of misprints per page = 0.27375.