The frequency distribution of marks obtained by 100 students in a certain subject is given below :Marks 0-10 10-20 20-30 30-40 No. of students 10 20 f 40
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Frequency Distribution Analysis
The question provides a frequency distribution of marks for 100 students and asks for the mean deviation about the arithmetic mean. The data is:
- Marks 0-10: 10 students
- Marks 10-20: 20 students
- Marks 20-30: f students
- Marks 30-40: 40 students
The total number of students is given as 100.
Determine Missing Frequency (f)
First, find the value of f using the total number of students:
Sum of students = \(10 + 20 + f + 40 = 100\)
\(70 + f = 100\)
\(f = 100 - 70 = 30\)
So, the frequency for the 20-30 marks class is 30.
Calculate Mean Deviation Components
To find the mean deviation about the arithmetic mean, we need class mid-points (\(x_i\)), frequencies (\(f_i\)), and the arithmetic mean (\(\bar{X}\)).
| Marks Interval | Frequency (\(f_i\)) | Mid-point (\(x_i\)) | \(f_i x_i\) |
|---|---|---|---|
| 0-10 | 10 | \( \frac{0+10}{2} = 5 \) | \(10 \times 5 = 50\) |
| 10-20 | 20 | \( \frac{10+20}{2} = 15 \) | \(20 \times 15 = 300\) |
| 20-30 | 30 | \( \frac{20+30}{2} = 25 \) | \(30 \times 25 = 750\) |
| 30-40 | 40 | \( \frac{30+40}{2} = 35 \) | \(40 \times 35 = 1400\) |
| Total | 100 | 2500 |
Calculate Arithmetic Mean (\(\bar{X}\))
Calculate the arithmetic mean using the formula:
\(\bar{X} = \frac{\sum f_i x_i}{\sum f_i}\)
\(\bar{X} = \frac{2500}{100} = 25\)
Calculate Mean Deviation
Now, calculate the absolute deviations from the mean (\(|x_i - \bar{X}|\)) and the weighted deviations (\(f_i |x_i - \bar{X}|\)).
| \(x_i\) | \(f_i\) | \(|x_i - \bar{X}|\) | \(f_i |x_i - \bar{X}|\) |
|---|---|---|---|
| 5 | 10 | |5 - 25| = 20 | \(10 \times 20 = 200\) |
| 15 | 20 | |15 - 25| = 10 | \(20 \times 10 = 200\) |
| 25 | 30 | |25 - 25| = 0 | \(30 \times 0 = 0\) |
| 35 | 40 | |35 - 25| = 10 | \(40 \times 10 = 400\) |
| Total | 100 | 800 |
The mean deviation about the arithmetic mean is calculated as:
\(MD = \frac{\sum f_i |x_i - \bar{X}|}{\sum f_i}\)
\(MD = \frac{800}{100} = 8\)
The Mean Deviation is 8.
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