Calculate the mean devation from the mean the following data:
`{:(,"Marks",0-10,10-20,20-30,30-40,40-50),(,"No. of stu.",5,8,15,16,6):}`

Answer

To calculate the mean deviation from the mean for the given data, we will follow these steps:

### Step 1: Identify the data
We have the following data:
- Marks intervals: 0-10, 10-20, 20-30, 30-40, 40-50
- Number of students (frequencies): 5, 8, 15, 16, 6

### Step 2: Calculate the midpoints (x_i) of each class interval
The midpoint for each class interval is calculated as follows:
- For 0-10: \( x_1 = \frac{0 + 10}{2} = 5 \)
- For 10-20: \( x_2 = \frac{10 + 20}{2} = 15 \)
- For 20-30: \( x_3 = \frac{20 + 30}{2} = 25 \)
- For 30-40: \( x_4 = \frac{30 + 40}{2} = 35 \)
- For 40-50: \( x_5 = \frac{40 + 50}{2} = 45 \)

### Step 3: Calculate \( f_i \times x_i \)
Now we multiply the frequency \( f_i \) by the corresponding midpoint \( x_i \):
- \( f_1 \times x_1 = 5 \times 5 = 25 \)
- \( f_2 \times x_2 = 8 \times 15 = 120 \)
- \( f_3 \times x_3 = 15 \times 25 = 375 \)
- \( f_4 \times x_4 = 16 \times 35 = 560 \)
- \( f_5 \times x_5 = 6 \times 45 = 270 \)

### Step 4: Calculate the sum of frequencies and the sum of \( f_i \times x_i \)
Now we sum the frequencies and the products:
- Total frequency \( \Sigma f_i = 5 + 8 + 15 + 16 + 6 = 50 \)
- Total of \( f_i \times x_i = 25 + 120 + 375 + 560 + 270 = 1350 \)

### Step 5: Calculate the mean (\( \bar{x} \))
The mean is calculated using the formula:
\[ \bar{x} = \frac{\Sigma (f_i \times x_i)}{\Sigma f_i} = \frac{1350}{50} = 27 \]

### Step 6: Calculate \( |x_i - \bar{x}| \)
Next, we calculate the absolute deviations from the mean:
- \( |x_1 - \bar{x}| = |5 - 27| = 22 \)
- \( |x_2 - \bar{x}| = |15 - 27| = 12 \)
- \( |x_3 - \bar{x}| = |25 - 27| = 2 \)
- \( |x_4 - \bar{x}| = |35 - 27| = 8 \)
- \( |x_5 - \bar{x}| = |45 - 27| = 18 \)

### Step 7: Calculate \( f_i \times |x_i - \bar{x}| \)
Now we multiply the frequencies by the absolute deviations:
- \( f_1 \times |x_1 - \bar{x}| = 5 \times 22 = 110 \)
- \( f_2 \times |x_2 - \bar{x}| = 8 \times 12 = 96 \)
- \( f_3 \times |x_3 - \bar{x}| = 15 \times 2 = 30 \)
- \( f_4 \times |x_4 - \bar{x}| = 16 \times 8 = 128 \)
- \( f_5 \times |x_5 - \bar{x}| = 6 \times 18 = 108 \)

### Step 8: Calculate the sum of \( f_i \times |x_i - \bar{x}| \)
Now we sum these products:
\[ \Sigma (f_i \times |x_i - \bar{x}|) = 110 + 96 + 30 + 128 + 108 = 472 \]

### Step 9: Calculate the mean deviation
Finally, we calculate the mean deviation using the formula:
\[ \text{Mean Deviation} = \frac{\Sigma (f_i \times |x_i - \bar{x}|)}{\Sigma f_i} = \frac{472}{50} = 9.44 \]

### Final Answer
The mean deviation from the mean is **9.44**.

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