The mean of the following distribution is
`{:("Class",0-5,5-10,10-15,15-20,20-25,25-30),("Frequ.",4,5,7,12,7,5):}`

Options

A. 15
B. 16
C. 17
D. 18

Answer

To find the mean of the given distribution, we will follow these steps:

### Step 1: Calculate the Mid-values (x) of the Class Intervals
The mid-value for each class interval is calculated using the formula:
\[ \text{Mid-value} (x) = \frac{\text{Lower Limit} + \text{Upper Limit}}{2} \]

- For the class interval 0-5:
\[ x_1 = \frac{0 + 5}{2} = \frac{5}{2} = 2.5 \]

- For the class interval 5-10:
\[ x_2 = \frac{5 + 10}{2} = \frac{15}{2} = 7.5 \]

- For the class interval 10-15:
\[ x_3 = \frac{10 + 15}{2} = \frac{25}{2} = 12.5 \]

- For the class interval 15-20:
\[ x_4 = \frac{15 + 20}{2} = \frac{35}{2} = 17.5 \]

- For the class interval 20-25:
\[ x_5 = \frac{20 + 25}{2} = \frac{45}{2} = 22.5 \]

- For the class interval 25-30:
\[ x_6 = \frac{25 + 30}{2} = \frac{55}{2} = 27.5 \]

### Step 2: Create a Table of Frequencies and Mid-values
Now, we will create a table with the mid-values and corresponding frequencies.

| Class Interval | Frequency (f) | Mid-value (x) |
|----------------|---------------|----------------|
| 0 - 5 | 4 | 2.5 |
| 5 - 10 | 5 | 7.5 |
| 10 - 15 | 7 | 12.5 |
| 15 - 20 | 12 | 17.5 |
| 20 - 25 | 7 | 22.5 |
| 25 - 30 | 5 | 27.5 |

### Step 3: Calculate \( fx \) for Each Class Interval
Now we will calculate \( fx \) (the product of frequency and mid-value) for each class interval.

- For the first interval:
\[ fx_1 = 4 \times 2.5 = 10 \]

- For the second interval:
\[ fx_2 = 5 \times 7.5 = 37.5 \]

- For the third interval:
\[ fx_3 = 7 \times 12.5 = 87.5 \]

- For the fourth interval:
\[ fx_4 = 12 \times 17.5 = 210 \]

- For the fifth interval:
\[ fx_5 = 7 \times 22.5 = 157.5 \]

- For the sixth interval:
\[ fx_6 = 5 \times 27.5 = 137.5 \]

### Step 4: Calculate \( \Sigma fx \) and \( \Sigma f \)
Now we will sum up all the \( fx \) values and the frequencies.

- \( \Sigma fx = 10 + 37.5 + 87.5 + 210 + 157.5 + 137.5 = 640 \)
- \( \Sigma f = 4 + 5 + 7 + 12 + 7 + 5 = 40 \)

### Step 5: Calculate the Mean
The mean is calculated using the formula:
\[ \text{Mean} = \frac{\Sigma fx}{\Sigma f} \]

Substituting the values we calculated:
\[ \text{Mean} = \frac{640}{40} = 16 \]

### Final Answer
Thus, the mean of the given distribution is **16**.

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