The ages of 40 students are given in the following table: Age(in yrs) 12 13 14 15 16 17 18 Frequency 2 4 6 9 8 7 4 Find the arithmetic mean.

Answer

To find the arithmetic mean of the ages of 40 students given in the table, we will follow these steps:

### Step 1: Set up the table
We have the ages (x) and their corresponding frequencies (f) as follows:

| Age (x) | Frequency (f) |
|---------|---------------|
| 12 | 2 |
| 13 | 4 |
| 14 | 6 |
| 15 | 9 |
| 16 | 8 |
| 17 | 7 |
| 18 | 4 |

### Step 2: Calculate the product of age and frequency (xf)
We will multiply each age by its corresponding frequency to get the xf column:

- For age 12: \( 12 \times 2 = 24 \)
- For age 13: \( 13 \times 4 = 52 \)
- For age 14: \( 14 \times 6 = 84 \)
- For age 15: \( 15 \times 9 = 135 \)
- For age 16: \( 16 \times 8 = 128 \)
- For age 17: \( 17 \times 7 = 119 \)
- For age 18: \( 18 \times 4 = 72 \)

Now, we can summarize this in a new table:

| Age (x) | Frequency (f) | xf |
|---------|---------------|------|
| 12 | 2 | 24 |
| 13 | 4 | 52 |
| 14 | 6 | 84 |
| 15 | 9 | 135 |
| 16 | 8 | 128 |
| 17 | 7 | 119 |
| 18 | 4 | 72 |

### Step 3: Sum the xf values and the frequencies
Now we will find the total of the xf column and the total of the frequency column:

- Total of xf: \( 24 + 52 + 84 + 135 + 128 + 119 + 72 = 614 \)
- Total of frequency (f): \( 2 + 4 + 6 + 9 + 8 + 7 + 4 = 40 \)

### Step 4: Calculate the arithmetic mean
The formula for the arithmetic mean is given by:

\[ \text{Mean} = \frac{\sum (xf)}{\sum (f)} \]

Substituting the values we found:

\[ \text{Mean} = \frac{614}{40} = 15.35 \]

### Final Answer
The arithmetic mean of the ages of the 40 students is **15.35** years.

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