A random sample of 20 people is classified in the following table according to their ages :
`|{:("Age","Frequency"),(15-25," 2"),(25-35," 4"),(35-45," 6"),(45-55," 5"),(55-65," 3"):}|`
What is the mean age of this group of people?

Options

A. `41.0`
B. `41.5`
C. `42.0`
D. `42.5`

Explanation

To find the mean age of the given group of people, we will follow these steps:

### Step 1: Determine the mid-values for each age group
The mid-value (or midpoint) for each age group can be calculated as follows:
- For the age group 15-25: Mid-value = (15 + 25) / 2 = 20
- For the age group 25-35: Mid-value = (25 + 35) / 2 = 30
- For the age group 35-45: Mid-value = (35 + 45) / 2 = 40
- For the age group 45-55: Mid-value = (45 + 55) / 2 = 50
- For the age group 55-65: Mid-value = (55 + 65) / 2 = 60

### Step 2: Create a table with mid-values and frequencies
Now, we can summarize the mid-values and their corresponding frequencies in a table:

| Age Group | Mid-value (x) | Frequency (f) |
|-----------|---------------|---------------|
| 15-25 | 20 | 2 |
| 25-35 | 30 | 4 |
| 35-45 | 40 | 6 |
| 45-55 | 50 | 5 |
| 55-65 | 60 | 3 |

### Step 3: Calculate \( x \times f \) for each age group
Next, we will calculate the product of mid-values and frequencies:

| Age Group | Mid-value (x) | Frequency (f) | \( x \times f \) |
|-----------|---------------|---------------|-------------------|
| 15-25 | 20 | 2 | 40 |
| 25-35 | 30 | 4 | 120 |
| 35-45 | 40 | 6 | 240 |
| 45-55 | 50 | 5 | 250 |
| 55-65 | 60 | 3 | 180 |

### Step 4: Calculate the summation of frequencies and \( x \times f \)
Now, we will calculate the total of frequencies and the total of \( x \times f \):

- Total frequency \( \sum f = 2 + 4 + 6 + 5 + 3 = 20 \)
- Total \( \sum (x \times f) = 40 + 120 + 240 + 250 + 180 = 830 \)

### Step 5: Calculate the mean age
Finally, we can calculate the mean age using the formula:

\[ \text{Mean} = \frac{\sum (x \times f)}{\sum f} = \frac{830}{20} = 41.5 \]

### Conclusion
The mean age of the group of people is **41.5 years**.

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