The mean of the following distribution is :
x : 4 6 9 10 15
y : 5 10 10 7 8

Question

Accepted Answer

The mean of a distribution is calculated by taking the sum of the products of each value and its corresponding frequency, and then dividing by the total number of observations.

Given the distribution:
- Values (x): 4, 6, 9, 10, 15
- Frequencies (y): 5, 10, 10, 7, 8

To find the mean, we first calculate the sum of the products of each value and its frequency:

$$
\text{Sum of products} = (4 \times 5) + (6 \times 10) + (9 \times 10) + (10 \times 7) + (15 \times 8)
$$

Calculating each term:
- $4 \times 5 = 20$
- $6 \times 10 = 60$
- $9 \times 10 = 90$
- $10 \times 7 = 70$
- $15 \times 8 = 120$

Adding these products together:
$$
20 + 60 + 90 + 70 + 120 = 360
$$

Next, we calculate the total number of observations by summing the frequencies:
$$
5 + 10 + 10 + 7 + 8 = 40
$$

Now, we find the mean by dividing the sum of the products by the total number of observations:
$$
\text{Mean} = \frac{360}{40} = 9
$$

Therefore, the mean of the distribution is 9, which corresponds to Option 2.

Explanation of why other options are incorrect:
1. **Option 1 (8):** This option is incorrect because the calculated mean is 9, not 8. The sum of the products divided by the total number of observations does not yield 8.
2. **Option 3 (10):** This option is incorrect because the calculated mean is 9, not 10. The division of the sum of products by the total number of observations results in 9.
3. **Option 4 (12):** This option is incorrect because the calculated mean is 9, not 12. The arithmetic calculation clearly shows that the mean is 9.

In conclusion, the correct answer is Option 2, as the mean of the given distribution is indeed 9.