The mean of the range, mode and median of the observations 15, 50, 120, 6, 80, 100, 10, 8, 10, 15, 15 is:

Question

Accepted Answer

To determine the mean of the range, mode, and median of the given observations, we need to calculate each of these statistical measures for the data set: 15, 50, 120, 6, 80, 100, 10, 8, 10, 15, 15.

1. **Range**: The range is the difference between the maximum and minimum values in the data set.
   - Maximum value = 120
   - Minimum value = 6
   - Range = 120 - 6 = 114

2. **Mode**: The mode is the value that appears most frequently in the data set.
   - The number 15 appears three times, which is more frequent than any other number.
   - Mode = 15

3. **Median**: The median is the middle value when the data set is arranged in ascending order.
   - Arranged data: 6, 8, 10, 10, 15, 15, 15, 50, 80, 100, 120
   - The middle value (6th value in this ordered list) is 15.
   - Median = 15

Now, we calculate the mean of the range, mode, and median:
- Mean = (Range + Mode + Median) / 3
- Mean = (114 + 15 + 15) / 3
- Mean = 144 / 3
- Mean = 48

Therefore, the mean of the range, mode, and median is 48, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2 (39)**: This option is incorrect because the calculated mean of 48 does not match 39. The arithmetic operations and values used in the calculation clearly show that the mean is 48.

- **Option 3 (52)**: This option is incorrect because the mean of 48 is less than 52. The calculations demonstrate that the correct mean is 48, not 52.

- **Option 4 (51)**: This option is incorrect because the mean of 48 is less than 51. The calculations confirm that the mean is 48, not 51.

In conclusion, the correct answer is Option 1, as the mean of the range, mode, and median is indeed 48.