What is the mean of the range, mode and median of the data given below?
5, 10, 3, 6, 4, 8, 9, 3, 15, 2, 9, 4, 19, 11, 4

Question

Accepted Answer

To determine why Option 2 is the correct answer, we need to calculate the mean of the range, mode, and median of the given data set: 5, 10, 3, 6, 4, 8, 9, 3, 15, 2, 9, 4, 19, 11, 4.

**Step 1: Calculate the Range**
- The range is the difference between the maximum and minimum values in the data set.
- Maximum value = 19
- Minimum value = 2
- Range = 19 - 2 = 17

**Step 2: Calculate the Mode**
- The mode is the number that appears most frequently in the data set.
- The numbers 3 and 4 both appear three times, but since 4 appears more frequently than any other number, it is the mode.
- Mode = 4

**Step 3: Calculate the Median**
- The median is the middle value when the data set is ordered from least to greatest.
- Ordered data set: 2, 3, 3, 4, 4, 4, 5, 6, 8, 9, 9, 10, 11, 15, 19
- The middle value (8th value in this ordered list) is 6.
- Median = 6

**Step 4: Calculate the Mean of the Range, Mode, and Median**
- Mean = (Range + Mode + Median) / 3
- Mean = (17 + 4 + 6) / 3 = 27 / 3 = 9

Therefore, the mean of the range, mode, and median is 9, which corresponds to Option 2.

**Explanation of Incorrect Options:**

- **Option 1 (8):** This option is incorrect because the calculated mean is 9, not 8. The arithmetic operations and values used in the calculation do not result in 8.

- **Option 3 (10):** This option is incorrect because the mean of the range, mode, and median is 9, not 10. The calculations clearly show that the sum of 17, 4, and 6 divided by 3 equals 9.

- **Option 4 (12):** This option is incorrect because the mean is 9, not 12. The values and calculations do not support 12 as a possible result.

In conclusion, the correct answer is Option 2, as the mean of the range, mode, and median of the given data set is indeed 9.