What is the mean of the mode, median and range of the data given below ?
17, 8, 11, 13, 9, 14, 10, 11, 16, 23, 15, 7

Question

Accepted Answer

To determine the mean of the mode, median, and range of the given data set, we need to calculate each of these statistical measures and then find their average.

**Data Set:** 17, 8, 11, 13, 9, 14, 10, 11, 16, 23, 15, 7

1. **Mode:** 
   - The mode is the number that appears most frequently in a data set. 
   - In this data set, the number 11 appears twice, while all other numbers appear only once. 
   - Therefore, the mode is 11.

2. **Median:**
   - The median is the middle value of a data set when it is ordered from least to greatest.
   - First, we order the data: 7, 8, 9, 10, 11, 11, 13, 14, 15, 16, 17, 23.
   - Since there are 12 numbers, the median will be the average of the 6th and 7th numbers.
   - The 6th and 7th numbers are 11 and 13, respectively.
   - Median = (11 + 13) / 2 = 12.

3. **Range:**
   - The range is the difference between the largest and smallest numbers in the data set.
   - The smallest number is 7, and the largest number is 23.
   - Range = 23 - 7 = 16.

4. **Mean of Mode, Median, and Range:**
   - Now, we calculate the mean of the mode, median, and range.
   - Mean = (Mode + Median + Range) / 3 = (11 + 12 + 16) / 3 = 39 / 3 = 13.

Therefore, the mean of the mode, median, and range is 13, which corresponds to Option 1.

**Explanation of Incorrect Options:**

- **Option 2 (12.5):** This option is incorrect because it does not accurately represent the mean of the mode, median, and range calculated above. The correct mean is 13, not 12.5.

- **Option 3 (13.5):** This option is incorrect because it is higher than the calculated mean of 13. The mean of the mode, median, and range is exactly 13, not 13.5.

- **Option 4 (12):** This option is incorrect because it is lower than the calculated mean of 13. The mean of the mode, median, and range is 13, not 12.

In conclusion, the correct answer is Option 1, as the mean of the mode, median, and range of the given data set is indeed 13.