The mean of the median, mode and range of the observations 6, 6, 9, 14, 8, 9, 9, 8 is

Options

A. 10.3
B. 10.5
C. 8.5
D. 8.8

Answer

To find the mean of the median, mode, and range of the observations 6, 6, 9, 14, 8, 9, 9, and 8, we will follow these steps:

### Step 1: Arrange the Data in Ascending Order
First, we need to arrange the given data in ascending order.

**Data:** 6, 6, 9, 14, 8, 9, 9, 8
**Ascending Order:** 6, 6, 8, 8, 9, 9, 9, 14

### Step 2: Find the Range
The range is calculated as the difference between the highest and lowest values in the dataset.

- **Highest Value:** 14
- **Lowest Value:** 6

**Range = Highest Value - Lowest Value = 14 - 6 = 8**

### Step 3: Find the Mode
The mode is the number that appears most frequently in the dataset.

- **Occurrences of Each Number:**
- 6 appears 2 times
- 8 appears 2 times
- 9 appears 3 times
- 14 appears 1 time

**Mode = 9** (as it appears most frequently, 3 times)

### Step 4: Find the Median
To find the median, we first note that there are 8 numbers (an even count). The median is the average of the two middle numbers.

- **Middle Positions:** Since n = 8, the middle positions are n/2 = 4th and (n/2) + 1 = 5th.
- **4th Value:** 8
- **5th Value:** 9

**Median = (8 + 9) / 2 = 17 / 2 = 8.5**

### Step 5: Calculate the Mean of Median, Mode, and Range
Now, we will calculate the mean of the median, mode, and range.

- **Values:**
- Median = 8.5
- Mode = 9
- Range = 8

**Mean = (Median + Mode + Range) / 3 = (8.5 + 9 + 8) / 3**

**Sum = 8.5 + 9 + 8 = 25.5**

**Mean = 25.5 / 3 = 8.5**

### Final Answer
The mean of the median, mode, and range of the observations is **8.5**.

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