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, 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: Calculate the Median
To find the median, we first determine the number of observations.

**Number of Observations (n):** 8 (which is even)

For an even number of observations, the median is calculated as:
\[ \text{Median} = \frac{\text{n/2 th term} + \text{(n/2 + 1) th term}}{2} \]

Here, \( n/2 = 4 \) and \( n/2 + 1 = 5 \).

The 4th term is 8 and the 5th term is 9. Therefore,
\[ \text{Median} = \frac{8 + 9}{2} = \frac{17}{2} = 8.5 \]

### Step 3: Calculate the Mode
The mode is the number that appears most frequently in the data set.

From the arranged data:
- 6 appears 2 times
- 8 appears 2 times
- 9 appears 3 times
- 14 appears 1 time

The most frequent number is 9, so:
\[ \text{Mode} = 9 \]

### Step 4: Calculate the Range
The range is calculated as the difference between the highest and lowest values in the data set.

**Highest Value:** 14
**Lowest Value:** 6
\[ \text{Range} = \text{Highest} - \text{Lowest} = 14 - 6 = 8 \]

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

\[ \text{Mean} = \frac{\text{Median} + \text{Mode} + \text{Range}}{3} = \frac{8.5 + 9 + 8}{3} \]
Calculating the sum:
\[ 8.5 + 9 + 8 = 25.5 \]
Now divide by 3:
\[ \text{Mean} = \frac{25.5}{3} = 8.5 \]

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

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