For the data given below, find the sum of mode, median and mean.
6, 3, 8, 4, 3, 11, 7, 4, 5, 4.

Options

A. 13.5
B. 13
C. 14
D. 14.5

Answer

To find the sum of the mode, median, and mean for the given data set \(6, 3, 8, 4, 3, 11, 7, 4, 5, 4\), we will follow these steps:

### Step 1: Arrange the data in ascending order
The first step is to sort the data from smallest to largest.

**Data in ascending order:**
\[3, 3, 4, 4, 4, 5, 6, 7, 8, 11\]

### Step 2: Calculate the Mode
The mode is the number that appears most frequently in the data set.

- The number \(3\) appears \(2\) times.
- The number \(4\) appears \(3\) times.
- The numbers \(5\), \(6\), \(7\), \(8\), and \(11\) appear \(1\) time each.

**Mode:**
The mode is \(4\) since it appears the most frequently (3 times).

### Step 3: Calculate the Median
The median is the middle value of the data set. Since there are \(10\) numbers (an even count), the median will be the average of the \(5^{th}\) and \(6^{th}\) terms.

- The \(5^{th}\) term is \(4\).
- The \(6^{th}\) term is \(5\).

**Median Calculation:**
\[ \text{Median} = \frac{4 + 5}{2} = \frac{9}{2} = 4.5 \]

### Step 4: Calculate the Mean
The mean is calculated by taking the sum of all the numbers and dividing it by the count of numbers.

**Sum of the data:**
\[ 3 + 3 + 4 + 4 + 4 + 5 + 6 + 7 + 8 + 11 = 55 \]

**Mean Calculation:**
\[ \text{Mean} = \frac{55}{10} = 5.5 \]

### Step 5: Calculate the Sum of Mode, Median, and Mean
Now, we will sum the mode, median, and mean.

**Sum Calculation:**
\[ \text{Sum} = \text{Mode} + \text{Median} + \text{Mean} = 4 + 4.5 + 5.5 \]
\[ \text{Sum} = 14 \]

### Final Answer
The sum of the mode, median, and mean is \(14\).

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