The arithmetic mean of the following numbers 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5. 5, 6, 6, 6, 6. 6, 6 and 7, 7, 7, 7. 7, 7. 7 is

Options

A. 4
B. 5
C. 14
D. 20

Answer

To find the arithmetic mean of the given numbers, we will follow these steps:

### Step 1: List the numbers and their frequencies
The numbers provided are:
- 1 appears 1 time
- 2 appears 2 times
- 3 appears 3 times
- 4 appears 4 times
- 5 appears 5 times
- 6 appears 6 times
- 7 appears 7 times

### Step 2: Calculate the sum of the numbers
To find the sum, we multiply each number by its frequency and then add them all together:
- Sum = (1 × 1) + (2 × 2) + (3 × 3) + (4 × 4) + (5 × 5) + (6 × 6) + (7 × 7)

Calculating each term:
- 1 × 1 = 1
- 2 × 2 = 4
- 3 × 3 = 9
- 4 × 4 = 16
- 5 × 5 = 25
- 6 × 6 = 36
- 7 × 7 = 49

Now, adding these results together:
- Total Sum = 1 + 4 + 9 + 16 + 25 + 36 + 49 = 140

### Step 3: Count the total number of observations
Next, we count the total number of observations:
- Total observations = 1 (for 1) + 2 (for 2) + 3 (for 3) + 4 (for 4) + 5 (for 5) + 6 (for 6) + 7 (for 7) = 28

### Step 4: Calculate the arithmetic mean
The arithmetic mean (average) is calculated using the formula:
\[ \text{Mean} = \frac{\text{Total Sum}}{\text{Total Observations}} \]
Substituting the values we have:
\[ \text{Mean} = \frac{140}{28} = 5 \]

### Final Answer
The arithmetic mean of the given numbers is **5**.

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