Calculate the mean deviation of the first five prime numbers.

1. Analyze the question

We need to find the mean deviation of the first five prime numbers. To do this, we must first identify the numbers, calculate their arithmetic mean, and then find the average of the absolute differences between each number and that mean.

2. Explain the concept

• Prime Numbers: Numbers greater than 1 that have only two factors: 1 and themselves.
• Arithmetic Mean ($\bar{x}$): The sum of the values divided by the number of values.
• Mean Deviation: The average of the absolute differences between each data point ($x$) and the mean ($\bar{x}$). The formula is: $$\text{Mean Deviation} = \frac{\sum |x - \bar{x}|}{n}$$

3. Show the solution

Step 1: Identify the first five prime numbers. The first five prime numbers are: 2, 3, 5, 7, and 11. Number of values ($n$) = 5.

Step 2: Calculate the mean ($\bar{x}$).

Step 3: Calculate the absolute deviations from the mean $|x - \bar{x}|$.

• $|2 - 5.6| = |-3.6| = 3.6$
• $|3 - 5.6| = |-2.6| = 2.6$
• $|5 - 5.6| = |-0.6| = 0.6$
• $|7 - 5.6| = |1.4| = 1.4$
• $|11 - 5.6| = |5.4| = 5.4$

Step 4: Calculate the mean of these deviations.

4. Identify the answer

Comparing our result to the given options, the correct value is 2.72. This matches option A.

5. Tip

When calculating mean deviation, remember that the result of $|x - \bar{x}|$ must always be positive. If you get a negative number, you have made a mistake in your subtraction. Also, ensure you don't confuse "Mean Deviation" with "Standard Deviation"—they are calculated using different formulas!