Step 1: Understanding the Concept:
The arithmetic mean (commonly known as the average) of a set of data is the sum of all values divided by the total count of values.
This is a fundamental concept in statistics used to find the "central tendency" of a data set.
The problem gives us the mean and the size of the set, which allows us to find the total sum of all elements.
Once the total sum is known, subtracting the known elements reveals the unknown value.
Key Formula or Approach:
The formula for Mean is:
\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]
Rearranging to find the total sum:
\[ \text{Total Sum} = \text{Mean} \times \text{Number of observations} \]
Step 2: Detailed Explanation:
Number of observations (\( n \)) = 5.
Mean (\( \bar{x} \)) = 24.
First, calculate the total sum of the five numbers:
\[ \text{Total Sum} = 24 \times 5 = 120 \]
Now, let's find the sum of the four known numbers.
Known numbers = 18, 22, 26, 30.
\[ \text{Sum of 4 numbers} = 18 + 22 + 26 + 30 \]
Let's add them systematically:
\[ 18 + 22 = 40 \]
\[ 26 + 30 = 56 \]
\[ 40 + 56 = 96 \]
Let the fifth number be represented by \( x \).
The total sum of all 5 numbers is:
\[ 96 + x = 120 \]
To find \( x \), subtract 96 from both sides:
\[ x = 120 - 96 \]
\[ x = 24 \]
We can verify this: \( \frac{18+22+26+30+24}{5} = \frac{120}{5} = 24 \). The calculation is correct.
Step 3: Final Answer:
The fifth number is 24.
This matches Option (C).