The average of 11 numbers is 7. If every number in the list is doubled, what will be the new average of these 11 numbers?

Introduction / Context:
This question checks your understanding of how the average behaves under simple linear transformations of the data, such as doubling every value. It illustrates that if each number in a list is multiplied by a constant, then the average is also multiplied by that constant. This is a fundamental property of averages that simplifies many calculations.

Given Data / Assumptions:

• There are 11 numbers.
• Their original average is 7.
• Each number is doubled, that is, multiplied by 2.
• We need to find the new average after doubling.

Concept / Approach:
Average is defined as total sum divided by number of items. If every number is multiplied by a constant factor k, the total sum is also multiplied by k while the number of items stays the same. Therefore, the new average is simply k times the original average. Here, k = 2.

Step-by-Step Solution:
Step 1: Original average = 7, number of numbers = 11.Step 2: Original total sum = 7 * 11 = 77.Step 3: After doubling each number, every term is multiplied by 2, so new total sum = 2 * 77 = 154.Step 4: The number of items remains 11.Step 5: New average = 154 / 11 = 14.

Verification / Alternative check:
Instead of computing totals, we can directly apply the rule that multiplying each value by 2 multiplies the average by 2. Hence new average = 2 * 7 = 14. This simple rule gives the same result, confirming that our understanding of the effect of scaling on averages is correct.

Why Other Options Are Wrong:
Option 7 is just the original average and would be correct only if the numbers were unchanged. Option 3.5 would correspond to halving the numbers, not doubling them. Options 10.5 and 21 correspond to multiplying the average by factors of 1.5 or 3 respectively, which do not match the given operation.

Common Pitfalls:

• Thinking the average remains unchanged regardless of operations on all data points.
• Confusing the effect of adding a constant with multiplying by a constant.
• Attempting to recompute each hypothetical new number instead of using the scaling property of averages.

Final Answer:
The new average of the 11 numbers is 14.