Calculating the New Average After Adding a Number to a Set

Understanding how to calculate the new average after adding a number to an existing set is a fundamental concept in statistics. This process involves simple arithmetic operations but requires a clear understanding of the relationship between sum, count, and average. Let's delve into the step-by-step process to calculate the new average when a number is added to a set of 12 numbers, where the average is initially 12.

The Initial Problem

We start with a set of 12 numbers having an average of 12. The goal is to determine the new average when 123 is added to the set.

Step 1: Determine the Initial Sum of the Numbers

To find the initial sum of the 12 numbers, we can multiply the average by the number of elements in the set:

Initial Sum 12 x 12 144

Step 2: Add the New Element to the Set

Next, we add the number 123 to the initial sum to get the new total:

New Total 144 123 267

Step 3: Calculate the New Average

The new average is the new total divided by the new number of elements in the set (which is now 13):

New Average 267 / 13 ≈ 20.54

Mathematical Representation and Proof

To summarize the process, we can represent it as:

New Average (12 x 12 123) / 13 (144 123) / 13 267 / 13 ≈ 20.54

This can be further detailed as an improper fraction or a mixed number:

267 / 13 20 7 / 13

To verify our solution, we can use the inverse operation:

13y - 123 144

Solving for y:

13y 144 123 13y 267 y 267 / 13 ≈ 20.54

This confirms that the new average is approximately 20.54.

Conclusion

Understanding how to calculate the new average after adding a number to a set is a critical skill in statistics. By following the steps outlined above, you can easily solve similar problems and ensure accurate results. Whether you need to find the new average in a practical context or for academic purposes, this method is both straightforward and reliable.

Key Takeaways:

The initial sum is the product of the average and the number of elements. The new total is the initial sum plus the added number. The new average is the new total divided by the new number of elements.

Feel free to explore more scenarios and apply these concepts in various contexts to deepen your understanding of averages and statistical calculations.