How to Calculate the Average of New Numbers After Including Them in a Set

Understanding how to calculate the average of new numbers after they are added to an existing set is a fundamental concept in statistics and data analysis. This tutorial will guide you through the process using two examples, illustrating how to determine the average of new numbers in different contexts.

Example 1: The Average of Five Numbers is 7. After Including Three New Numbers, the Average Becomes 8.5

Let's start with a practical scenario: the average of five numbers is 7, and when three new numbers are included, the average of the eight numbers becomes 8.5. We need to find the average of these three new numbers.

### Step 1: Calculate the Sum of the Original Five Numbers

The average of the five numbers is 7. Therefore, the sum of these five numbers is:

[ text{Sum of five numbers} 5 times 7 35 ]

### Step 2: Calculate the Sum of All Eight Numbers

The new average of the eight numbers is 8.5. Therefore, the sum of the eight numbers is:

[ text{Sum of eight numbers} 8 times 8.5 68 ]

### Step 3: Calculate the Sum of the Three New Numbers

Let ( S ) be the sum of the three new numbers. We know that the sum of the eight numbers is the sum of the five numbers plus the sum of the three new numbers:

[ text{Sum of eight numbers} text{Sum of five numbers} S ]

Substituting the known values:

[ 68 35 S ]

Solving for ( S ):

[ S 68 - 35 33 ]

### Step 4: Calculate the Average of the Three New Numbers

The average of the three new numbers is:

[ text{Average of three new numbers} frac{S}{3} frac{33}{3} 11 ]

Example 2: The Average of 5 Numbers is 9. When 3 New Numbers Are Added, the Average of the 8 Numbers Is 11

Now let's consider another example: the average of five numbers is 9, and when three new numbers are added, the average of the eight numbers becomes 11. We need to find the average of the three new numbers.

### Step 1: Calculate the Sum of the 5 Original Numbers

The sum of the five numbers is:

[ text{Sum of 5 numbers} 5 times 9 45 ]

### Step 2: Calculate the Sum of the 8 Numbers

The sum of the eight numbers is:

[ text{Sum of 8 numbers} 8 times 11 88 ]

### Step 3: Calculate the Sum of the Three New Numbers

The sum of the three new numbers is:

[ text{Sum of 3 new numbers} 88 - 45 43 ]

The average of the three new numbers is:

[ text{Average of 3 new numbers} frac{43}{3} 14.33 ]

Conclusion

Calculating the average of new numbers after they are added to an existing set can be done systematically using these steps. Understanding how to apply this concept in different scenarios is crucial for data analysis and statistics.

Keytakeaways:

Step 1: Calculate the sum of the original set of numbers. Step 2: Calculate the sum of the new set of numbers, including the original and the new numbers. Step 3: Calculate the sum of the new numbers alone by subtracting the sum of the original set from the sum of the new set. Step 4: Calculate the average of the new numbers by dividing the sum of the new numbers by the number of new numbers.

By following these steps, you can easily determine the average of new numbers added to an existing set, ensuring accurate and reliable data analysis.