Calculating the Average of Remaining Numbers

This problem asks us to find the average of a set of numbers after two specific numbers have been removed from the original set. We start with the information given about the initial set of 20 numbers.

Understanding the Initial Data

• Total number of elements initially: 20
• Average of these 20 numbers: 32

The average of a set of numbers is calculated using the formula:

$\text{Average} = \frac{\text{Sum of numbers}}{\text{Number of numbers}}$

Calculating the Total Sum of the Original Numbers

Using the formula, we can find the total sum of the 20 numbers:

$\text{Sum of 20 numbers} = \text{Average} \times \text{Number of numbers}$

$\text{Sum of 20 numbers} = 32 \times 20 = 640$

So, the sum of the original 20 numbers is 640.

Removing Two Numbers

Two numbers, 29 and 31, are removed from the original set of 20 numbers.

• The two numbers removed are 29 and 31.
• The sum of these two numbers is $29 + 31 = 60$.
• The number of remaining numbers is $20 - 2 = 18$.

Calculating the Sum of the Remaining Numbers

To find the sum of the remaining 18 numbers, we subtract the sum of the removed numbers from the total sum of the original 20 numbers.

$\text{Sum of remaining 18 numbers} = \text{Sum of 20 numbers} - \text{Sum of 2 removed numbers}$

$\text{Sum of remaining 18 numbers} = 640 - 60 = 580$

The sum of the remaining 18 numbers is 580.

Calculating the Average of the Remaining Numbers

Now we need to find the average of these 18 remaining numbers. We use the average formula again:

$\text{Average of remaining 18 numbers} = \frac{\text{Sum of remaining 18 numbers}}{\text{Number of remaining numbers}}$

$\text{Average of remaining 18 numbers} = \frac{580}{18}$

Let's perform the division:

$\frac{580}{18} = \frac{290}{9} \approx 32.2222...$

The question asks for the average correct up to two decimals. Rounding 32.2222... to two decimal places gives us 32.22.

Therefore, the average of the remaining 18 numbers is approximately 32.22.

Revision Table: Key Concepts

Additional Information: Average Calculations

The average, also known as the mean, is a fundamental concept in statistics used to represent a typical value in a set of data. It is sensitive to outliers (extremely high or low values).

• When values are added to or removed from a dataset, the total sum changes, which in turn affects the average.
• If the removed values are higher than the original average, the new average of the remaining numbers will likely decrease.
• If the removed values are lower than the original average, the new average of the remaining numbers will likely increase.
• In this problem, the original average was 32. The removed numbers (29 and 31) average $\frac{29+31}{2} = 30$, which is lower than the original average. As expected, the average of the remaining numbers (32.22) is slightly higher than the original average.