The average of twenty-five numbers is 54. The average of the first 13 numbers and that of the last 13 numbers is 52.8 and 62.2, respectively. If the 13 th number is excluded, then what is the average of the remaining numbers (correct to one decimal place)?
50.2
Calculating Averages and Excluding a Number
The problem asks us to find the average of a set of numbers after excluding a specific number. We are given the average of the total set, and the averages of the first and last parts of the set, which overlap at the number being excluded.
Understanding the Given Information
We are given:
- Total number of elements: 25
- Average of the 25 numbers: 54
- Number of elements in the first group: 13
- Average of the first 13 numbers: 52.8
- Number of elements in the last group: 13
- Average of the last 13 numbers: 62.2
The first 13 numbers and the last 13 numbers together cover $13 + 13 = 26$ positions. Since there are only 25 numbers in total, this overlap means the 13th number (from the beginning or the end) is counted in both the first 13 and the last 13.
Step-by-Step Calculation
Step 1: Calculate the total sum of the 25 numbers.
The sum of a set of numbers is equal to the average multiplied by the number of elements.
Sum of 25 numbers = Average of 25 numbers $\times$ Total number of numbers
Sum of 25 numbers $= 54 \times 25 = 1350$
Step 2: Calculate the sum of the first 13 numbers.
Sum of first 13 numbers = Average of first 13 numbers $\times$ Number of first 13 numbers
Sum of first 13 numbers $= 52.8 \times 13 = 686.4$
Step 3: Calculate the sum of the last 13 numbers.
Sum of last 13 numbers = Average of last 13 numbers $\times$ Number of last 13 numbers
Sum of last 13 numbers $= 62.2 \times 13 = 808.6$
Step 4: Determine the value of the 13th number.
The sum of the first 13 numbers and the sum of the last 13 numbers combined includes the 13th number twice. The total sum of 25 numbers includes the 13th number only once. Therefore, the sum of the first 13 and last 13 minus the total sum of 25 will give us the value of the 13th number.
Sum of (first 13 numbers + last 13 numbers) $= 686.4 + 808.6 = 1495$
Value of the 13th number = (Sum of first 13 + Sum of last 13) - Sum of 25 numbers
Value of the 13th number $= 1495 - 1350 = 145$
Step 5: Calculate the sum of the remaining 24 numbers.
If we exclude the 13th number from the total set of 25 numbers, we are left with 24 numbers. The sum of these 24 numbers is the total sum of 25 minus the value of the 13th number.
Sum of remaining 24 numbers = Sum of 25 numbers - Value of the 13th number
Sum of remaining 24 numbers $= 1350 - 145 = 1205$
Step 6: Calculate the average of the remaining 24 numbers.
The average of the remaining numbers is the sum of these numbers divided by the count of these numbers.
Average of remaining 24 numbers = Sum of remaining 24 numbers / Number of remaining numbers
Average of remaining 24 numbers $= 1205 / 24$
Average of remaining 24 numbers $\approx 50.20833...$
Rounding the average to one decimal place, we get 50.2.
Summary of Calculations
| Description | Calculation | Result |
|---|---|---|
| Total Sum of 25 numbers | $54 \times 25$ | 1350 |
| Sum of First 13 numbers | $52.8 \times 13$ | 686.4 |
| Sum of Last 13 numbers | $62.2 \times 13$ | 808.6 |
| Sum of First 13 + Last 13 | $686.4 + 808.6$ | 1495 |
| Value of 13th Number | $1495 - 1350$ | 145 |
| Sum of Remaining 24 numbers | $1350 - 145$ | 1205 |
| Average of Remaining 24 numbers | $1205 / 24$ | $\approx 50.20833$ |
| Average (rounded to 1 decimal) | 50.2 |
The average of the remaining numbers when the 13th number is excluded is 50.2.
Revision Table: Average Calculations
| Concept | Formula | Notes |
|---|---|---|
| Average | Sum of elements / Number of elements | Measure of central tendency |
| Sum of elements | Average $\times$ Number of elements | Useful for total calculations |
| Finding an overlapping element | (Sum of overlapping groups) - (Total Sum) | Applies when an element is counted twice in group sums |
Additional Information: Properties of Averages
- The average is sensitive to outliers. A single very large or very small number can significantly affect the average.
- When a number is removed from a set, the new average depends on the value of the removed number relative to the original average. If a number greater than the average is removed, the new average will be lower. If a number less than the average is removed, the new average will be higher.
- In this problem, the original average is 54, and the excluded 13th number is 145. Since 145 is much greater than 54, removing it causes the average of the remaining numbers (50.2) to be lower than the original average.
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