To analyze what happens to the median and mean of the set of data (the numbers of trading cards owned by the 10 middle-school students), let's first define the median and the mean:
- Median: The median is the middle number in a sorted list. If there is an even number of data points, the median is the average of the two middle numbers.
- Mean: The mean is the average of all the numbers, calculated by summing all the numbers and dividing by the number of data points.
Given the ordered data:
- Data Set: 294, 402, 429, 437, 458, 486, 517, 533, 551, 573
(A) What happens to the median?
Since there are 10 numbers (an even count), the median will be the average of the 5th and 6th numbers in the ordered list.
- The 5th number is 458.
- The 6th number is 486.
Median Calculation: \[ \text{Median} = \frac{458 + 486}{2} = \frac{944}{2} = 472 \]
If we were to add another number to the dataset, the median would adjust depending on the new number's position. However, without a new data point, the median remains at 472.
(B) What happens to the mean?
To find the mean, we sum all the numbers and divide by the total count (which remains 10).
Mean Calculation: First, calculate the sum of the values: \[ 294 + 402 + 429 + 437 + 458 + 486 + 517 + 533 + 551 + 573 \]
Calculating the sum step-by-step:
- \(294 + 402 = 696\)
- \(696 + 429 = 1125\)
- \(1125 + 437 = 1562\)
- \(1562 + 458 = 2020\)
- \(2020 + 486 = 2506\)
- \(2506 + 517 = 3023\)
- \(3023 + 533 = 3556\)
- \(3556 + 551 = 4107\)
- \(4107 + 573 = 4680\)
Now, divide the total sum by the number of students (10): \[ \text{Mean} = \frac{4680}{10} = 468 \]
Summary:
- Mean: The mean is 468.
- Median: The median is 472.
As datasets change (for example, if you add or remove a data point), both the mean and median can change based on the nature of the new data. If you provide more data points, I can show how exactly the median and mean would adjust.
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