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Chapter 1: Problem 9

a. Compute the mean and median for the list: 5,18,22,46 , 80,105,110 b. Change one of the entries in the list in part (a) so that the median stays the same but the mean increases.

Short Answer

Expert verified
Mean: 55.14, Median: 46. Change 5 to 50, new mean: 61.57, median: 46.

Step by step solution

01

Organize the Data

List of numbers: 5, 18, 22, 46, 80, 105, 110. Arrange the numbers in ascending order. Since they are already in order, proceed to the next step.
02

Compute the Mean

To find the mean, sum all the numbers and then divide by the number of entries. Sum: 5 + 18 + 22 + 46 + 80 + 105 + 110 = 386. Mean: \(\frac{386}{7} \approx 55.14\)
03

Find the Median

To determine the median, locate the middle number in the ordered list. As there are 7 numbers, the 4th number is the median. Median: 46.
04

Change One Entry for Same Median, Higher Mean

Identify the effect of changing an entry: pick a number larger than the current entry but ensure the order around the median remains. Change 5 to 50 (a higher number without disturbing the order around the median). New list: 18, 22, 46, 50, 80, 105, 110.
05

Compute New Mean

Sum with the changed number: 18 + 22 + 46 + 50 + 80 + 105 + 110 = 431. New Mean: \(\frac{431}{7} \approx 61.57\).
06

Confirm the Median

Check if the new list's median remains 46: new ordered list is 18, 22, 46, 50, 80, 105, 110. Median unchanged: 46.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistical Analysis
Statistical analysis is a key part of understanding data. It involves collecting, reviewing, and interpreting data to uncover patterns and trends. In our example, we use two important metrics: the mean and the median. The mean gives us an average value, while the median shows the middle value in a list. Both metrics provide a snapshot of the data but highlight different aspects. Mean is sensitive to extreme values, while the median remains stable even if some data points are very large or very small.
Data Organization
Organizing data is crucial before performing any calculations. It ensures clarity and accuracy. For our problem, the numbers are already sorted: 5, 18, 22, 46, 80, 105, 110. This simplifies our calculations for the mean and median. Always arrange data in ascending or descending order to find medians easily. This step minimizes errors and makes analysis straightforward.
Mean Calculation
The mean, or average, is calculated by summing all the data points and dividing by the number of data points. Using our list: 5, 18, 22, 46, 80, 105, 110:
Sum: 5 + 18 + 22 + 46 + 80 + 105 + 110 = 386
Number of entries: 7
Mean: \(\frac{386}{7} \approx 55.14\)
This mean represents an average value from our dataset. If we need to increase the mean while keeping the median the same, we can adjust one of the numbers, ensuring the median remains undisturbed.
Median Determination
The median is the middle value in a data set when it is ordered. For an odd number of entries, it is the middle one. For our list: 5, 18, 22, 46, 80, 105, 110:
Number of entries: 7
Middle position: 4th value
Median: 46
Even when we change one of the entries to increase the mean, we must ensure that the median value stays the same. By changing 5 to 50 (keeping order: 18, 22, 46, 50, 80, 105, 110), the median remains at 46, but the new sum is 431, raising the mean to \(\frac{431}{7} \approx 61.57\).

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