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Difficulty: MediumData Distributions and Measures

A small design firm employs 1010 people. The mean of their annual salaries is $50,000\$50,000, and the median is $48,000\$48,000. The employee with the highest salary of $95,000\$95,000 receives a $10,000\$10,000 raise, while the salaries of all other employees remain unchanged. What are the mean and the median of the salaries after the raise?

  1. A
    Mean: $50,000\$50,000; Median: $49,000\$49,000
  2. Mean: $51,000\$51,000; Median: $48,000\$48,000Answer
  3. C
    Mean: $51,000\$51,000; Median: $49,000\$49,000
  4. D
    Mean: $60,000\$60,000; Median: $48,000\$48,000

Answer

The new mean salary is $51,000\$51,000 and the new median salary is $48,000\$48,000.
The correct option correctly identifies that the mean salary increases to $51,000\$51,000 and the median salary remains at $48,000\$48,000. Since the sum of the salaries increases by $10,000\$10,000 and there are 1010 employees, the mean increases by $10,000÷10=$1,000\$10,000 \div 10 = \$1,000, resulting in a new mean of $51,000\$51,000. Because the raise is given to the employee who already earns the highest salary, the sorted order of the salaries is unaffected, meaning the middle values remain unchanged, so the median remains $48,000\$48,000.

Step-by-Step Solution

1
Calculate the new mean salary by finding the effect of the salary raise on the average.
The total sum of all salaries increases by $10,000\$10,000. Since there are 1010 employees, the mean salary increases by $10,00010=$1,000\frac{\$10,000}{10} = \$1,000. Therefore, the new mean is $50,000+$1,000=$51,000\$50,000 + \$1,000 = \$51,000.
The mean is calculated as the sum of all values divided by the number of values. A change in the sum changes the mean proportionally.
2
Determine the new median salary by analyzing if the order of the dataset has changed.
The raise is given to the employee who already has the highest salary of $95,000\$95,000. Since this remains the highest salary (now $105,000\$105,000), the sorted order of the 1010 salaries does not change. The median is the average of the 5th and 6th values in the sorted list, which remain unchanged. Thus, the median remains $48,000\$48,000.
The median represents the middle value of a sorted dataset and is insensitive to changes in extreme values that do not alter the sorted sequence.

Key Concept

Understanding how individual data point adjustments affect the mean and median of a distribution.
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Topics

Data Distributions and Measures

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