The median salary for professional group A is \$40,610. The median salary for professional group B is \$40,810. :
Column A: The median salary for groups A and B combined
Column B: \$40,710
Column A: The median salary for groups A and B combined
Column B: \$40,710
Show Hint
- Quantity A is greater
- Quantity B is greater
- The two quantities are equal
- The relationship cannot be determined from the information given
The Correct Option is D
Solution and Explanation
The median is the middle value in a sorted dataset. To find the median of a combined group, we need to know the number of individuals in each group and the distribution of their salaries. Knowing only the medians of the individual groups is not sufficient.
Step 2: Detailed Explanation:
Let's analyze why we cannot determine the combined median. The combined median depends on how the two salary distributions interleave when merged.
Scenario 1: Assume each group has 3 people.
Group A salaries: \{\$40,000, \$40,610, \$50,000\}. Median is \$40,610.
Group B salaries: \{\$30,000, \$40,810, \$60,000\}. Median is \$40,810.
Combined list: \{\$30,000, \$40,000, \$40,610, \$40,810, \$50,000, \$60,000\}.
The combined median is the average of the two middle values: \((\$40,610 + \$40,810) / 2 = \$40,710\).
In this case, Column A = Column B.
Scenario 2: Assume Group A has 3 people and Group B has only 1.
Group A salaries: \{\$40,000, \$40,610, \$50,000\}. Median is \$40,610.
Group B salary: \{\$40,810\}. Median is \$40,810.
Combined list: \{\$40,000, \$40,610, \$40,810, \$50,000\}.
The combined median is the average of the two middle values: \((\$40,610 + \$40,810) / 2 = \$40,710\).
In this case, Column A = Column B.
Scenario 3: Change the distribution.
Group A salaries: \{\$40,600, \$40,610, \$40,620\}. Median is \$40,610.
Group B salaries: \{\$40,800, \$40,810, \$40,820\}. Median is \$40,810.
Combined list: \{\$40,600, \$40,610, \$40,620, \$40,800, \$40,810, \$40,820\}.
The combined median is \((\$40,620 + \$40,800) / 2 = \$40,710\).
It seems that in many simple cases it equals \$40,710. However, consider this:
Group A salaries: \{\$10,000, \$10,000, \$40,610, \$100,000, \$100,000\}. Median is \$40,610.
Group B salaries: \{\$40,810\}. Median is \$40,810.
Combined list: \{\$10,000, \$10,000, \$40,610, \$40,810, \$100,000, \$100,000\}.
The median is again \$40,710. The key is the relative sizes of the groups.
Let Group A have 101 members, all earning \$40,610. Median = \$40,610.
Let Group B have 1 member, earning \$40,810. Median = \$40,810.
The combined group has 102 members. 101 are \$40,610 and one is \$40,810. The sorted list has \$40,610 in positions 1 through 101 and \$40,810 in position 102. The middle values are at positions 51 and 52. Both are \$40,610. The combined median is \$40,610, which is less than \$40,710.
Since we can construct a case where the median is less than \$40,710 and a case where it could be greater (by swapping the group sizes), the relationship cannot be determined.
Step 3: Final Answer:
Without knowing the number of people in each group or the distribution of salaries, the combined median cannot be determined. It can be less than, greater than, or equal to \$40,710.
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