Ordered sale prices are $210,000, $220,000, $220,000, $220,000, $240,000, $255,000 and $310,000. Which mode/median pair is correct?
Correct Answer
D) Mode $220,000; median $220,000
The value $220,000 occurs three times, more often than any other price, so it is the mode. The fourth of seven ordered observations is also $220,000, so it is the median. These are different definitions—frequency versus ordered position—and they need not agree in another data set. The highest observation is not the mode simply because it is largest.
Why This Is the Correct Answer
Option D is correct because $220,000 appears three times, more often than any other price, which makes it the mode. With seven ordered observations, the median is the fourth one, and the fourth price is also $220,000. The two measures happen to agree here, but they rest on different definitions: frequency versus ordered position.
Why the Other Options Are Wrong
Option A: Mode $240,000; median $220,000
Option A gives a mode of $240,000, but $240,000 appears only once, while $220,000 appears three times. The median of $220,000 is right, but the mode is wrong, so the pair fails.
Option B: Mode $210,000; median $255,000
Option B gives a mode of $210,000 and a median of $255,000. $210,000 is simply the lowest price and appears only once, and $255,000 is the sixth of seven ordered values, not the fourth (middle) one, which is $220,000.
Option C: Mode $310,000; median $220,000
Option C gives a mode of $310,000, but that is just the highest price and it appears only once. The median of $220,000 is right, but the largest value is not the mode simply because it is largest.
Mode is most, median is middle
MOde sounds like MOst (the most frequent value), and MEDian sounds like the MEDian strip in the middle of a road (the middle value).
How to use: Use this when a question asks you to pick out the mode, the median, or both from a list of prices.
Exam Tip
Count repeats first to find the mode, then use position (n + 1) / 2 on the ordered list to find the median. Do not assume the largest or smallest value is the mode.
Common Mistakes to Avoid
- -Choosing the highest or lowest price as the mode
- -Taking the median from the wrong position in the ordered list
- -Assuming the mode and the median must be different values
Concept Deep Dive
Analysis
This question tests whether you can tell the mode from the median. The mode is the value that occurs most often, and the median is the middle value once the data are in order. Count how often each price appears, then locate the middle position of the seven ordered prices.
Background Knowledge
The mode is the most frequently occurring value in a data set, and the median is the value with half the data below it and half above it. For an odd number of ordered observations, the median is the single middle observation, found at position (n + 1) / 2. These measures of central tendency can match or differ depending on the data.
Real-World Application
An appraiser reviewing seven recent sales in a tract subdivision notices that three identical models sold for $220,000. Reporting both the most common price and the middle price helps describe the typical sale in the neighborhood section of the report.
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