Last month 15 homes were sold in Town X. The average (arithmetic mean) sale price of the homes was $150,000 and the median sale price was $130,000. Which of the following statements must be true?

I. At least one of the homes was sold for more than $165,000.
II. At least one of the homes was sold for more than $130,0000 and less than $150,000
III. At least one of the homes was sold for less than $130,000.

(A) I only
(B) II only
(C) III only
(D) I and II
(E) I and III

This question is from the GMAT prep tests 1 and 2 so if you haven't done those yet skip this one for now. You're given average and median and with those two pieces of information need to make some inferences. Know your basic GMAT content. There shouldn't be any confusion about fundamental statistics terms, average, median, mean, and standard deviation. For standard deviation you don't need to know the formula or how to calculate it but should have a strong understanding of what it measures.

OK, back to the question. What does it mean that the median sale prices was $130,000? Well, median is a spatial measurement. It's a location: the middle of the set. For an odd set it's just the middle number (in this case the 8th). For an even set it is the average of the two middle numbers. OK, so if you have a median of $130,000 in a 15 number set then it must be that 7 numbers are equal to or below $130,000 and 7 numbers are equal to or above above $130,000. With that information alone none of the statements MUST be true. They all COULD be true.

I. At least one of the homes was sold for more than $165,000. Sure. All of the houses above the median could have been sold for $1.2 million. Or not. They could all have been sold for $140,000. There's nothing limiting this.
II. At least one of the homes was sold for more than $130,0000 and less than $150,000 Same idea as statement one. Easy to have home in that range or not.
III. At least one of the homes was sold for less than $130,000. All of the homes could have been sold at the median or you could have 7 houses below and 7 above. There a variety of ways to make this true or not true.

The average sale price being $150,000 also doesn't allow us to commit to any of the statements. They still all COULD be true.

I. At least one of the homes was sold for more than $165,000. With the mean at $150,000 we know that the balance of things ends up at $150,000 but how we get there is completely unknown. You could have 14 houses sell for $1 and the 15 house sell for way above $150,000 so the average balances at $150,000. You could have all 15 houses sell for $150,000. Again, there many ways to do this.

II. At least one of the homes was sold for more than $130,0000 and less than $150,000 Same idea as statement one. Easy to have home in that range or not.
III. At least one of the homes was sold for less than $130,000. All of the homes could have been sold at the average ($150,000) or you could have a home sold for less than $130,000 with at least one of the other houses above the average to balance out to the average.

 

Putting the mean and median together it becomes clear that at least some of houses need to be above $130,000 in order to create a mean of $150,000. That puts a shadow over statement III. It doesn't seem that we any limitations on the downside. Our main concern is having enough $'s above the median to balance things out to $150,000. Statement II has the same issue. There's nothing to limit the sales prices that are above the mean to the range of $130,000 and $150,000. With that we can pick A.

But how do we prove that at least one of the homes was sold for more than $165,000? Let's say we maximize the value of the houses to the left of the median. To do that we'd make them all equal the median at $130,000. If you have 7 houses at $130,000 what do the 7 houses above the median have to be to balance to a mean of $150,000? There's an equal number of houses (7 and 7) so just think of it as one house at $130,000. What does the other house have to be: $170,000. Add those and divide by two and you get $150,000. But, you might say, what about the median? We didn't count that. No, we didn't. But will the median at $130,000 make it so the $170,000 is higher or lower? We're adding more weight to the $130,000 side so the $170,000 would need to be increased to compensate. At $170,000 we're already clearing the $165,000 line from statement I so adding the median just pushes us further in that direction. Again, A MUST be true.

Additional GMAT Statistics Example Questions!

Here's a GMAT statistics question (median and range) which will teach you how to organize a whole bunch of unknowns without using variable. Also a good one to work on maximizing a value.

A set of 15 different integers has median of 25 and a range of 25. What is greatest possible integer that could be in this set?

Here's a mini-statistics puzzle from the GMAT Official Guide. It has a somewhat simple solution. Still, it's surprising how often it comes up in GMAT tutoring sessions.

If Q is an odd number and the median of Q consecutive integers is 120, what is the largest of these integers?

Here's a challenging statistics, max/min question which we review in just about every GMAT preparation. It's a great one to review not only to understand how to organize a whole bunch of potential variables but to understand how to put context on a max/min scenario.

Seven pieces of rope have an average (arithmetic mean) length of 68 centimeters and a median length of 84 centimeters. If the length of the longest piece of rope is 14 centimeters more than 4 times the length of the shortest piece of rope, what is the maximum possible length, in centimeters, of the longest piece of rope?

Meat and potatoes GMAT statistics question. Not the toughest but it is still very important. Be ready to interpret a chart/graph. What isn't challenging in practice can quickly become a mess on test day.

The table above shows the distribution of test scores for a group of management trainees. Which score interval contains the median of the 73 scores?

Middle difficulty DS statistics range question from GMAT question of the day

And here's a very challenging GMAT statistics range question from GMAT question of the day that also incorporates a bunch of algebra and number properties. Stay organized!

Good luck and happy studies!

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