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.