Median - GMAT Quantitative

Median of A = 3

Mean of A = 3.2

Median of B = 5

Mean of B = 4.8

The four numbers appear in ascending order, so their median must be the arithmetic mean of the middle two numbers. Therefore,

In a data set of 19 elements, 19 being odd, the median is the element that occurs in the position when they are arranged in ascending (or descending) order. Neither statement alone tells us what that middle element is. But put together, they tell us what the ninth and eleventh elements would be when arranged in ascending (or descending) order. Since both are 72, the element between them - the tenth element, and, thus, the median - must be 72.

The answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.

The range tells us the difference between the maximum and the minimum values, but provides no information about the median. The mode indicates the number that appears most frequently in the data set. While it is possible that the median is 17, it is impossible to determine the median from the data provided.

Statement 1 alone would not be helpful.

Example 1: If and , the list, in descending order, is and the median would be .

Example 2: If and , the list, in descending order, is and the median would be .

In contrast, if Statement 2 is true, since and , and . Regardless of their relationship, this makes the fourth-highest number, and, therefore, the median.

Assume Statement 1 alone. The median of five numbers (an odd number) is the number in the middle when they are arranged in ascending order. By Statement 1, this number is , so the question is answered.

Statement 2 alone, however, gives that the mean is . It is possible that the mean and the median can be one and the same or two different numbers.

Case 1:

The mean is

making this consistent with Statement 2.

The median is the middle element, .

Case 2:

again, making this consistent with Statement 2.

The median is the middle element, .

Assume Statement 1 alone. By the Addition Property of Inequality, if

,

then can be added to each quantity to give an equivalent inequality:

.

By extension, the continued inequality in Statement 1 is equivalent to

.

A similar argument holds for Statement 2. By the Multiplication Property of Inequality, if

, then

and .

By extension, the continued inequality in Statement 2 is also equivalent to

.

The median of five numbers (an odd number) is the number in the middle when they are arranged in ascending order. Therefore, it can be established from either statement alone that of the five, the median is .

Both statements can be shown to be equivalent to the continued inequality

by way of the Multiplication Property of Inequality. We will demonstrate this as follows:

Multiply each expression in

(Statement 1)

by :

.

Multiply each expression in

(Statement 2)

by :

The median of five numbers (an odd number) is the number in the middle when they are arranged in ascending order. Therefore, it can be established from either statement alone that of the five, the median is .

If , then .

If , then

The two statements are equivalent. This is enough to allow the mean to be found:

The answer is that either statement alone is sufficient to answer the question.

If we only assume Statement 1, then by examining these two cases, we show that we do not know whether the median changes.

Suppose the original set is

If we only know that one of the elements added is 50, then it is possible that the median does not change - this happens if the other element added is 70:

has median 60.

It is also possible that the median does change - this happens if the other element added is 50.

has median 55.

A similar argument can be used to demonstrate that Statement 2 is insufficient.

However, suppose we know both. 60 is the sixth-highest element of the old data set, and, since one element greater than 60 and one less than 60 are added, 60 is the seventh-highest element of the new thirteen-element set. We therefore know the median did not change.

The median of a data set with thirteen elements is the seventh-highest element; the median of a data set with fifteen elements is the eighth-highest.

The two statements together provide insufficient information, as we show with two examples.

The data set

has thirteen elements and median 75; after 30 and 40 are added, the set is

,

which has median 75.

By contrast, the data set

has thirteen elements and median 75; after 30 and 40 are added, the set is

which has median 74.

Both sets started out with median 75, but the addition of 30 and 40 changed one median and not the other.

Statement 1 is sufficient to prove that . If , then each of 6 and 7 occurs four times, more than any other element, making the set bimodal. If , then no element other than 6 occurs more than three times, giving the set only one mode.

Statement 2 is insufficient. The median of this data set, which has fifteen elements, is its eighth-greatest element. This happens if .

For example, if , the set becomes

If , the set becomes

The median of both sets is 7.

The range tells us the difference between the maximum and the minimum values, but provides no information about the median. The mode indicates the number that appears most frequently in the data set. While it is possible that the median is 17, it is impossible to determine the median from the data provided.

Statement 1 alone would not be helpful.

Example 1: If and , the list, in descending order, is and the median would be .

Example 2: If and , the list, in descending order, is and the median would be .

In contrast, if Statement 2 is true, since and , and . Regardless of their relationship, this makes the fourth-highest number, and, therefore, the median.

Looking at statements (1) and (2) separately, we cannot get the median number since we don’t know the 50th percentile. However, putting the two statements together, we know that the 50th percentile is 5. So the median is 5.

Assume Statement 1 alone. By the Addition Property of Inequality, if

,

then can be added to each quantity to give an equivalent inequality:

.

By extension, the continued inequality in Statement 1 is equivalent to

.

A similar argument holds for Statement 2. By the Multiplication Property of Inequality, if

, then

and .

By extension, the continued inequality in Statement 2 is also equivalent to

.

The median of five numbers (an odd number) is the number in the middle when they are arranged in ascending order. Therefore, it can be established from either statement alone that of the five, the median is .

Median of A = 3

Mean of A = 3.2

Median of B = 5

Mean of B = 4.8

The four numbers appear in ascending order, so their median must be the arithmetic mean of the middle two numbers. Therefore,

In a data set of 19 elements, 19 being odd, the median is the element that occurs in the position when they are arranged in ascending (or descending) order. Neither statement alone tells us what that middle element is. But put together, they tell us what the ninth and eleventh elements would be when arranged in ascending (or descending) order. Since both are 72, the element between them - the tenth element, and, thus, the median - must be 72.

The answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.