GMAT Math : Descriptive Statistics

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Example Question #6 : Dsq: Calculating Median

What is the median of the following numbers?

$\frac{1}{4},\frac{1}{5}, \frac{1}{6},\frac{1}{7},\frac{1}{8},\frac{1}{A},\frac{1}{B}$

Statement 1: B > A

Statement 2: A > 8 and B > 8

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

Statement 1 alone would not be helpful.

Example 1: If A = 2 and B = 3 , the list, in descending order, is $\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5}, \frac{1}{6},\frac{1}{7},\frac{1}{8}$ and the median would be $\frac{1}{5}$ .

Example 2: If A = 9 and B = 10 , the list, in descending order, is $\frac{1}{4},\frac{1}{5}, \frac{1}{6},\frac{1}{7},\frac{1}{8},\frac{1}{9},\frac{1}{10}$ and the median would be $\frac{1}{7}$ .

In contrast, if Statement 2 is true, since A > 8 and B > 8 , $\frac{1}{A} < \frac{1}{8}$ and $\frac{1}{B} < \frac{1}{8}$ . Regardless of their relationship, this makes $\frac{1}{7}$ the fourth-highest number, and, therefore, the median.

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Example Question #7 : Dsq: Calculating Median

Consider the data set

$\left \{ 2, 4, 5, 6, 6, 6, 6, 7, 7, 8, 9, 10, 11, N, N\right \}$

What is the value of ?

Statement 1: The data set is bimodal.

Statement 2: The median of the data set is 7.

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

Statement 1 is sufficient to prove that N = 7 . If N = 7 , then each of 6 and 7 occurs four times, more than any other element, making the set bimodal. If $N \neq 7$ , 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 $N \geq 7$ .

For example, if N = 7 , the set becomes

$\left \{ 2, 4, 5, 6, 6, 6, 6, 7, 7, 7,7, 8, 9, 10, 11\right \}$

If N = 8 , the set becomes

$\left \{ 2, 4, 5, 6, 6, 6, 6, 7, 7, 8,8, 8, 9, 10, 11\right \}$

The median of both sets is 7.

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Example Question #8 : Dsq: Calculating Median

A data set comprises thirteen elements, the median of which is 75. Two new elements are added to the data set. Does the median change?

Statement 1: One of the elements added to the data set is 30.

Statement 2: One of the elements added to the data set is 40.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

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

$\left \{ 75,75,75,75,75,75,75,75,75,75,75,75,75\right \}$

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

$\left \{30, 40 ,75,75,75,75,75,75,75,75,75,75,75,75,75\right \}$ ,

which has median 75.

By contrast, the data set

$\left \{69, 70, 71,72,73,74, 75, 76, 77, 78, 79, 80, 81\right \}$

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

$\left \{30, 40, 69, 70, 71,72,73,74, 75, 76, 77, 78, 79, 80, 81\right \}$

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.

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Example Question #9 : Dsq: Calculating Median

A data set comprises eleven elements, the median of which is 60. Two new elements are added to the data set.

Does the median change?

Statement 1: One of the elements added to the set is 50.

Statement 2: One of the elements added to the set is 70.

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

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

$\left \{35, 40, 45, 50, 55, 60, 65, 70, 75, 80,85\right \}$

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:

$\left \{35, 40, 45, 50,50, 55, 60, 65, 70,70, 75, 80,85\right \}$

has median 60.

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

$\left \{35, 40, 45, 50,50,50, 55, 60, 65, 70, 75, 80,85\right \}$

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.

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Example Question #10 : Dsq: Calculating Median

Given five distinct positive integers - A,B,C,D,E - which of them is the median?

Statement 1: $AB < BE< B^{2}< BD< BC$

Statement 2: $\frac{A}{B} < \frac{E}{B} <1< \frac{D}{B} <\frac{ C}{B}$

Possible Answers:

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

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

A < E< B< D< C

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

Multiply each expression in

$AB < BE< B^{2}< BD< BC$

(Statement 1)

by $\frac{1}{B}$ :

$AB \cdot \frac{1}{B} < BE \cdot \frac{1}{B}< B^{2} \cdot \frac{1}{B}< BD \cdot \frac{1}{B}< BC \cdot \frac{1}{B}$

A < E< B< D< C .

Multiply each expression in

$\frac{A}{B} < \frac{E}{B} <1< \frac{D}{B} <\frac{ C}{B}$

(Statement 2)

by :

$\frac{A}{B} \cdot B < \frac{E}{B} \cdot B <1 \cdot B < \frac{D}{B} \cdot B <\frac{ C}{B} \cdot B$

A < E< B< D< C

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 .

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Example Question #61 : Descriptive Statistics

Given five distinct positive integers - A,B,C,D,E - which of them is the median?

Statement 1: A+ E < B + E< 2E < D + E< C + E

Statement 1: -2A > -2B > -2E > -2D > -2C

Possible Answers:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

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

A+ E < B + E ,

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

A+ E+ (-E) < B + E + (-E)

A < B .

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

A<B< E< D< C .

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

-2A > -2B , then

$-2A \cdot \left ( - \frac{1}{2}\right ) < -2B \cdot \left ( - \frac{1}{2}\right )$

and A < B .

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

A<B< E< D< C .

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Example Question #62 : Descriptive Statistics

Given five distinct positive integers - A,B,C,D,E - which of them is the median?

Statement 1: A < E < C < D < B

Statement 2: $\frac{A+B+C+D+E}{5} = C$

Possible Answers:

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

Explanation:

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: A = 25, B = 26, C= 27,D= 28, E= 29

The mean is

$\frac{A+B+C+D+E}{5} = \frac{25+26+27+28+29}{5} = \frac{135}{5} = 27 = C$

making this consistent with Statement 2.

The median is the middle element, C = 27 .

Case 2: A = 1, B = 2, C= 27,D= 3, E= 102

$\frac{A+B+C+D+E}{5} = \frac{1+2+27+3+102}{5} = \frac{135}{5} = 27 = C$

again, making this consistent with Statement 2.

The median is the middle element, D = 3 .

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Example Question #63 : Descriptive Statistics

What is the median of a data set comprising nineteen elements?

Statement 1: When arranged in ascending order, the ninth element is 72.

Statement 2: When arranged in descending order, the ninth element is 72.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is not sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is not sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

In a data set of 19 elements, 19 being odd, the median is the element that occurs in the $\frac{19+1}{2}=10th$ 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.

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GMAT Math : Descriptive Statistics

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #6 : Dsq: Calculating Median

Example Question #7 : Dsq: Calculating Median

Example Question #8 : Dsq: Calculating Median

Example Question #9 : Dsq: Calculating Median

Example Question #10 : Dsq: Calculating Median

Example Question #61 : Descriptive Statistics

Example Question #62 : Descriptive Statistics

Example Question #63 : Descriptive Statistics

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