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## Example Questions

### Example Question #1 : Dsq: Calculating Median

What is the median number of students assigned per workshop at School R?

(1) 30% of the workshops at School R have 6 or more students assigned to each workshop.

(2) 40% of the workshops at School R have 4 or fewer students assigned to each workshop.

**Possible Answers:**

E. Statements (1) and (2) TOGETHER are NOT sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

**Correct answer:**

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

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.

### Example Question #2 : Dsq: Calculating Median

The median of the numbers , , , and is . What is equal to?

**Possible Answers:**

None of the other answers are correct.

**Correct answer:**

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

### Example Question #2 : Dsq: Calculating Median

What is the mean of this set?

1)

2)

**Possible Answers:**

**Correct answer:**EITHER statement ALONE is sufficient to answer the question.

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.

### Example Question #3 : Dsq: Calculating Median

Data Sufficiency Question

The mean of 8 numbers is 17. Is the median also 17?

1. The range of the numbers is 11.

2. The mode is 17.

**Possible Answers:**

each statement alone is sufficient

both statements taken together are sufficient to answer the question, but neither statement alone is sufficient

statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question

statements 1 and 2 together are not sufficient, and additional data is needed to answer the question

statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question

**Correct answer:**

statements 1 and 2 together are not sufficient, and additional data is needed 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.

### Example Question #5 : Dsq: Calculating Median

On Monday, 40 people are asked to rate the quality of product A on a seven point scale (1=very poor, 2=poor.....6=very good, 7=excellent).

On Tuesday, a different group of 40 is asked to rate the quality of product B using the same seven point scale.

The results for product A:

7 votes for category 1 (very poor);

8 votes for category 2;

10 votes for 3;

6 vote for 4;

4 votes for 5;

3 votes for 6;

2 votes for 7;

The results for product B:

2 votes for category 1 (very poor);

3 votes for category 2;

4 votes for 3;

6 vote for 4;

10 votes for 5;

8 votes for 6;

7 votes for 7;

It appears that B is the superior product.

Which one of the following statements is true?

**Possible Answers:**

The mean score of product A is greater than the mean score of product B.

The median score for product A is less than the mean score for product A.

The median score of product B is less than the mean score of product B.

The median score of product A is greater than the mean score of product A.

The median score of product A is greater than the median score of product B.

**Correct answer:**

The median score for product A is less than the mean score for product A.

Median of A = 3

Mean of A = 3.2

Median of B = 5

Mean of B = 4.8

### Example Question #6 : Dsq: Calculating Median

What is the median of the following numbers?

Statement 1:

Statement 2: and

**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.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 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:**

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

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.

### Example Question #6 : Dsq: Calculating Median

Consider the data set

What is the value of ?

Statement 1: The data set is bimodal.

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

**Possible Answers:**

EITHER 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.

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

BOTH statements TOGETHER are insufficient 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:**

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

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.

### Example Question #7 : 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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

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.

### 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.

**Correct answer:**

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.

### Example Question #10 : Dsq: Calculating Median

Given five distinct positive integers - - which of them is the median?

Statement 1:

Statement 2:

**Possible Answers:**

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

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

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 .

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