Descriptive Statistics - GMAT Quantitative

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Question

Data sufficiency question- do not actually solve the question

Find the mean of a set of 5 numbers.

1. The sum of the numbers is 72.

2. The median of the set is 15.

Answer

Statement 2 does not provide enough information about the mean as it can vary greatly from the median. Statement 1 is sufficient to calculate the mean, because even though it is impossible to calculate the set of numbers, the mean is calculated by dividing the sum by the total number of incidences in the set.

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Question

How much greater is the average of the integers from 500 to 700 than the average of the integers from 60 to 90?

Answer

In this case, average is also the middle value.

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Question

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.

Answer

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.

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Question

What is the median of the following numbers?

Statement 1:

Statement 2: and

Answer

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.

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Question

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.

Answer

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.

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Question

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

Statement 1:

Statement 1:

Answer

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 .

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Question

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?

Answer

Median of A = 3

Mean of A = 3.2

Median of B = 5

Mean of B = 4.8

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Question

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

Answer

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

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Question

What is the mean of this set?

Answer

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.

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Question

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.

Answer

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.

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Question

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.

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.

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Question

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.

Answer

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.

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Question

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.

Answer

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.

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Question

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

Statement 1:

Statement 2:

Answer

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, .

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Question

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

Statement 1:

Statement 2:

Answer

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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Question

What is the mode of a data set with ten data values?

  1. The value 15 occurs four times in the data set.

  2. The value 16 occurs three times in the data set.

Answer

If we are given only that 15 occurs four times in the data set, it is possible that another number can occur up to six times; similarly, if we are given only that 16 occurs three times, it is possible that another number can occur up to seven times. Either way, the mode - the most frequently occurring data value - cannot be determined.

However, if we know both facts, then no other data value can occur more than three times, so 15 must be the mode.

Therefore, the answer is that both statements are sufficient, but not one alone.

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Question

Give the median of the data set

,

where and are integers.

Answer

Suppose but is unknown. The above set, with known elements ordered, is

The median is the arithmetic mean of the two middle elements, when the elements are ordered.

Regardless of the value of , the two middle elements must both be 84, making the median 84.

Now, suppose but is unknown. The above set, with known elements ordered, is

The median cannot be determined with certainty. For example, if , as stated before, the median is 84. But if , the middle elements are 84 and 87, making the median 85.5.

The answer is that Statement 1 alone is sufficent to answer the question, but Statement 2 alone is not.

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Question

The set

is bimodal. What is equal to?

Answer

If we know that , then the set is known to have one 24, five 26's, three 27's, two 28's, and one 29. The only way the set can have two modes is for and ; this makes 27 occur five times, just as frequently as 26.

If we know , however, the set is known to have one 24, four 26's, four 27's, two 28's, and one 29. There are two ways for the set to have two modes (26 and 27): for and , or for and .

The answer is that Statement 1 alone is sufficient to answer the question, but not Statement 2.

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Question

Consider this data set:

Which of the following statements correctly compares the median and the mode?

Answer

The median of a data set with an even number of elements is the arithmetic mean of the two elements that fall in the middle when the elements are arranged in ascending order. These two elements are both 6, so 6 is the median.

The mode of a data set is the element that occurs most frequently. Since 6 appears thre times, 7 appears two times, and all other elements appear once each, the mode is 6.

Therefore, the median and the mode are equal.

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Question

Consider the data set

What is the value of ?

Statement 1: The data set is bimodal.

Statement 2: The mean of the data set is 6.5.

Answer

The data set has four 6's and no more than two of any other element - and there cannot be more than four of any other element regardless of the value of - so 6 must be one of the modes. For the set to be bimodal, there must be four of another element. Since occurs twice, it must be set to a number known to occur exactly two other times. There are, however, two choices, 5 and 7, so Statement 1 is insufficient.

Statement 2 is sufficient, as seen below:

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