### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #10 : Mode

Compare the mean and the mode in the following set of data:

**Possible Answers:**

The mean is greater than the mode.

It is not possible to compare the mean and the mode based on the information given.

The mode is greater than the mean.

The mean and the mode are equal.

**Correct answer:**

The mean is greater than the mode.

The mode of a set of data is the value which occurs most frequently, which in this case is .

The mean of a set of data is given by the sum of the data, divided by the total number of values in the set. So we can write:

So the mean is greater than the mode.

### Example Question #11 : How To Find Mode

Consider the following set of data where is an unknown value:

Which one is greater:

**Possible Answers:**

is greater

and are equal.

It is not possible to compare the mean and the mode based on the information given.

is greater

**Correct answer:**

and are equal.

The mode of a set of data is the value which occurs most frequently. The value of appears at least four times in the set. So regardless of the value of , none of the other values can appear more than two times. So is the mode.

### Example Question #141 : Data Analysis

A data set has six known quantities and one unknown quantity, as follows:

Which is the greater quantity?

(A) The mode of the set if

(b) The mode of the set if

**Possible Answers:**

(B) is greater

It is impossible to determine which is greater from the information given

(A) and (B) are equal

(A) is greater

**Correct answer:**

(A) is greater

The mode of a data set is the element that occurs most frequently.

If , the set becomes .

The mode is 35, which occurs three times, more than any other element.

If , the set becomes .

The mode is 25, which occurs three times, more than any other element.

(A) is greater.

### Example Question #11 : Mode

Consider the data set .

Which of the following elements replaces the box to make the data set bimodal?

(A)

(B)

(C)

**Possible Answers:**

(C) only

None of (A), (B), or (C) are correct.

(A) only

(A), (B), and (C) are all correct.

(B) only

**Correct answer:**

None of (A), (B), or (C) are correct.

At current, 40 appears four times, more than any other value. For a set to be bimodal, it needs to have *two* modes; that is, another value would have to appear four times as well. Regardless of whether 30, 50, 60, or any other value replaces the box, this is impossible. The set cannot be made bimodal by adding one element.

### Example Question #142 : Data Analysis

Both

and

are data sets with 2 as their only mode.

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(a) and (b) are equal

(b) is the greater quantity

**Correct answer:**

It cannot be determined which of (a) and (b) is greater

The mode of a data set is the element that occurs the most frequently in the set.

In the data set , if , it appears four times, more than any other element. If , 1 and 2 each appear three times, making the set bimodal, so 2 is not the *only *mode, but one of two modes. If is equal to any other number, 2 appears three times, again, more than any other element. Therefore, can be any number except 1.

In the data set , each of 1, 2, and 3 appears twice. The only value of that makes 2 appear the most times, and therefore, the only mode , is .

We therefore know that , but we do not know ; we do not know which one is the greater.

### Example Question #143 : Data Analysis

is a data set with exactly one mode.

is a data set with exactly one mode.

is a data set with exactly one mode.

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(b) is the greater quantity

(a) and (b) are equal

**Correct answer:**

(a) and (b) are equal

The mode of a set is the element that occurs in the set most frequently. A set without any repeated element is considered to have no mode.

For there to be a mode in the set , it must hold that ; the value of is the mode of this set. Similarly, ..

If , then, in the set , two of the elements in appear twice, giving the set two modes. If , then the value they are equal to appears three times as opposed to the other values, which appear one time each. That gives the set exactly one mode.

### Example Question #144 : Data Analysis

and

are both bimodal sets.

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(b) is the greater quantity

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(a) and (b) are equal

**Correct answer:**

(b) is the greater quantity

The mode of a set is the element that occurs in the set most frequently; a set is bimodal if two elements tie for this.

Consider . If , then 1 and each appear twice, more than any other element; this makes the set bimodal. If , the set has one mode, 1, since it appears three times, more than any other element. If has any other value, then the set has one mode, 1. Therefore, . For similar reasons, . Therefore, .

### Example Question #1 : Range

Examine this stem-and-leaf diagram for a set of data:

Which is the greater quantity?

(a) The range of the data?

(b)

**Possible Answers:**

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

(b) is greater

**Correct answer:**

(a) and (b) are equal

The "stem" of this data set represents the tens digits of the data values; the "leaves" represent the units digits.

The range of a data set is the difference of the high and low values. The highest value represented is 87 (7 is the last "leaf" in the bottom, or, 8, row); the low value is 47 (7 is the first "leaf" in the top, or, 4, row). The difference is , which is the range.

### Example Question #2 : How To Find Range

Consider the set of numbers:

Quantity A: The sum of the median and mode of the set

Quantity B: The range of the set

**Possible Answers:**

The two quantities are equal.

The relationship cannot be determined from the information given.

Quantity B is greater.

Quantity A is greater.

**Correct answer:**

Quantity A is greater.

Quantity A: The median (middle number) is , and the mode (most common number) is , so the sum of the two numbers is .

Quantity B: The range is the smallest number subtracted from the largest number, which is .

Quantity A is larger.

### Example Question #1 : How To Find Range

In the following set of data compare the median and the range:

**Possible Answers:**

The median and the range are equal

The median is greater than the range

The range is greater than the median

It is not possible to compare the mean and the mode based on the information given

**Correct answer:**

The range is greater than the median

The median is the average of the two middle values of a set of data with an even number of values. So we have:

The range is the difference between the lowest and the highest values. So we have:

So the range is greater than the median.

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