### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #1 : Median

The median of seven consecutive integers is 129. What is the least integer?

**Possible Answers:**

It cannot be determined from the information given.

**Correct answer:**

The median of seven (an odd number) integers is the one in the middle when the numbers are arranged in ascending order; in this case, it is the fourth lowest. Since the seven integers are consecutive, the lowest integer is three less than the median, or .

### Example Question #2 : Median

Consider the data set

Which is the greater quantity?

(a) The mean of this data set

(b) The median of this data set

**Possible Answers:**

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

(a) is greater

**Correct answer:**

(a) is greater

(a) The mean of this data set is the sum divided by 10:

(b) The median of a data set with ten elements is the arithmetic mean of the fifth-highest and sixth-highest elements:

The set, arranged, is

The median is

This makes (a) the greater quantity

### Example Question #3 : How To Find Median

Which is the greater quantity?

(a) The median of the data set

(b) The median of the data set

**Possible Answers:**

(a) and (b) are equal.

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

**Correct answer:**

(a) and (b) are equal.

Each data set has ten elements, so the median in each case is the arithmetic mean of the fifth-highest and sixth-highest elements. In each data set, these elements are 10 and 10, so the median of each set is 10. Therefore, both quantities are equal.

### Example Question #3 : Median

Consider the data set .

Which is the greater quantity?

(a) The mean of this set

(b) The median of this set

**Possible Answers:**

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

**Correct answer:**

(b) is greater.

(a) The mean of this set is .

(b) Since there are elements, the median of this set is the seventh-highest number, which is .

Therefore (b) is the greater quantity.

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

Which is the greater quantity?

(a) The mean of the first ten prime numbers

(b) The median of the first ten prime numbers

**Possible Answers:**

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

**Correct answer:**

(a) is greater.

The first ten primes form the data set:

(a) Add these primes, and divide by :

(b) The median of a data set with ten elements is the arithmetic mean of the fifth-highest and sixth-highest elements. These are and , so the median is

.

(a) is the greater quantity.

### Example Question #5 : Median

A data set with nine elements has mean 10 and median 10. A new data set is formed with these nine elements, plus two new elements, 2 and 18.

Which is the greater quantity?

(a) The mean of the new data set

(b) The median of the new data set

**Possible Answers:**

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

(a) is greater.

**Correct answer:**

(a) and (b) are equal.

(a) The mean of nine elements is , so, if is their sum, and . The sum of the new data set is . Since the new set has elements, its mean is .

(b) The median of the nine elements is , so, when they are ranked, the fifth-highest element is . Since is less than and is greater than , when they are added to the set, is the sixth-highest of eleven elements, which is the median.

Therefore, both are equal to .

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

A data set has twelve elements; the mean and the median of the set are both 50.

A new data set is formed by increasing each element by 5. Which is the greater quantity?

(a) The mean of the new data set

(b) The median of the new data set

**Possible Answers:**

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

**Correct answer:**

(a) and (b) are equal

(a) Since each element of the old set is increased by 5, the sum of the elements is increased by . This increases the mean by , to 55.

(b) The median of the old set is the mean of the sixth- and seventh-highest elements. Since each element of the old set is increased by 5, these elements remain the sixth- and seventh-highest elements; their sum is increased by 10, and their mean is increased by 5, to 55.

The mean and the median of the new set are equal.

### Example Question #7 : Median

Which is the greater quantity?

(a) The median of the data set

(b)

**Possible Answers:**

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

(b) is greater

**Correct answer:**

(a) and (b) are equal

The median of a data set with five elements is its third-greatest element, which here is .

### Example Question #8 : How To Find Median

Which is the greater quantity?

(a) The median of the data set

(b)

**Possible Answers:**

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

(a) is greater

**Correct answer:**

(a) and (b) are equal

The median of a data set with seven elements is its fourth-greatest element, which here is .

### Example Question #8 : Median

A data set has nine elements. Four of the elements are greater than 50; four are less than 50.

Which is the greater quantity?

(a) The median of the data set

(b) 50

**Possible Answers:**

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

**Correct answer:**

(a) and (b) are equal.

If four of the elements are greater than 50 and four are less than 50, then the fifth-highest element, which is the median of a nine-element set, *must* be 50.