Data Analysis - ISEE Middle Level Quantitative Reasoning

Card 0 of 3225

Question

Which is the greater quantity?

(A) The number of modes of the data set $\left \{ 57, 43, 98, 103, 22, 45, 73 \right \}$

(B)

Answer

The mode of a data set is the element that appears most frequently. The given data set has no repeated elements; it is considered to have no modes. (B) is greater.

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Question

Which is the greater quantity?

(A) The median of the data set $\left\{ 1, 10, 100, 1000, 10000 \right \}$

(B) The median of the data set $\left\{ 80, 90, 100, 110, 120 \right \}$

Answer

Each data set has five elements; the median is the element in the middle after the elements are arranged in ascending sequence, which both are. In each case, the median will be the third-highest element. Since in both data sets, this element is 100, the medians are equal.

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Question

$\left \{ 11,10,9,10,15,12,10 \right \}$

Find the range of the data set above.

Answer

To find the range, you must subtract the lowest term from the greatest term in the data set. The greatest term is 15. The lowest term is 9. Therefore, to find the range, you calculate . The answer is 6.

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Question

Give the mode(s) of this data set:

$\left \{ 1,3,4,4,6,7,7,7,9,10 \right \}$

Answer

The mode of a data set is the value or values that appears the most frequently. 7 appears three times, 4 appears two times, and no other value appears more than once. 7 is therefore the only mode.

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Question

Define two sets as follows:

$A = \left \{ 8, 16, 24, 32, 40, 48, 56...\right \}$

$B = \left \{ 1, 4, 9, 16, 25, 36, 49,...\right \}$

Which of the following is not an element of the set $A \cup B$ ?

Answer

$A \cup B$ is the union of and , the set of all elements that appear in either set_._ Therefore, we are looking to eliminate the elements in and those in to find the element in neither set.

is the set of all multiples of 8. We can eliminate two choices as mulitples of 8:

$272 \div 8 = 34$ , so $272 \in A$

$344\div 8 = 43$ , so $344 \in A$

is the set of all perfect square integers. We can eliminate two additional choices as perfect squares:

$\sqrt{225}=15$ , so $225 \in B$

$\sqrt{441}=21$ , so $441 \in B$

All four of the above are therefore elements of $A \cup B$ .

420, however is in neither set:

$420 \div 8 = 52 \textrm{ R } 4$ , so $420 \notin A$

and

$20 = \sqrt{400}<\sqrt{420} < \sqrt{441} = 21$ , so $420 \notin B$

Therefore, $420 \notin A \cup B$ , making this the correct choice.

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Question

$\left \{ 11,10,9,10,15,12,10 \right \}$

Find the range of the data set above.

Answer

To find the range, you must subtract the lowest term from the greatest term in the data set. The greatest term is 15. The lowest term is 9. Therefore, to find the range, you calculate . The answer is 6.

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Question

Give the mode(s) of this data set:

$\left \{ 1,3,4,4,6,7,7,7,9,10 \right \}$

Answer

The mode of a data set is the value or values that appears the most frequently. 7 appears three times, 4 appears two times, and no other value appears more than once. 7 is therefore the only mode.

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Question

Define two sets as follows:

$A = \left \{ 8, 16, 24, 32, 40, 48, 56...\right \}$

$B = \left \{ 1, 4, 9, 16, 25, 36, 49,...\right \}$

Which of the following is not an element of the set $A \cup B$ ?

Answer

$A \cup B$ is the union of and , the set of all elements that appear in either set_._ Therefore, we are looking to eliminate the elements in and those in to find the element in neither set.

is the set of all multiples of 8. We can eliminate two choices as mulitples of 8:

$272 \div 8 = 34$ , so $272 \in A$

$344\div 8 = 43$ , so $344 \in A$

is the set of all perfect square integers. We can eliminate two additional choices as perfect squares:

$\sqrt{225}=15$ , so $225 \in B$

$\sqrt{441}=21$ , so $441 \in B$

All four of the above are therefore elements of $A \cup B$ .

420, however is in neither set:

$420 \div 8 = 52 \textrm{ R } 4$ , so $420 \notin A$

and

$20 = \sqrt{400}<\sqrt{420} < \sqrt{441} = 21$ , so $420 \notin B$

Therefore, $420 \notin A \cup B$ , making this the correct choice.

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Question

Give the mode of the data set $\left \{ 2, 4, 8, 16, 32, 64\right \}$

Answer

No element in the data set appears more than once; for an element to have a mode, that element must appear at least twice. The set has no mode.

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Question

Consider the data set $\left \{ 20, 30, 30, 40, 40, 40, 40, 50, 50, 60, \square \right \}$

Which of the following elements replaces the box to make 30 the mode of the data set?

Answer

Regardless of what value replaces the box, 40 will still be the most frequently appearing element in the set (four times, more than any other element) - that is, 40 will be the mode.

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Question

Which is the greater quantity?

(A) The midrange of the data set $\left\{ 1, 10, 100, 1000, 10000 \right \}$

(B) The midrange of the data set $\left\{ 80, 90, 100, 110, 120 \right \}$

Answer

The midrange of a data set is the mean of its least and greatest elements. The midrange of the first data set is $\left (10,000 + 1 \right ) \div 2 = 5,000.5$ ; that of the second data set is $(80 + 120) \div 2 = 100$ . (A) is greater.

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Question

Give the median of the data set $\left \{ 2, 4, 8, 16, 32, 64\right \}$

Answer

The median of a data set with an even number of elements is the mean of the two elements in the middle when they are arranged in ascending order. The data set has six elements, so the median is the mean of the third-highest and third-lowest elements:

$M = \frac{8+16}{2} = \frac{24}{2} = 12$

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Question

$\left \{ 11,10,9,10,15,12,10 \right \}$

Find the range of the data set above.

Answer

To find the range, you must subtract the lowest term from the greatest term in the data set. The greatest term is 15. The lowest term is 9. Therefore, to find the range, you calculate . The answer is 6.

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Question

Give the mode of the data set below.

$\left \{ 20, 21, 23, 24, 25, 26, 27, 28, 30, 34\right \}$

Answer

The mode is the term that is repeated most often in a given set. In a set in which no element is repeated, such as this one, there is no mode.

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Question

Give the mode(s) of this data set:

$\left \{ 1,3,4,4,6,7,7,7,9,10 \right \}$

Answer

The mode of a data set is the value or values that appears the most frequently. 7 appears three times, 4 appears two times, and no other value appears more than once. 7 is therefore the only mode.

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Question

$\left \{ 2,3,4,5,6,7,8,9\right \}$

The above set represents the numbered cards in a standard deck of cards. What value is missing?

Answer

The numbered cards in a deck range from to .

Since is not in the above set, it is the missing NUMBER card from the deck.

King and Ace are incorrect choices as they are called face cards in a deck of cards.

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Question

$\left \{ 14,17,20,15,14,16,18,14 \right \}$

Which measure of central tendency would you use to find the term that is most occuring?

Answer

Mode is the number that occurs most often, so that is the correct answer. Median is the middle number, while mean is the average of the set. Range is the difference between the greatest and least numbers.

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Question

Which is the greater quantity?

(A) The median of the data set $\left\{ 1, 10, 100, 1000, 10000 \right \}$

(B) The median of the data set $\left\{ 80, 90, 100, 110, 120 \right \}$

Answer

Each data set has five elements; the median is the element in the middle after the elements are arranged in ascending sequence, which both are. In each case, the median will be the third-highest element. Since in both data sets, this element is 100, the medians are equal.

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Question

Which is the greater quantity?

(A) The midrange of the data set $\left\{ 1, 10, 100, 1000, 10000 \right \}$

(B) The midrange of the data set $\left\{ 80, 90, 100, 110, 120 \right \}$

Answer

The midrange of a data set is the mean of its least and greatest elements. The midrange of the first data set is $\left (10,000 + 1 \right ) \div 2 = 5,000.5$ ; that of the second data set is $(80 + 120) \div 2 = 100$ . (A) is greater.

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Question

$\left \{ 14,17,20,15,14,16,18,14 \right \}$

Which measure of central tendency would you use to find the term that is most occuring?

Answer

Mode is the number that occurs most often, so that is the correct answer. Median is the middle number, while mean is the average of the set. Range is the difference between the greatest and least numbers.

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