How to find range - ISEE Upper Level (grades 9-12) Quantitative Reasoning

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Question

Consider the set of numbers: $\left { 2,4,5,5,6,6,6,10,12\right }$

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

Quantity B: The range of the set

Answer

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.

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Question

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

$\left.\begin{matrix} 4\ 5\ 6\ 7\ 8 \end{matrix}\right|\begin{matrix} \textrm{7 9}; ; ; ; ; ; ; ; ; ; ;; ; ; ; ; ; ; ; ; ; ; ; ;\ \textrm{4 4 7 7}; ; ; ; ; ; ; ; ; ; ;; ; ; ; ; ; ; \ \textrm{0 1 2 2 4 5 5 8 8 9}\ \textrm{3 5 5 8}; ; ; ; ; ; ; ; ; ; ;; ; ; ; ; ; ; \ \textrm{4 7}; ; ; ; ; ; ; ; ; ; ;; ; ; ; ; ; ; ; ; ; ; ; ; \end{matrix}$

Which is the greater quantity?

(a) The range of the data?

(b)

Answer

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.

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Question

Consider the following set of data:

$\left { 2,4,4,5,6,10 \right }$

Compare and .

: The sum of the median and the mean of the set

: The range of the set

Answer

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:

$Mean = \frac{2+4+4+5+6+10}{6}=\frac{31}{6}\approx 5.17$

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

$Median=\frac{4+5}{2}=4.5$

So we have:

$Median+Mean\approx 4.5+5.17\approx 9.67$

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

$Range = Maximum\ Value-Minimum\ Value=10-2=8$

Therefore is greater than .

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Question

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

$\left { 2,4,7,8\right }$

Answer

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

$Median=\frac{4+7}{2}=5.5$

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

$Range = Maximum\ Value-Minimum\ Value=8-2=6$

So the range is greater than the median.

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Question

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

$\left { 4,3,8,8\right }$

Answer

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:

$Mean = \frac{4+3+8+8}{4}=\frac{23}{4}=5.75$

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

$Range = Maximum\ Value-Minimum\ Value=8-3=5$

So the mean is greater than the range.

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Question

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

$\left { 1,1,1,2,2,2,2,3,3\right }$

Answer

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

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

$Range = Maximum\ Value-Minimum\ Value=3-1=2$

So the range is equal to the mode.

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