Descriptive Statistics - GMAT Quantitative

Card 0 of 856

Question

Give the median of the following data set in terms of :

$\left {a -1, a +2, a-3, a+ 4, a -5, a+6, a-7, a+8 \right }$

Answer

The data set can be arranged from least to greatest as follows:

$\left {a-7, a-5, a-3, a-1, a+2, a+4, a+6, a+8 \right }$

The median of a data set with eight elements is the mean of its fourth-highest and fourth-lowest elements, which are and . Add and divide by two:

$\frac{(a-1) + (a+2)}{2} = \frac{2a+1}{2} = a + \frac{1}{2}$

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Question

What is the range for the following set:

Answer

The range is the difference between the highest and lowest number.

First sort the set:

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Question

Consider the following set of numbers:

85, 87, 87, 82, 89

What is the range?

Answer

The range is the difference between the maximum and minimum value.

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Question

A large group of students is given a standardized test. The following information is given about the scores:

Mean: 73.8

Standard deviation: 6.3

Median: 71

25th percentile: 61

75th percentile: 86

Highest score: 100

Lowest score: 12

What is the interquartile range of the tests?

Answer

The interquartile range of a data set is the difference between the 75th and 25th percentiles:

$\textrm{IQR }= 86-61=25$

All other given information is extraneous to the problem.

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Question

What is the range for the following data set:

Answer

The range is the highest value number minus the lowest value number in a sorted data set:

We need to sort the data set:

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Question

Below is the stem-and-leaf display of a set of test scores.

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

What is the range of this set of scores?

Answer

The range of a data set is the difference of the highest and lowest scores,

The numbers in the "stem" of this display represent tens digits of the test scores, and the numbers in the "leaves" represent the units digits. The highest and lowest scores represented are 87 and 42, so the range is their difference: .

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Question

Consider the data set $\left { 1, -2, 3, -4, 5, -6, 7, -8, 9, -10 \right }$ .

What is its midrange?

Answer

The midrange of a data set is the arithmetic mean of its greatest element and least element. Here, those elements are and , so we can find the midrange as follows:

$\left [9 + (-10) \right ]\div 2 = -1\div 2 = -0.5$

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Question

Below is the stem-and-leaf display of a set of test scores.

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

What is the interquartile range of these test scores?

Answer

The numbers in the "stem" of this display represent tens digits of the test scores, and the numbers in the "leaves" represent the units digits. This stem-and-leaf display represents twenty scores.

The interquartile range is the difference of the third and first quartiles.

The third quartile is the median of the upper half, or the upper ten scores. This is the arithmetic mean of the fifth- and sixth-highest scores. These scores are 73 and 69, so the mean is $\left (73 +69 \right )\div 2 = 71$ .

The first quartile is the median of the lower half, or the lower ten scores. This is the arithmetic mean of the fifth- and sixth-lowest scores. Both of these scores are the same, however - 57.

The interquartile range is therefore the difference of these numbers:

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Question

Set 1: 5, 13, -2, -1, 19, 27

Set 2: 6, -3, 23, 15, m, 1

What should the value of be if we want the ranges of both sets of number to be equal?

Answer

The range of a set of numbers is the difference between the highest number and the lowest number in the set.

The range of set 1 is:

The range of the second set, ignoring the value of m is:

We need to either subtract 3 from the lowest number in set Set 2 or add 3 to the highest number in Set 2 to get the value of m such that the range of both sets are equal.

$m=(-3)-3=-6 \rightarrow range =23-(-6)=29$

or

$m=23+3=26 \rightarrow range = 26-(-3)=29$

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Question

Calculate the range of the following set of data:

Answer

The range of a set of data is the difference between its highest value and its lowest value, as this describes the range of values spanned by the set. A quick way to calculate the range is to locate the lowest value in the set and subtract it from the highest value, but let's arrange the set in increasing order to visualize the problem first:

Now we can see that the lowest value in the set is 9, and the highest value in the set is 27, so the range of the set is:

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Question

Calculate the range of the following set of data:

$\begin{Bmatrix} 41, 28, 37, 41, 29, 33, 31, 40, 53, 49, 38, 42 \end{Bmatrix}$

Answer

The range of a set of data is the difference between its smallest and greatest values. We can first look through the set for the greatest value, which we can see is 53. We then look through the set for the smallest value, which we can see is 28. The range of the set is then:

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Question

Determine the mean for the following set of numbers.

Answer

To find the range, simply subract the smallest number from the largest. Therefore:

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Question

Find the range of the following set of numbers:

$\textup{4,6,7,7,8,10,28}$

Answer

To find range, subtract the smallest number from the largest number. Thus,

$\textup{range}=28-4=24$

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Question

Find the range of the following set of numbers.

1,1,2,7,8,10,11

Answer

To find the range, you m ust subtract the smallest number from the largest. Thus,

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Question

Find the range of the following data set:

Answer

Find the range of the following data set:

Range is as simple as finding the diffference between the largest and smallest terms in a set. So, let's find our largest and smallest terms.

Largest: 989

Smallest: 2

Next, let's calculate the range:

So our answer should be 987

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Question

.

Give the midrange of the set $\left { A, B, C, D, E,F\right }$ .

Answer

The midrange of a set is the arithmetic mean of the greatest and least values, which here are and . This makes the midrange $\frac{A+F}{2}$ .

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Question

Salaries for employees at ABC Company: 1 employee makes $25,000 per year, 4 employees make $40,000 per year, 2 employees make $50,000 per year and 5 employees make $75,000 per year.

What is the average (arithmetic mean) salary for the employees at ABC Company?

Answer

The average is found by calculating the total payroll and then dividing by the total number of employees. $\frac{(1\cdot 25,000)+(4\cdot 40,000)+(2\cdot 50,000)+(5\cdot 75,000)}{1+4+2+5}$

$\frac{25,000+160,000+100,000+375,000}{12} = \frac{660,000}{12}= $55,000$

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Question

A bowler had an average (arithmetic mean) score of 215 on the first 5 games she bowled. What must she bowl on the 6th game to average 220 overall?

Answer

For the first 5 games the bowler has averaged 215. The equation to calculate the answer is

$\frac{(215\cdot 5)+x}{6}=220$

where $\dpi{100} \small x$ is the score for the sixth game. Next, to solve for the score for the 6th game $\dpi{100} \small (x)$ multiply both sides by 6:

$(215\cdot 5)+x =1,320$

which simplifies to:

$1,075+x =1,320$

After subtracting 1,075 from each side we reach the answer:

$x =1,320 - 1, 075 = 245$

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Question

Five friends go out to a restaurant to celebrate the birthday of one of them. The cost of the meal for all of them including tax is . They decide to leave a tip and to split the total amount equally. If the person celebrating his or her birthday does not pay, how much do each of the other four friends pay?

Answer

The total cost of the meal including tax and the tip is:

$280\times(1+0.15)=322$

They split the total amount equally between four of them, so each of those four people then pays:

$\frac{322}{4}=80.5$

Each of the four friends pays .

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Question

Ashley averaged a score of 87 on her first 5 tests. She scored a 93 on her 6th test. What is her average test score, assuming all 6 tests are weighted equally?

Answer

We can't just average 87 and 93! This will give the wrong answer! The average formula is $\dpi{100} \small average = \frac{sum}{number\ of\ terms}$ .

For the first 5 tests, $\dpi{100} \small 87=\frac{sum}{5}$ . Then $\dpi{100} \small sum=87\times 5=435$ .

Now combine that with the 6th test to find the overall average.

$\dpi{100} \small average = \frac{435+93}{6}=88$

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