College Statistics Quiz: Measures Of Center

20 questions · exam conditions

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Measures Of CenterQuestion 1 of 20

A set contains five distinct positive integers. The mean of the set is 10, and the median is 8. What is the maximum possible value for the largest integer in the set?

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College Statistics Quiz

College Statistics Quiz: Measures Of Center

Practice Measures Of Center in College Statistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Measures Of Center, giving you a quick way to practice the rules, question types, and explanations that matter most for College Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A set contains five distinct positive integers. The mean of the set is 10, and the median is 8. What is the maximum possible value for the largest integer in the set?

26
30 (correct answer)
32
40

Explanation: The sum of the five integers is

5 \times 10 = 50

. Since the median is 8 and the integers are distinct, the set can be represented as

\{x_1, x_2, 8, x_4, x_5\}

, where

x_1 < x_2 < 8 < x_4 < x_5

. To maximize

x_5

, the other integers must be minimized. The smallest distinct positive integers for

x_1

and

x_2

are 1 and 2. The smallest distinct integer for

x_4

that is greater than 8 is 9. The sum is

1 + 2 + 8 + 9 + x_5 = 50

, which simplifies to

20 + x_5 = 50

. Therefore, the maximum possible value for

x_5

is 30.

Question 2

A histogram of employee salaries at a large corporation is bimodal. Which of the following is the most plausible explanation for this observation?

The data contain a few extremely high salaries that skew the distribution.
The sample was a mix of two distinct employee groups with different pay scales. (correct answer)
The mean and the median of the salaries are significantly different.
Salaries are reported as rounded values, creating artificial peaks.

Explanation: Bimodality in a distribution often indicates that the sample is composed of two different subpopulations. In the context of salaries, this could be two distinct groups of employees, such as entry-level staff and senior management, each with its own central tendency, creating two peaks in the overall distribution.

Question 3

The mean salary at a company is $70,000, and the median salary is $55,000. This year, every employee receives a $5,000 raise. What are the new mean and median salaries?

Mean $75,000, Median $55,000
The new values cannot be determined without the number of employees.
Mean $70,000, Median $60,000
Mean $75,000, Median $60,000 (correct answer)

Explanation: This question tests how transformations affect measures of central tendency. When every data point in a distribution is increased by the same constant amount, both the mean and median shift by that exact amount. The mean is calculated by summing all values and dividing by the number of observations. If you add $5,000 to every salary, the sum increases by $5,000 times the number of employees, so the new mean becomes $70,000 + $5,000 = $75,000. Similarly, the median is the middle value when data is arranged in order. Adding $5,000 to every salary shifts every position by $5,000, including the middle position, so the new median becomes $55,000 + $5,000 = $60,000. Choice A incorrectly assumes only the mean changes while the median stays constant. This reflects a misunderstanding of how the median responds to uniform shifts. Choice B suggests we need additional information, but the number of employees is irrelevant since we're adding the same amount to each salary regardless of company size. Choice C mistakenly keeps the mean unchanged while only adjusting the median, which reverses the correct transformation principle. Remember this key rule: adding or subtracting a constant to every data point shifts both the mean and median by that same constant. This applies to any linear transformation of the form "new value = old value + constant." Watch for similar questions involving temperature conversions, wage adjustments, or score modifications where this principle appears frequently on statistics exams.

Question 4

For a dataset $\{x_1, x_2, ..., x_n\}$ , the value of $c$ that minimizes the expression $\sum_{i=1}^{n} (x_i - c)^2$ is the mean, $\bar{x}$ . If a dataset has a severe right skew due to a large outlier, how would the value of $c$ that minimizes $\sum_{i=1}^{n} |x_i - c|$ relate to $\bar{x}$ ?

It would be greater than $\bar{x}$ .
It would be less than $\bar{x}$ . (correct answer)
It would be equal to $\bar{x}$ .
The relationship cannot be determined from the given information.

Explanation: The value of

c

that minimizes the sum of absolute differences

\sum |x_i - c|

is the median of the dataset. For a right-skewed distribution, the mean is pulled in the direction of the long tail (the large outlier), making it greater than the median. Therefore, the value that minimizes the sum of absolute differences (the median) would be less than the value that minimizes the sum of squared differences (the mean,

\bar{x}

Question 5

For the dataset $\{2, 5, 5, 11, 12\}$ , which measure of central tendency is equal to 5, and what property does this value have for the dataset?

The mean, which minimizes the sum of squared deviations from the center.
The median, which minimizes the sum of absolute deviations from the center. (correct answer)
The mode, which is always the most resistant measure to outliers.
The midrange, which is the most efficient estimate of the center.

Explanation: First, calculate the measures of center. The mean is

(2+5+5+11+12)/5 = 7

. The median is the middle value of the ordered set, which is 5. The mode is the most frequent value, which is 5. The value 5 represents both the median and the mode. A key property of the median is that it is the value

c

that minimizes the sum of absolute deviations,

\sum |x_i - c|

Question 6

A real estate agent notes that for houses sold in a particular suburb last month, the mean sale price was $410,000 and the median sale price was $350,000. Which of the following is the most likely reason for the significant difference between the mean and median?

The distribution of sale prices is skewed to the left.
The distribution of sale prices is skewed to the right. (correct answer)
The distribution of sale prices is approximately symmetric.
The mode of the sale prices must be approximately $380,000.

Explanation: When the mean is significantly greater than the median, it indicates that the distribution is skewed to the right. This is caused by a few very high-value outliers (in this case, very expensive houses) that pull the mean upwards but have little effect on the median.

Question 7

A dataset of daily high temperatures for a city has a mean of 25°C and a median of 22°C. What will be the new mean and median if these temperatures are converted to degrees Fahrenheit using the formula $F = 1.8C + 32$ ?

Mean = 77°F, Median = 71.6°F (correct answer)
Mean = 77°F, Median = 74°F
Mean = 45°F, Median = 39.6°F
Mean = 57°F, Median = 54°F

Explanation: When a dataset undergoes a linear transformation (multiplying by a constant and adding a constant), the measures of center are transformed in the same way. New Mean:

1.8 \times 25 + 32 = 45 + 32 = 77

°F. New Median:

1.8 \times 22 + 32 = 39.6 + 32 = 71.6

°F.

Question 8

A set contains five distinct positive integers. The mean of the set is 10, and the median is 8. What is the maximum possible value for the largest integer in the set?

26
30 (correct answer)
32
40

Explanation: The sum of the five integers is

5 \times 10 = 50

. Since the median is 8 and the integers are distinct, the set can be represented as

\{x_1, x_2, 8, x_4, x_5\}

, where

x_1 < x_2 < 8 < x_4 < x_5

. To maximize

x_5

, the other integers must be minimized. The smallest distinct positive integers for

x_1

and

x_2

are 1 and 2. The smallest distinct integer for

x_4

that is greater than 8 is 9. The sum is

1 + 2 + 8 + 9 + x_5 = 50

, which simplifies to

20 + x_5 = 50

. Therefore, the maximum possible value for

x_5

is 30.

Question 9

The mean of a set of 8 numbers is 12. The mean of another set of 12 numbers is 20. If the two sets are combined, one number is removed, and the mean of the remaining 19 numbers is 17. What was the value of the number that was removed?

13 (correct answer)
16
16.8
23

Explanation: First, find the sum of each original set. Set 1 sum:

8 \times 12 = 96

. Set 2 sum:

12 \times 20 = 240

. The sum of the combined set of 20 numbers is

96 + 240 = 336

. The sum of the final set of 19 numbers is

19 \times 17 = 323

. The value of the number that was removed is the difference between these two sums:

336 - 323 = 13

Question 10

A student's final grade is determined by a weighted average: homework is worth 20%, two midterms are worth 25% each, and the final exam is worth 30%. A student has a homework average of 95, a first midterm score of 80, and a second midterm score of 88. What score must the student achieve on the final exam to earn an overall grade of exactly 88?

88
89
90 (correct answer)
91

Explanation: Let F be the score on the final exam. The weighted average equation is

0.20(95) + 0.25(80) + 0.25(88) + 0.30(F) = 88

. Calculate the contribution from the known scores:

19 + 20 + 22 = 61

. The equation becomes

61 + 0.30(F) = 88

. Subtracting 61 from both sides gives

0.30(F) = 27

. Dividing by 0.30 gives

F = 27 / 0.30 = 90

Question 11

The median of the set of numbers $\{x, 3x, 6x, 10x, 11x\}$ , where $x > 0$ , is 24. If the smallest number in the set, $x$ , is replaced by $x - 2$ , what is the new median of the set?

16
22
24 (correct answer)
26

Explanation: First, find the value of x. Since

x > 0

, the set is already ordered. The median is the third value,

6x

. So,

6x = 24

, which means

x = 4

. The original set is

\{4, 12, 24, 40, 44\}

. The smallest number, 4, is replaced by

x - 2 = 4 - 2 = 2

. The new set is

\{2, 12, 24, 40, 44\}

. The new set is still ordered, and the number of elements is unchanged. The median is still the third value, which remains 24.

Question 12

Consider the set of data $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, N\}$ , where N is an unknown integer. For which value of N would the mean and median of the set S be equal?

5.5
10
11 (correct answer)
55

Explanation: The sum of the first 10 integers is

\frac{10(11)}{2} = 55

. The sum of the set S is

55 + N

. The mean is

\frac{55+N}{11}

. The median depends on the value of N. For the mean and median to be equal, the distribution must be symmetric. The set

\{1, ..., 10\}

is symmetric around 5.5. To maintain symmetry for the 11-element set S, N must be placed such that the new center is the median. Let's test the options. If

N=11

, the ordered set is

\{1, 2, ..., 11\}

. The median is the 6th element, which is 6. The mean is

\frac{55+11}{11} = \frac{66}{11} = 6

. Since the mean (6) equals the median (6),

N=11

is the correct value.

Question 13

In a class, the mean score on a test was 75.2. The mean score for the 12 male students was 72. If there are 20 students in total, what was the mean score for the female students?

77.0
78.4
79.5
80.0 (correct answer)

Explanation: The total sum of scores for all 20 students is

20 \times 75.2 = 1504

. The total sum of scores for the 12 male students is

12 \times 72 = 864

. The number of female students is

20 - 12 = 8

. The total sum of scores for the female students is the total sum minus the male students' sum:

1504 - 864 = 640

. The mean score for the female students is their total sum divided by their number:

640 / 8 = 80.0

Question 14

A sample of 10 data points has a mean of 15.0. If a data point with a value of 24 is removed from the sample, what will be the mean of the remaining 9 data points?

12.6
13.5
14.0 (correct answer)
15.0

Explanation: The sum of the original 10 data points is given by the mean times the number of points:

10 \times 15.0 = 150

. When the data point with value 24 is removed, the new sum is

150 - 24 = 126

. The new sample has 9 data points. The new mean is the new sum divided by the new count:

126 / 9 = 14.0

Question 15

Mean = 77°F, Median = 71.6°F (correct answer)
Mean = 77°F, Median = 74°F
Mean = 45°F, Median = 39.6°F
Mean = 57°F, Median = 54°F

Explanation: When a dataset undergoes a linear transformation (multiplying by a constant and adding a constant), the measures of center are transformed in the same way. New Mean:

1.8 \times 25 + 32 = 45 + 32 = 77

°F. New Median:

1.8 \times 22 + 32 = 39.6 + 32 = 71.6

°F.

Question 16

A marketing researcher is analyzing consumer data. For which of the following variables would the mode be the most appropriate measure of central tendency to report?

The annual income of consumers in dollars.
The age of consumers in years.
Consumers' satisfaction rating on a scale from 1 to 5.
The brand of smartphone owned by consumers. (correct answer)

Explanation: The brand of smartphone is categorical (nominal) data. For nominal data, where values are labels without a natural order, the mean and median are not meaningful. The mode, which identifies the most frequent category, is the only appropriate measure of central tendency.

Question 17

It would be greater than $\bar{x}$ .
It would be less than $\bar{x}$ . (correct answer)
It would be equal to $\bar{x}$ .
The relationship cannot be determined from the given information.

Explanation: The value of

c

that minimizes the sum of absolute differences

\sum |x_i - c|

\bar{x}

Question 18

13 (correct answer)
16
16.8
23

Explanation: First, find the sum of each original set. Set 1 sum:

8 \times 12 = 96

. Set 2 sum:

12 \times 20 = 240

. The sum of the combined set of 20 numbers is

96 + 240 = 336

. The sum of the final set of 19 numbers is

19 \times 17 = 323

. The value of the number that was removed is the difference between these two sums:

336 - 323 = 13

Question 19

88
89
90 (correct answer)
91

Explanation: Let F be the score on the final exam. The weighted average equation is

0.20(95) + 0.25(80) + 0.25(88) + 0.30(F) = 88

. Calculate the contribution from the known scores:

19 + 20 + 22 = 61

. The equation becomes

61 + 0.30(F) = 88

. Subtracting 61 from both sides gives

0.30(F) = 27

. Dividing by 0.30 gives

F = 27 / 0.30 = 90

Question 20

The distribution of sale prices is skewed to the left.
The distribution of sale prices is skewed to the right. (correct answer)
The distribution of sale prices is approximately symmetric.
The mode of the sale prices must be approximately $380,000.