Multiple Choice

1. The mean of the five numbers $\\{8, 12, x, 15, 20\\}$ is $14$. What is the value of $x$?

   $16$

   $10$

   $15$

   $12$

2. A set of five numbers, when arranged in ascending order, is $\\{12, 18, x, 25, 30\\}$. If the median of this set is $20$, what is the value of $x$?

   $20$

   $25$

   $22$

   $18$

3. The set of numbers $\\{7, 10, 13, x, 10, 7\\}$ must have a unique mode of $10$. What is the value of $x$?

   $9$

   $7$

   $13$

   $10$

4. The numbers in a data set are $\\{45, 30, 60, x\\}$. The range of the set is $35$. If $x$ is the smallest number in the set, what is the value of $x$?

   $30$

   $45$

   $25$

   $20$

5. A student took $4$ tests and had an average score of $85$. After taking a fifth test, their average score increased to $87$. What score did the student get on the fifth test?

   $95$

   $89$

   $91$

   $87$

6. The numbers $\\{5, 10, x, 20\\}$ are arranged in ascending order. If the median of the set is $12.5$, what is the value of $x$?

   $14$

   $10$

   $12$

   $15$

7. A set of numbers is $\\{15, 22, 10, x\\}$. The range of the set is $20$. If $x$ is the largest number in the set, what is the value of $x$?

   $22$

   $25$

   $10$

   $30$

8. The data set $\\{2, 5, 8, x, 5, 2\\}$ must have a unique mode of $5$. What is the value of $x$?

   $2$

   $8$

   $10$

   $5$

9. The mean of three numbers is $18$. Two of the numbers are $14$ and $20$. What is the third number?

   $18$

   $20$

   $22$

   $16$

10. A set of numbers is $\\{3, x, 15, 18, 25\\}$. When these numbers are arranged in ascending order, the median is $15$. Which of the following is a possible value for $x$?

    $12$

    $17$

    $20$

    $16$

11. A dataset consists of the numbers $10, 12, 14, 16, 18$. If the number $20$ is added to this dataset, how do the mean and median change?

    Both the mean and median increase by $1$.

    Both the mean and median remain unchanged.

    The mean decreases and the median increases by $1$.

    The mean increases by $1$ and the median decreases.

12. Consider the dataset $5, 6, 7, 8, 9$. If the number $50$ is added to this dataset, which measure of central tendency is most significantly affected?

    The mean, as it changes from $7$ to approximately $14.17$.

    The range, as it increases from $4$ to $45$.

    Both the mean and median are affected equally.

    The median, as it changes from $7$ to $7.5$.

13. A dataset is given as $2, 3, 3, 4, 5, 6$. If one of the $3$s is removed from the dataset, how do the mode and range change?

    The mode changes from $3$ to having no mode, and the range remains unchanged.

    The mode changes from $3$ to $2$ and the range decreases.

    Both the mode and range remain unchanged.

    The mode remains $3$ and the range increases.

14. For the dataset $10, 15, 20, 25, 30$, the largest value, $30$, is mistakenly recorded and should have been $40$. How does this correction affect the median and the range of the dataset?

    The median remains unchanged, and the range decreases.

    Both the median and range remain unchanged.

    The median remains unchanged, and the range increases by $10$.

    The median increases, and the range increases.

15. A set of exam scores is $50, 60, 70, 80$. If $5$ bonus marks are added to each score, what happens to the mean and range of the scores?

    The mean increases by $5$ and the range increases by $5$.

    Both the mean and range remain unchanged.

    The mean increases by $5$ and the range remains unchanged.

    The mean remains unchanged and the range increases by $5$.

16. A dataset has values $2, 4, 6, 8$. If each value in the dataset is multiplied by $3$, how do the mean and median change?

    The mean is multiplied by $3$ and the median increases by $3$.

    Both the mean and median are multiplied by $3$.

    The mean is multiplied by $3$ but the median remains unchanged.

    Both the mean and median increase by $3$.

17. When an extreme outlier is added to a dataset, which of the following measures of central tendency is typically the *least* affected?

    The mode.

    The median.

    The range.

    The mean.

18. The dataset of ages of students in a small club is $15, 16, 16, 17, 18$. A new student, aged $17$, joins the club. How does this addition affect the mode and median of the dataset?

    The mode becomes bimodal ($16$ and $17$) and the median increases to $16.5$.

    The mode remains $16$ and the median remains $16$.

    The mode becomes $17$ and the median decreases to $16$.

    The mode remains $16$ and the median increases to $16.5$.

19. A dataset has values $10, 20, 30, 40, 50$. If the smallest value ($10$) and the largest value ($50$) are removed, how does the range of the dataset change?

    The range increases by $10$.

    The range increases by $20$.

    The range remains unchanged.

    The range decreases by $20$.

20. A student's scores on four tests are $70, 75, 80, 85$. The student takes a fifth test and scores $80$. What happens to the mean and median of the student's scores?

    The mean increases by $0.5$ and the median increases by $2.5$.

    The mean decreases, and the median remains unchanged.

    Both the mean and median remain unchanged.

    The mean increases by $0.5$ and the median decreases.

21. A set of five numbers is $\\{12, 23, 15, x, y\\}$. The mean of these five numbers is 18. If the value of $x$ is doubled (while $y$ remains unchanged), the new mean of the five numbers becomes 22. What is the value of $y$?

    15

    25

    20

    10

22. A set of six distinct integers is arranged in ascending order as $x_1, x_2, x_3, x_4, x_5, x_6$. The mean of this set is 15 and the median is 14. The smallest number is 5 and the largest number is 25. What is the sum $x_3 + x_4$?

    28

    27

    29

    26

23. A dataset of 10 numbers has a mean of 45. If two numbers, 30 and 60, are removed from the dataset, and a new number, $y$, is added, what must be the value of $y$ for the mean of the new dataset (which now has 9 numbers) to remain 45?

    45

    40

    30

    35

24. A dataset contains the numbers $\\{3, 5, 5, 8, 10, 12, 12, 15\\}$. If two numbers are removed from the dataset, and a new number is added, resulting in a dataset of 7 numbers, which of the following changes to the mode is NOT possible?

    The new dataset has a unique mode of 5.

    The new dataset has a unique mode of 12.

    The new dataset has no mode.

    The new dataset has two modes, 8 and 10.

25. A set of 10 distinct integers has a range of 40. The smallest number in the set is 10. If the largest number is removed and replaced by a new number $x$, the range of the new set (which still has 10 distinct integers) becomes 30. Which of the following statements about $x$ is true?

    $x < 10$

    $x=40$

    $x > 40$

    $x \le 40$

26. A set of 5 distinct positive integers $S = \\{a, b, c, d, e\\}$ is arranged in ascending order. The mean of the set is 15. The median is 14. If the smallest number, $a$, is removed, and a new number $f$ is added such that $f > e$, what is the new median of the 5-number set $\\{b, c, d, e, f\\}$?

    $b$

    $e$

    $c$

    $d$

27. A set of 6 distinct integers has a mean of 15. Its range is 20. The smallest number in the set is 8. If the largest number is removed and replaced by a new number $x$, the mean of the new set becomes 14, and the median remains 13. What is the value of $x$?

    22

    24

    18

    20

28. A set of 5 positive integers has a range of 18, a mode of 10, and a mean of 15. The integers are $a, b, c, d, e$ in ascending order. If $a=7$, what is the value of $e$?

    25

    23

    22

    24

Numerical

29. Elise looked at her calendar to see how many hours she had worked in the past 5 days.

    

30. A newspaper researched how many grocery stores there are in each town.

    

31. The city recorded how many fire hydrants there are on each street. What is the mean of the Data?

    

32. Some children compared how many thank-you notes they wrote last month. What is the mean of the numbers?

    

33. The readers of a magazine reported how many winter hats they owned. What is the range of the numbers?

    

34. The parks department compared how many basketball hoops there are at each park. What is the mean of the numbers?

    

35. Erin looked at her calendar to figure out how much time she spent babysitting each month. What is the mean of the numbers?