Descriptive Statistics and Data Representations Practice Questions — ACT

The table below shows the number of books read by 7 students during their summer vacation:

Student	Books Read
Amy	16
Ben	8
Chris	3
Diana	12
Ethan	14
Fiona	6
Gabe	11

What is the median number of books read by these students?

8

10

11

12

13

Show answer & explanation

Answer: 11

Answer

11

To find the median of a data set, the values must first be arranged in order from least to greatest: 3, 6, 8, 11, 12, 14, 16. Since there are 7 values, the median is the single middle value, which is 11.

Step-by-Step Solution

1

Sort the data set from least to greatest.

3, 6, 8, 11, 12, 14, 16

The median of a data set is the middle value when the numbers are ordered.

2

Identify the middle value of the sorted list.

11

Since there are 7 data points (an odd number), the median is the 4th value, which has exactly 3 values below it and 3 values above it.

Key Concept

Finding the median of a set of data

Question 2Question →

A student earned scores of $85$ , $90$ , $88$ , and $92$ on the first four quizzes of the semester. What score must the student earn on the fifth quiz to have an average (arithmetic mean) score of exactly $90$ for all five quizzes?

1

5

89

90

95

Show answer & explanation

Answer: 95

Answer

The student must earn a score of 95 on the fifth quiz.

To find the score needed on the fifth quiz to achieve an average of

90

, first find the total number of points required. An average of

90

on

5

quizzes requires a total sum of

5 \times 90 = 450

points. The sum of the first four quiz scores is

85 + 90 + 88 + 92 = 355

points. Subtracting the current sum from the required total sum (

450 - 355

) yields

95

, which is the score the student must earn on the fifth quiz.

Step-by-Step Solution

1

Calculate the sum of the student's scores on the first four quizzes.

85 + 90 + 88 + 92 = 355

To find the total number of points the student has earned so far.

2

Calculate the total sum of points required to achieve an average of

90

over five quizzes.

5 \times 90 = 450

The arithmetic mean formula is Mean = Sum / Count, which can be rearranged to Sum = Mean * Count.

3

Subtract the sum of the first four quiz scores from the required total sum of five quiz scores.

450 - 355 = 95

The difference represents the score needed on the fifth quiz to achieve the target average.

Key Concept

Finding a missing value given a target arithmetic mean

Question 3Question →

A researcher records the following list of seven temperatures (in degrees Fahrenheit) during a week, where $x$ and $y$ are unknown values:

88, 92, 75, 85, x, y, 90

The mean temperature for the seven days is $85.0^\circ\text{F}$ , and the range of the temperatures is $22.0^\circ\text{F}$ . If the highest temperature of the week is $y$ and the lowest temperature of the week is $x$ , what is the median temperature of the week?

82.5^\circ\text{F}

85.0^\circ\text{F}

86.5^\circ\text{F}

88.0^\circ\text{F}

90.0^\circ\text{F}

Show answer & explanation

Answer: $88.0^\circ\text{F}$

Answer

88.0^\circ\text{F}

To find the median temperature, we must first find the values of

x

and

y

. Since the mean of the seven temperatures is

85.0^\circ\text{F}

, their sum is

7 \times 85.0 = 595.0

. The sum of the five known temperatures is

88 + 92 + 75 + 85 + 90 = 430.0

. Therefore, the sum of the two unknown temperatures is

x + y = 595.0 - 430.0 = 165.0

. We are also given that the range of the temperatures is

22.0^\circ\text{F}

, and that

y

is the highest and

x

is the lowest temperature, which means

y - x = 22.0

. Solving this system of equations (

y + x = 165.0

and

y - x = 22.0

) yields

x = 71.5

and

y = 93.5

. Placing the seven temperatures in ascending order gives

71.5, 75, 85, 88, 90, 92, 93.5

. The median is the middle value (the 4th value) in this sorted list, which is

88.0^\circ\text{F}

.

Step-by-Step Solution

1

Determine the sum of the temperatures using the given mean of

85.0^\circ\text{F}

.

Total sum =

595.0

Since the mean of 7 temperatures is

85.0

, the sum of all temperatures must be

7 \times 85.0 = 595.0

.

2

Set up an equation for the sum of the unknown temperatures

x

and

y

.

x + y = 165.0

The sum of the five known temperatures is

88 + 92 + 75 + 85 + 90 = 430.0

. Thus,

x + y = 595.0 - 430.0 = 165.0

.

3

Set up and solve the system of equations with the range constraint to find

x

and

y

.

x = 71.5

and

y = 93.5

We are given that

y

is the maximum and

x

is the minimum, so the range is

y - x = 22.0

. Solving the system

y + x = 165.0

and

y - x = 22.0

by adding the equations gives

2y = 187.0 \implies y = 93.5

. Substituting back gives

x = 71.5

.

4

Sort the seven temperatures in ascending order and identify the median value.

Sorted list:

71.5, 75, 85, 88, 90, 92, 93.5

; Median =

88.0

For an odd number of data points (7), the median is the 4th value when the list is sorted. The 4th value in the sorted list is

88.0

.

Key Concept

Calculating the median of a dataset containing unknown values by deriving and solving a system of linear equations based on the mean and the range.

Alternative Method

Instead of setting up and solving the system of equations algebraically, one can test the median by using the fact that the sum of the deviations from the mean (

85

) must equal

0

. The deviations of the known numbers from

85

are:

(88-85) + (92-85) + (75-85) + (85-85) + (90-85) = 3 + 7 - 10 + 0 + 5 = 5

. Therefore, the sum of the deviations of

x

and

y

from

85

must be

-5

:

(x-85) + (y-85) = -5

, which simplifies to

x + y - 170 = -5

, or

x + y = 165

. Since

y - x = 22

, we can quickly find

x = 71.5

and

y = 93.5

, then sort the list to find the median.

Estimated Time:3m 0s

Question 4Question →

A local community center surveyed a group of $20$ students about the number of hours they volunteered last month. The table below displays the results of the survey, where $a$ and $b$ represent the number of students in their respective categories.

Hours Volunteered	Number of Students
$2$	$a$
$5$	$b$
$8$	$4$
$12$	$3$
$15$	$2$

If the mean number of hours volunteered per student for this group is $6.6$ , what is the median number of hours volunteered per student for these $20$ students?

2.0

3.5

5.0

6.6

8.0

Show answer & explanation

Answer: 5.0

Answer

The median number of hours volunteered per student is 5.0.

The correct answer is 5.0. To find this, we first establish two equations from the given information: the total count of students (

a + b + 9 = 20 \implies a + b = 11

) and the mean of the data (

\frac{2a + 5b + 32 + 36 + 30}{20} = 6.6 \implies 2a + 5b = 34

). Solving this system of equations gives

a = 7

and

b = 4

. To find the median of the 20 values, we look at the 10th and 11th sorted values. Since there are seven 2s (positions 1–7) and four 5s (positions 8–11), both the 10th and 11th values are 5, making the median 5.0.

Step-by-Step Solution

1

Set up an equation for the total number of students.

a + b = 11

The total number of students in the survey is 20. Summing the frequencies gives

a + b + 4 + 3 + 2 = 20

, which simplifies to

a + b = 11

.

2

Set up an equation for the mean of the dataset.

2a + 5b = 34

The mean of the data is 6.6. Using the formula for the weighted mean:

\frac{2a + 5b + 8(4) + 12(3) + 15(2)}{20} = 6.6

. Multiplying both sides by 20 gives

2a + 5b + 98 = 132

, which simplifies to

2a + 5b = 34

.

3

Solve the system of linear equations for

a

and

b

.

a = 7

and

b = 4

Multiply the first equation by 2 to get

2a + 2b = 22

. Subtract this from

2a + 5b = 34

to find

3b = 12 \implies b = 4

. Substitute

b = 4

back into the first equation to find

a = 7

.

4

Find the median of the 20 sorted values.

5.0

With 20 data points, the median is the average of the 10th and 11th data values in ascending order. Since there are seven 2s followed by four 5s, the 10th and 11th values are both 5. The average of 5 and 5 is 5.0.

Key Concept

Determining the median of a grouped frequency distribution by solving for missing frequencies using the total count and the mean.

Question 5Question →

A dataset consists of $7$ positive integers sorted in non-decreasing order. The mean of the dataset is $20$ , the median is $18$ , and the unique mode is $15$ . If the range of the dataset is $22$ , what is the minimum possible value of the largest number in the dataset?

27

29

30

35

37

Show answer & explanation

Answer: 30

Answer

30

The correct answer is 30 because minimizing the largest value requires minimizing the smallest value under the sum and uniqueness constraints. When the smallest value is 8, the largest value is 30, which allows for a valid, sorted sequence of positive integers where 15 is the unique mode: {8, 15, 15, 18, 26, 28, 30}.

Step-by-Step Solution

1

Set up the variables for the sorted dataset.

Let the seven positive integers in non-decreasing order be

x_1, x_2, x_3, x_4, x_5, x_6, x_7

. Since the median is the 4th value,

x_4 = 18

. The mean is

20

, so the sum of all elements is

7 \times 20 = 140

.

Establishing standard notation and utilizing the definitions of median and mean.

2

Incorporate the range into the sum equation.

x_1 + x_2 + x_3 + 18 + x_5 + x_6 + x_7 = 140

. Since the range is

22

, the largest number is

x_7 = x_1 + 22

. Substituting this into the sum gives

2x_1 + x_2 + x_3 + x_5 + x_6 = 100

.

Simplifying the system of equations by expressing the largest element in terms of the smallest element.

3

Apply the mode constraint.

The unique mode is

15

. Since

x_4 = 18

and the dataset is sorted,

15

must be in the lower half of the dataset and must repeat. Thus,

x_2 = x_3 = 15

. The equation simplifies to

2x_1 + 30 + x_5 + x_6 = 100 \implies 2x_1 + x_5 + x_6 = 70

.

Using the properties of the mode and sorted list to lock the values of the second and third elements.

4

Set up inequalities to minimize the largest value.

To minimize

x_7 = x_1 + 22

, we must minimize

x_1

, which is equivalent to maximizing

x_5

and

x_6

. Since

15

is the unique mode with a frequency of

2

, no other value can repeat. Thus,

x_5, x_6,

and

x_7

must be distinct from each other and larger than the median

18

. This means

x_5 \ge 19

. The upper bounds are

x_6 \le x_7 - 1 = x_1 + 21

and

x_5 \le x_7 - 2 = x_1 + 20

.

Applying the uniqueness constraint of the mode to restrict the values of the upper half of the dataset.

5

Solve for the minimum integer value of the smallest element.

Substitute the maximum bounds of

x_5

and

x_6

into the equation:

2x_1 + (x_1 + 20) + (x_1 + 21) \ge 70 \implies 4x_1 + 41 \ge 70 \implies 4x_1 \ge 29 \implies x_1 \ge 7.25

. Since

x_1

must be an integer, the minimum possible value is

x_1 = 8

. This yields a minimum largest value of

x_7 = 8 + 22 = 30

.

Solving the inequality system to find the smallest valid integer boundaries.

Key Concept

Descriptive Statistics and Optimization

Question 6Question →

A list of $15$ positive integers is sorted in non-decreasing order. The minimum value in the list is $10$ , the maximum value is $50$ , and the median is $30$ . If the unique mode of the list is $20$ , what is the maximum possible sum of the $15$ integers in the list?

488

503

506

508

518

Show answer & explanation

Answer: 508

Answer

508

The maximum sum is achieved by setting the frequency of the unique mode

20

to its maximum limit of

6

, occupying positions

2

through

7

(since the first element is

10

and the median at position

8

is

30

). This allows any other value to appear at most

5

times. To maximize the sum, we assign the maximum value

50

to

5

positions (positions

11

to

15

), and the next highest value

49

to the remaining

2

positions in the upper half (positions

9

and

10

). Summing these values gives

10 + 6(20) + 30 + 2(49) + 5(50) = 508

.

Step-by-Step Solution

1

Identify fixed elements based on the median, minimum, and maximum constraints.

For

15

sorted integers, the median is the 8th term:

x_8 = 30

. The minimum is

x_1 = 10

, and the maximum is

x_{15} = 50

.

Establishing these boundary points helps constrain the values of the other elements in the sorted list.

2

Determine the maximum possible frequency of the unique mode

20

.

Since

x_8 = 30

and the list is sorted,

20

can only occupy positions

2

through

7

(a maximum of

6

occurrences).

To maximize the overall sum, we want to maximize the frequency of the mode so that other large values (like

50

) can also appear as many times as possible.

3

Determine the maximum allowed frequency of other values under the unique mode constraint.

With the frequency of

20

set to

6

, any other number can appear at most

5

times.

Since

20

is the unique mode, no other value in the list can have a frequency equal to or greater than

6

.

4

Assign the largest possible values to the remaining positions in the upper half (

x_9

to

x_{14}

).

Set

x_{11} = x_{12} = x_{13} = x_{14} = x_{15} = 50

(frequency of

5

). The remaining two positions,

x_9

and

x_{10}

, are set to the next largest integer,

49

.

This maximizes the sum of the upper half of the list without violating the maximum frequency limit of

5

for any non-mode value.

5

Sum the elements of the optimal list.

Sum =

10 + 6(20) + 30 + 2(49) + 5(50) = 10 + 120 + 30 + 98 + 250 = 508

.

Adding all fifteen optimal values together gives the absolute maximum possible sum.

Key Concept

Using central tendency constraints (median, mode) and range limits to optimize a dataset sum.

Question 7Question →

A set of $10$ distinct positive integers has a median of $25$ and a range of $30$ . What is the greatest possible value of the mean of these $10$ integers?

25

29

31

33

37

Show answer & explanation

Answer: 33

Answer

The greatest possible value of the mean of the 10 integers is 33.

The correct answer is 33. To find the greatest possible mean of the 10 distinct positive integers, we must maximize their sum. Let the sorted integers be

x_1 < x_2 < x_3 < x_4 < x_5 < x_6 < x_7 < x_8 < x_9 < x_{10}

. The median is the average of the 5th and 6th terms, so

\frac{x_5 + x_6}{2} = 25

, or

x_5 + x_6 = 50

. Since the integers are distinct, the maximum value for

x_5

is 24, which forces

x_6 = 26

. To maximize the sum of the first five terms, they should be consecutive integers ending at 24:

x_1 = 20, x_2 = 21, x_3 = 22, x_4 = 23, x_5 = 24

. Since the range is 30, the maximum value is

x_{10} = x_1 + 30 = 20 + 30 = 50

. To maximize the remaining terms in the upper half, we choose the largest possible distinct integers less than 50:

x_7 = 47, x_8 = 48, x_9 = 49

. The maximum sum is

20 + 21 + 22 + 23 + 24 + 26 + 47 + 48 + 49 + 50 = 330

, yielding a maximum mean of

330 / 10 = 33

.

Step-by-Step Solution

1

Define the variables and apply the median constraint.

Let the 10 sorted distinct positive integers be

x_1 < x_2 < x_3 < x_4 < x_5 < x_6 < x_7 < x_8 < x_9 < x_{10}

. The median is the average of the 5th and 6th terms:

\frac{x_5 + x_6}{2} = 25

, which means

x_5 + x_6 = 50

.

Since the number of terms is even, the median is the average of the two middle terms.

2

Maximize the first five integers to find the maximum value of the first term.

Since the integers are distinct, we must have

x_5 < x_6

. With

x_5 + x_6 = 50

, the maximum possible integer value for

x_5

is 24 (which makes

x_6 = 26

). To maximize the sum, we make the preceding terms as large as possible:

x_4 = 23

,

x_3 = 22

,

x_2 = 21

, and

x_1 = 20

.

To maximize the mean, we must maximize the sum of all terms, which requires making each term as large as possible within the distinct integer constraints.

3

Apply the range constraint to find the maximum value of the last term.

The range is 30, so

x_{10} - x_1 = 30

. Since the maximum value of

x_1

is 20, the maximum possible value for

x_{10}

is

20 + 30 = 50

.

The range of a dataset is the difference between the maximum and minimum values.

4

Maximize the remaining terms in the upper half of the dataset.

We have

x_6 = 26

. The remaining terms must satisfy

26 < x_7 < x_8 < x_9 < x_{10} = 50

. To maximize the sum, we choose the largest possible distinct integers for these slots:

x_9 = 49

,

x_8 = 48

, and

x_7 = 47

.

This maximizes the sum of the upper half of the dataset under the constraint that the maximum value is 50.

5

Calculate the maximum sum and the resulting maximum mean.

Sum =

20 + 21 + 22 + 23 + 24 + 26 + 47 + 48 + 49 + 50 = 330

. Mean =

\frac{330}{10} = 33

.

The mean is calculated by dividing the sum of the elements by the number of elements.

Key Concept

Maximizing the mean of a bounded dataset using median, range, and distinctness constraints.

Question 8Question →

During a basketball season, a player scored the following number of points in their first five games: $18$ , $9$ , $24$ , $14$ , and $15$ . What is the median number of points the player scored in these five games?

15

16

24

68

80

Show answer & explanation

Answer: $15$

Answer

15

To find the median of a dataset, the values must first be arranged in numerical order. Sorting the scores

18

,

9

,

24

,

14

, and

15

from least to greatest yields:

9

,

14

,

15

,

18

, and

24

. Since there is an odd number of scores (

5

scores), the median is the middle value, which is the third number in the sorted list. The third number is

15

.

Step-by-Step Solution

1

Arrange the given list of scores in ascending order.

The sorted list of scores is

9

,

14

,

15

,

18

, and

24

.

Before finding the median of a dataset, the values must be sorted from least to greatest.

2

Identify the middle value in the sorted list of

5

numbers.

Since there are

5

numbers, the middle value is the third number, which is

15

.

For an odd number of data points, the median is the exact middle value of the sorted list.

Key Concept

Finding the median of an odd number of data points by first sorting the dataset in ascending order.

Estimated Time:45s

Question 9Question →

A store manager records the number of laptop computers sold each day during a 5-day work week. The daily sales, in chronological order from Monday to Friday, are 2, 10, 9, 4, and 5. What is the positive difference between the mean and the median of the number of laptops sold daily?

3

1

4

17

21

Show answer & explanation

Answer: 1

Answer

The positive difference between the mean and the median of the daily laptop sales is 1.

To find the correct answer, we first calculate the mean of the dataset by summing the numbers (

2 + 10 + 9 + 4 + 5 = 30

) and dividing by the count (

30 / 5 = 6

). Next, we find the median by sorting the dataset (

2, 4, 5, 9, 10

) and selecting the middle value, which is

5

. The positive difference between the mean and the median is

6 - 5 = 1

.

Step-by-Step Solution

1

Calculate the mean of the daily laptop sales.

The sum of the laptop sales is

2 + 10 + 9 + 4 + 5 = 30

. Since there are 5 days, the mean is

\frac{30}{5} = 6

.

The mean is calculated by dividing the sum of all values by the total number of values.

2

Find the median of the daily laptop sales.

Sorting the sales in ascending order gives

2, 4, 5, 9, 10

. The middle value (the 3rd number) is

5

.

The median of a set of numbers is the middle value when the numbers are arranged in order.

3

Calculate the positive difference between the mean and the median.

The positive difference is

6 - 5 = 1

.

Subtract the smaller value from the larger value to find the positive difference.

Key Concept

Calculating and comparing descriptive statistics (mean and median) of a dataset.

Question 10Question →

An employee at a retail store recorded the number of hours they worked each day for one week in the table below.

Day	Hours Worked
Monday	8
Tuesday	2
Wednesday	10
Thursday	5
Friday	3
Saturday	7
Sunday	7

What is the median number of hours worked per day for this week?

5

6

7

8

36

Show answer & explanation

Answer: 7

Answer

The median number of hours worked per day for this week is 7.

The median of a set of data is the middle value when the numbers are arranged in order. Sorting the given hours worked from least to greatest gives: 2, 3, 5, 7, 7, 8, 10. Since there are 7 data points, the middle value is the 4th value, which is 7.

Step-by-Step Solution

1

Arrange the daily hours worked in ascending order.

The sorted list of hours is: 2, 3, 5, 7, 7, 8, 10.

Finding the median of a data set requires organizing the values from least to greatest.

2

Identify the middle value in the ordered list of 7 values.

The 4th value in the sorted list is 7.

Since the data set has an odd number of values (7), the median is the single middle value, located at the 4th position.

Key Concept

The median of a data set is the middle value when the data are arranged in numerical order.

Question 11Question →

A local shipping warehouse records the number of packages shipped during each hour of a 6-hour morning shift in the table below:

Hour	Packages Shipped
Hour 1	$15$
Hour 2	$27$
Hour 3	$18$
Hour 4	$32$
Hour 5	$12$
Hour 6	$22$

What is the median number of packages shipped per hour during this shift?

20

21

22

25

29

Show answer & explanation

Answer: $20$

Answer

The median number of packages shipped per hour is

20

.

The correct answer is

20

. To find the median, we first arrange the numbers in ascending order:

12, 15, 18, 22, 27, 32

. Since there is an even number of values (

6

), the median is the average of the two middle values,

18

and

22

. The average is

\frac{18 + 22}{2} = 20

.

Step-by-Step Solution

1

Sort the package counts in ascending order.

The sorted dataset is

12, 15, 18, 22, 27, 32

.

Finding the median of a dataset requires the data to be ordered from least to greatest.

2

Identify the two middle numbers in the ordered dataset.

Since there is an even number of data points (

6

values), the two middle numbers are the 3rd and 4th values:

18

and

22

.

For an even number of data points, the median is the average of the two central values.

3

Calculate the average of the two middle numbers.

\frac{18 + 22}{2} = \frac{40}{2} = 20

.

The average of

18

and

22

yields the median value of

20

.

Key Concept

Calculating the median of a dataset with an even number of values.

Question 12Question →

The table below shows the distribution of the number of books read by a group of students during a summer reading program.

Number of Books Read	Number of Students
1	1
2	4
3	8
4	$x$
5	4

If the median number of books read by the students is 3, what is the maximum possible value for the mean number of books read by these students?

3.0

3.2

3.4

3.6

3.8

Show answer & explanation

Answer: 3.4

Answer

The maximum possible value for the mean number of books read is 3.4.

To maximize the mean, we want to maximize the number of students who read 4 books, which is

x

. However,

x

is constrained by the requirement that the median must remain 3. The total number of students is

17 + x

. The 3s occupy sorted positions 6 through 13. For the median to be 3, the middle position(s) of the sorted list must not exceed position 13. When

N = 17+x

is odd (so

x

is even), the median position is

\frac{18+x}{2}

. Setting

\frac{18+x}{2} \le 13

yields

x \le 8

. For

x = 8

, the total number of students is 25, and the 13th student is the last one who read 3 books, making the median 3. If

x = 9

, the total is 26, and the median is the average of the 13th (3) and 14th (4) values, which is 3.5. Therefore, the maximum integer value of

x

is 8. The mean with

x=8

is the sum of all books read divided by the total number of students:

(1 \times 1 + 2 \times 4 + 3 \times 8 + 4 \times 8 + 5 \times 4) / 25 = 85 / 25 = 3.4

.

Step-by-Step Solution

1

Express the total number of students,

N

, in terms of

x

.

N = 1 + 4 + 8 + x + 4 = 17 + x

To calculate the mean and locate the median, we need the total count of data points.

2

Determine the range of positions that the value 3 occupies in the sorted dataset.

Positions 6 through 13.

Since the ratings are sorted, the single 1 is at position 1, the four 2s are at positions 2–5, and the eight 3s are at positions 6–13.

3

Find the maximum integer value of

x

that keeps the median at 3.

x = 8

If

x = 8

, the total number of students is

25

. The median is the 13th value, which is 3. If

x = 9

, the total is

26

, and the median is the average of the 13th (3) and 14th (4) values, which is 3.5.

4

Calculate the mean of the dataset when

x = 8

.

Mean =

85 / 25 = 3.4

The sum of the books read is

1(1) + 2(4) + 3(8) + 4(8) + 5(4) = 85

. Dividing this by the total of 25 students gives the maximum mean.

Key Concept

Finding the maximum mean of a frequency distribution given a median constraint by setting up inequalities for the median position.

Question 13Question →

The frequency table below shows the number of siblings reported by a group of $10$ students in a class.

Number of Siblings	Number of Students
$0$	$2$
$1$	$4$
$2$	$3$
$3$	$1$

What is the mean number of siblings per student for this group?

1.0

1.5

1.3

2.5

3.25

Show answer & explanation

Answer: $1.3$

Answer

The mean number of siblings per student is

1.3

.

To find the mean number of siblings per student, first determine the total number of siblings by multiplying each sibling category by its frequency:

(0 \times 2) + (1 \times 4) + (2 \times 3) + (3 \times 1) = 13

. Next, determine the total number of students by summing the frequencies:

2 + 4 + 3 + 1 = 10

. Finally, divide the total siblings by the total students:

\frac{13}{10} = 1.3

.

Step-by-Step Solution

1

Find the total number of students by adding the frequencies in the second column.

2 + 4 + 3 + 1 = 10

students

This sum represents the total number of observations in the dataset.

2

Calculate the total number of siblings by multiplying each sibling count by the number of students reporting that count, then summing the products.

(0 \times 2) + (1 \times 4) + (2 \times 3) + (3 \times 1) = 0 + 4 + 6 + 3 = 13

siblings

This gives the sum of all data values across the entire group.

3

Divide the total number of siblings by the total number of students to find the mean.

\frac{13}{10} = 1.3

The mean is calculated as the sum of all values divided by the number of observations.

Key Concept

Calculating the mean from a frequency table.

Estimated Time:1m 0s

Question 14Question →

During a science experiment, a student measured the temperature changes of a liquid in degrees Celsius at seven different intervals. The recorded changes were $15^\circ\text{C}$ , $6^\circ\text{C}$ , $20^\circ\text{C}$ , $24^\circ\text{C}$ , $7^\circ\text{C}$ , $4^\circ\text{C}$ , and $8^\circ\text{C}$ . What is the positive difference between the median and the mean of these seven recorded temperatures?

4

6

8

12

16

Show answer & explanation

Answer: 4

Answer

4

To find the correct positive difference, we first find the mean by dividing the sum of the temperatures by 7:

(15 + 6 + 20 + 24 + 7 + 4 + 8) / 7 = 84 / 7 = 12

. We then sort the temperatures in ascending order:

4, 6, 7, 8, 15, 20, 24

. The median is the middle value of this sorted list, which is 8. The positive difference between the median and the mean is

|8 - 12| = 4

.

Step-by-Step Solution

1

Calculate the mean of the temperatures.

Mean = 12

Sum the seven values (

15 + 6 + 20 + 24 + 7 + 4 + 8 = 84

) and divide by the total number of data points (7) to get

84 / 7 = 12

.

2

Find the median of the temperatures.

Median = 8

Sort the values in ascending order:

4, 6, 7, 8, 15, 20, 24

. Since there are 7 data points, the median is the fourth value, which is 8.

3

Calculate the positive difference between the median and the mean.

Positive difference = 4

Subtract the median from the mean (or vice versa) and take the absolute value:

|8 - 12| = 4

.

Key Concept

Calculating and comparing the mean and median of a dataset, including the necessity of sorting data before finding the median.

Question 15Question →

A geology student collected a group of rocks and recorded their weights, in grams, in the table below. One frequency, $f$ , is missing.

Weight (grams)	Frequency
25	5
10	4
20	$f$
30	2
15	6

If the mean weight of the rocks in the group is exactly $18.75$ grams, what is the median weight, in grams, of the rocks?

3

10

15

17.5

20

Show answer & explanation

Answer: 17.5

Answer

17.5

The correct answer is 17.5. By setting up the weighted mean equation, we find that the missing frequency

f

is 3. This means there are 20 rocks in total. When sorted in ascending order, the weights are four 10s, six 15s, three 20s, five 25s, and two 30s. The 10th rock weighs 15 grams and the 11th rock weighs 20 grams. The median is the average of these two middle values: (15 + 20) / 2 = 17.5 grams.

Step-by-Step Solution

1

Set up an equation for the mean weight using the frequencies and weights from the table.

The sum of the weights is

25(5) + 10(4) + 20(f) + 30(2) + 15(6) = 315 + 20f

. The total number of rocks is

5 + 4 + f + 2 + 6 = 17 + f

. The mean is

\frac{315 + 20f}{17 + f} = 18.75

.

To find the missing frequency

f

using the given mean.

2

Solve the equation for

f

.

315 + 20f = 18.75(17 + f) \implies 315 + 20f = 318.75 + 18.75f \implies 1.25f = 3.75 \implies f = 3

.

To determine the number of rocks that weigh 20 grams.

3

Find the total number of rocks and identify the median position.

The total number of rocks is

17 + 3 = 20

. Since the total number is even, the median is the average of the 10th and 11th values when the weights are sorted in ascending order.

The median of an even number of data points is the average of the two middle values.

4

List the sorted weights and find the 10th and 11th values.

The sorted weights are: four 10s, six 15s (making 10 rocks total), followed by three 20s. Thus, the 10th rock weighs 15 grams and the 11th rock weighs 20 grams.

To find the values of the two middle rocks.

5

Calculate the average of the 10th and 11th values.

Median =

\frac{15 + 20}{2} = 17.5

grams.

To find the median weight.

Key Concept

Calculating the median from a frequency distribution table after finding a missing frequency using the mean.

Estimated Time:2m 30s

Question 16Question →

A gymnast competed in 5 events during a state meet, earning scores of $14.2$ , $13.8$ , $14.5$ , $13.5$ , and $14.0$ . She has one event remaining. If her goal is to have an average (arithmetic mean) score of exactly $14.1$ for all 6 events, what score must she earn on her final event?

14.0

14.1

14.5

14.6

15.1

Show answer & explanation

Answer: 14.6

Answer

14.6

To find the score needed on the final event, we first calculate the total sum of the scores from the first 5 events:

14.2 + 13.8 + 14.5 + 13.5 + 14.0 = 70.0

. Next, we find the total sum required for all 6 events to average exactly

14.1

:

14.1 \times 6 = 84.6

. Finally, we subtract the sum of the first 5 events from this target total:

84.6 - 70.0 = 14.6

. Therefore, the required score on the final event is 14.6.

Step-by-Step Solution

1

Calculate the sum of the scores from the first 5 events.

14.2 + 13.8 + 14.5 + 13.5 + 14.0 = 70.0

To find the missing score, we need to compare the current total score to the target total score.

2

Calculate the target total score for all 6 events to achieve an average of

14.1

.

14.1 \times 6 = 84.6

Since the average is the sum divided by the number of events, the total sum is the average multiplied by 6.

3

Subtract the sum of the first 5 events from the target total sum to find the required final score.

84.6 - 70.0 = 14.6

The difference between the required total sum and the current sum represents the score needed on the final event.

Key Concept

Calculating a missing value in a data set to achieve a target arithmetic mean.

Question 17Question →

A student volunteer recorded the hours they spent working at a community garden and the number of plants watered each day, as shown in the table below:

Day	Hours Volunteered	Plants Watered
Monday	4	32
Tuesday	5	45
Wednesday	6	48
Thursday	4	36
Friday	8	80

What was the average (mean) number of hours volunteered per day by the student during this five-day period?

5.0

5.4

20.6

27.0

48.2

Show answer & explanation

Answer: 5.4

Answer

5.4 hours per day

The correct answer is 5.4 hours. To find the mean, sum the daily hours volunteered (

4 + 5 + 6 + 4 + 8 = 27

) and divide by the number of days (

5

), resulting in

5.4

hours per day.

Step-by-Step Solution

1

Calculate the total hours volunteered by summing the values in the Hours Volunteered column.

4 + 5 + 6 + 4 + 8 = 27

hours

To find the average, we first find the sum of all daily hours.

2

Divide the total hours by the number of days, which is 5.

27 \div 5 = 5.4

hours per day

The mean is defined as the sum of the data values divided by the total number of data points.

Key Concept

Calculating the arithmetic mean from a frequency table or data set.

Question 18Question →

A track athlete recorded the following times, in seconds, for five 100-meter sprint trials:

12.4,\ 11.8,\ 13.1,\ 12.0,\ 12.2

What is the median time, in seconds, for these five trials?

1.3

12.0

12.2

12.3

13.1

Show answer & explanation

Answer: 12.2

Answer

The median time for the five trials is 12.2 seconds.

To find the median, first arrange the five times in ascending order: 11.8, 12.0, 12.2, 12.4, and 13.1. Since there are 5 values, the median is the middle (3rd) value, which is 12.2.

Step-by-Step Solution

1

Order the data set in ascending numerical order.

11.8, 12.0, 12.2, 12.4, 13.1

The median is defined as the middle value of an ordered data set.

2

Identify the middle (third) position in the 5-item ordered list.

The third value is

12.2

.

For an odd number of data points (

n = 5

), the median is the value at position

\frac{n+1}{2} = 3

.

Key Concept

Calculating the median of a finite data set

Question 19Question →

A local library recorded the number of books read by students participating in a summer reading program. The results are summarized in the frequency table below.

Number of Books Read	Number of Students
$2$	$4$
$3$	$6$
$5$	$5$
$8$	$3$
$10$	$2$

What is the mean number of books read per student for this group?

4.75

5.00

5.60

19.00

4.00

Show answer & explanation

Answer: $4.75$

Answer

The mean number of books read per student is

4.75

.

The value

4.75

is correct because finding the mean from a frequency table requires calculating the weighted sum of all data values divided by the total frequency. Multiplying each book count by its student count gives

2 \times 4 + 3 \times 6 + 5 \times 5 + 8 \times 3 + 10 \times 2 = 95

total books. Dividing

95

by the total student count of

20

yields

4.75

.

Step-by-Step Solution

1

Calculate the total number of students.

Total students =

4 + 6 + 5 + 3 + 2 = 20

.

To find the mean per student, the total number of participants is needed for the denominator.

2

Calculate the total number of books read by multiplying each value by its frequency and summing.

Total books =

(2 \times 4) + (3 \times 6) + (5 \times 5) + (8 \times 3) + (10 \times 2) = 8 + 18 + 25 + 24 + 20 = 95

.

The mean calculation requires the aggregate total of all books read across all students.

3

Divide the total number of books by the total number of students.

Mean =

\frac{95}{20} = 4.75

.

The mean of a frequency distribution is given by the total sum divided by the total frequency.

Key Concept

Weighted Mean from a Frequency Table

Estimated Time:1m 30s

Question 20Question →

A biology class recorded the height, in centimeters, of $15$ plant seedlings after two weeks of growth. The frequency table below summarizes the recorded measurements:

Height (cm)	Number of Seedlings
$3$	$2$
$4$	$4$
$5$	$5$
$6$	$3$
$7$	$1$

What is the positive difference between the mean height and the median height, in centimeters, of these $15$ seedlings?

0.0

0.2

0.5

0.8

1.2

Show answer & explanation

Answer: $0.2$

Answer

The positive difference between the mean height and the median height is

0.2

centimeters.

To calculate the mean height, multiply each height by its frequency, sum the products (

72

), and divide by the total number of seedlings (

15

), giving

4.8

cm. To find the median, locate the middle value (

8^{\text{th}}

value) in the ordered dataset of

15

measurements, which is

5.0

cm. The positive difference between

5.0

cm and

4.8

cm is

0.2

cm.

Step-by-Step Solution

1

Calculate the total sum of all seedling heights using the frequency table

(3 \times 2) + (4 \times 4) + (5 \times 5) + (6 \times 3) + (7 \times 1) = 6 + 16 + 25 + 18 + 7 = 72

cm

Each height value must be multiplied by its corresponding frequency count to find the total sum.

2

Calculate the weighted mean height

Mean =

\frac{72}{15} = 4.8

cm

The sum of all heights (

72

) is divided by the total number of seedlings (

15

).

3

Determine the median height of the

15

seedlings

Median =

5.0

cm

For

15

ordered values, the median is the

8^{\text{th}}

data point. Accumulating counts from smallest height:

2

(at height 3)

+ 4

(at height 4)

= 6

values. The next

5

values (positions 7 through 11) all have a height of 5. Therefore, the

8^{\text{th}}

value is

5.0

cm.

4

Find the positive difference between the mean and the median

|4.8 - 5.0| = 0.2

cm

Subtracting the mean (

4.8

) from the median (

5.0

) gives a positive difference of

0.2

cm.

Key Concept

Descriptive Statistics and Data Representations

Estimated Time:1m 30s

Descriptive Statistics and Data Representations

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