QUIZ SCORES IN MRS. ARCH’S CLASS Quiz Score Number of Students 60 9 70 7 80 4 90 5 100 3 In the table above, what is the mean quiz score?

QUIZ SCORES IN MRS. ARCH’S CLASS

2024 Handbook

2024 Test A Math · Question 28 of 57

Question

QUIZ SCORES IN
MRS. ARCH’S CLASS

Quiz Score	Number of Students
60	9
70	7
80	4
90	5
100	3

In the table above, what is the mean quiz score?

Show the answer

Correct answer: C

First, find the total number of students who took the quiz.

$9 + 7 + 4 + 5 + 3 = 28$

Multiply each quiz score by the number of students who received that score. Add those products together to find the total number of points scored.

$(60 \times 9) + (70 \times 7) + (80 \times 4) + (90 \times 5) + (100 \times 3) = 2{,}100$

To find the mean quiz score, divide the total number of points by the number of students who took the quiz.

$2{,}100 \div 28 = 75$

Question analysis

A frequency table question: the table lists five quiz scores from 60 to 100 and how many students in Mrs. Arch's class earned each. Students find the mean quiz score for the whole class. Each score has to count once for every student who earned it, and the class size has to be added up from the table.

NYS standards: 6.SP.5c range, center, and overall pattern
6.SP.5a number of observations
Grade level: Grade 6 content, by NYS standard
Compared with grade 7: At grade 7 A mean from a frequency table is grade 6 content in a three-step form that grade 7 students meet regularly. Five rows of products make it slightly longer than a typical exercise, but the arithmetic is friendly.
Estimated difficulty: Moderate: several steps, or a close reading with one strong distractor.
Thinking it demands: Depth of knowledge 2, skills and concepts · mainly apply
What makes it hard: Reading a figure, table or graph moderate
Many steps minor
Error-prone arithmetic minor
Solution: 3 steps
Reading level: Question and choices: about grade 2, 19 words.
Students, last 30 days: Not enough answers yet to show (it takes 10).

Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

The table invites a shortcut: average the five scores in the first column and get a round number that is a choice. That ignores the fact that more students earned the low scores than the high ones, which pulls the true mean down. Weighting each score by its count is a five-row calculation with friendly numbers, and the other choices catch students who confuse the mean with the most common or the middle score.

Reading a figure, table or graph moderate Understanding the question
Each row of the table stands for several students, 9 with 60, 7 with 70 and so on, so a score counts once per student and the class size, 28, must be added up. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
Many steps minor Working it out
Five products, 540, 490, 320, 450 and 300, must be found and added to 2,100 before dividing by the 28 students. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)
Error-prone arithmetic minor Working it out
The last step, 2,100 ÷ 28, is not a familiar fact; noticing that 28 × 75 = 2,100, or testing the choices, keeps it quick. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Multiply each score by its number of students, add the products for the total points, add the student counts, and divide the points by the students.

Traps

Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.
Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.

The wrong answers, classified

A Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)
Reveals a surface strategy Expected pull: weak
Picks the score that the most students earned, 60, the mode of the class, instead of computing the mean.
B Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)
Reveals a misconception Expected pull: moderate
Finds the median instead: in the ordered list of 28 scores, the 14th and 15th are both 70.
D Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Reveals a misconception Expected pull: strong
Averages the five listed scores, (60 + 70 + 80 + 90 + 100) ÷ 5 = 80, without weighting each by its number of students.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Mean from a table: $\text{mean} = \frac{\text{total points}}{\text{number of students}}$; The mean is the average score. Total points is every student’s score added up. Each score in the table counts once for every student who got it. Number of students is how many took the quiz.

Sort out the table
The left column lists quiz scores. The right column says how many students got each score. For example, 9 students scored 60. You need the mean quiz score, which is the average.
Count the students
$9 + 7 + 4 + 5 + 3 = 28$
28 students took the quiz. So there are 28 scores in all, not just 5.
Find the total points
Each score counts once for each student who got it. So multiply each score by its number of students.
Score Students Points
60 9 $60 \times 9 = 540$
70 7 $70 \times 7 = 490$
80 4 $80 \times 4 = 320$
90 5 $90 \times 5 = 450$
100 3 $100 \times 3 = 300$
$540 + 490 + 320 + 450 + 300 = 2{,}100$
The class scored 2,100 points in all.
Divide to find the mean
$\text{mean} = \frac{\text{total points}}{\text{number of students}}$
$\text{mean} = \frac{2{,}100}{28} = 75$
To divide, try $28 \times 70 = 1{,}960$ . That leaves $2{,}100 - 1{,}960 = 140$ , and $140 = 28 \times 5$ . So the mean is $70 + 5 = 75$ . Watch out: do not just average the five scores in the table. That gives 80, choice D. Also, 70 is the median and 60 is the most common score, choices B and A. So the answer is C, 75.

Score	Students	Points
60	9	$60 \times 9 = 540$
70	7	$70 \times 7 = 490$
80	4	$80 \times 4 = 320$
90	5	$90 \times 5 = 450$
100	3	$100 \times 3 = 300$

Check your answer

More students scored low, 60 or 70, than high. So the mean should be below 80, the middle of the listed scores. 75 is below 80. Also, $28 \times 75 = 2{,}100$ , the total points.

The original DOE question

Question 85 as printed on page 83 of the DOE handbook. — As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 83, question 85. Open page 83 of the handbook (PDF)