Jack scored a mean of 15 points

2018 Handbook

2018 Test B Math · Question 25 of 57

Also in 2025 Practice Test B Math, question 25.

Question

Jack scored a mean of 15 points per game in his first 3 basketball games. In his 4th game, he scored 27 points. What is his mean score for the first 4 games?

Show the answer

Correct answer: C

Jack scored a mean of 15 points per game in each of the first 3 games, so he earned a total of 45 points for the first 3 games. Use that information to calculate the mean over the four games:

$\frac{45+27}{4}=\frac{72}{4}=18$

Question analysis

This statistics question gives Jack's mean score for his first three basketball games and his score in the fourth, and asks for his mean over all four games. Students recover a total from a mean, add the new score and divide again. The earlier mean stands for three games, not one, which is easy to forget.

NYS standards: 6.SP.5c range, center, and overall pattern
Grade level: Grade 6 content, by NYS standard
Compared with grade 7: At grade 7 Finding a mean is grade 6 content, but working back from a mean to a total and then updating it is a common grade 7 test item, with a tempting shortcut that gives the wrong value.
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: Turning words into math moderate
Something you must already know minor
Solution: 3 steps
Reading level: Question and choices: about grade 1, 29 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 tempting shortcut, averaging 15 and 27, is quick, feels reasonable and gives a choice, so the question rewards students who slow down. Its error is weighting: the 15 stands for three games and the 27 for one. Rebuilding the total from the mean and dividing by four games takes only a moment longer.

Turning words into math moderate Understanding the question
The phrase “a mean of 15 points per game in his first 3 games” has to be turned into a total, 45 points, before the fourth game's 27 can be added. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
Something you must already know minor Working it out
Students must know that a mean is a total divided by a count, so a mean cannot be averaged with a single score as if both had equal weight. (Anderson & Krathwohl, 2001)

Strategy: Multiply the first mean by 3 to get the points so far, add the fourth game's points, and divide the new total by 4.

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.

The wrong answers, classified

A Numbers taken from the question (Hegarty, Mayer & Monk, 1995)
Reveals a surface strategy Expected pull: weak
Gives back the first mean, 15, as if one high-scoring game could not change the average.
B Little reason to choose it (Haladyna & Downing, 1993; Rodriguez, 2005)
Reveals a surface strategy Expected pull: weak
No standard computation gives 17; it lies between the old mean and the right one, so it mostly draws students who estimate.
D Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Reveals a misconception Expected pull: strong
Averages the old mean and the new score, (15 + 27) ÷ 2, giving one game the same weight as the first three together.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Mean (average): $\text{mean} = \frac{\text{sum of the values}}{\text{number of values}}$; The mean is the average. Add all the values, here the points in each game. Then divide by how many values there are, here the number of games. Turned around, it also says: sum = mean × number of values.

Write down what you know
Jack’s mean for his first 3 games was 15 points. The mean is the average: the total shared out equally among the games.
In game 4 he scored 27 points. You need his mean for all 4 games.
Find the total for 3 games
A mean of 15 over 3 games means the total is 3 times 15. You need the total, because a new mean is built from the total.
$15 \times 3 = 45$
He scored 45 points in the first 3 games.
Add the fourth game
Now add the 27 points from game 4 to get the total for all 4 games.
$45 + 27 = 72$
He scored 72 points in 4 games.
Divide by 4 games
$\text{mean} = \frac{\text{sum of the values}}{\text{number of values}} = \frac{72}{4} = 18$
Watch out: Do not average 15 and 27. That gives 21, choice D. It treats one game as if it counted as much as three.
So the answer is C, 18.

Check your answer

The mean should go up a little, since 27 is above 15. It went from 15 to 18, so that makes sense. Also, $18 \times 4 = 72$ , which matches the total points.

The original DOE question

Question 82 as printed on page 157 of the DOE handbook. — As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 157, question 82. Open page 157 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D).