SAT · Math · Problem-Solving and Data Analysis

One-variable data: distributions and measures of center and spread

One-variable data: distributions and measures of center and spread is part of Problem-Solving and Data Analysis in the SAT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.

Question types you will see here include effect of an outlier, compare spread from two data sets, mean from a frequency table and missing value given the mean.

Common traps: claiming the median changes as much as the mean; ignoring that range depends only on extremes; assuming the standard deviation stays fixed.

Sample questions

Question 1
The numbers of books read by 10 students last summer were: 18, 23, 2, 28, 32, 33, 3, 2, 24, 29. What is the median of the data?
- A.
  97/5
- B.
  47/2
- C.
  49/2
- D.
  33
Answer: B. 47/2
Order the data: 2, 2, 3, 18, 23, 24, 28, 29, 32, 33. The two middle values are 23 and 24; their mean is 47/2.
Question 2
Class A has 20 students with a mean exam score of 72. Class B has 30 students with a mean exam score of 82. What is the mean score of all 50 students?
- A.
  76
- B.
  77
- C.
  78
- D.
  82
Answer: C. 78
Weight each mean by class size: (20 × 72 + 30 × 82)/50 = (1,440 + 2,460)/50 = 78 (choice C). Choice A swaps the weights, using 30 for class A and 20 for class B; Choice B averages 72 and 82 without weighting; Choice D reports only class B's mean.
Question 3
A youth football team recorded the number of goals it scored in each of its 20 matches.
Goals scored Number of matches
0 3
1 5
2 6
3 4
4 2
What is the mean number of goals scored per match?
- A.
  1.85
- B.
  2
- C.
  4
- D.
  7.4
Answer: A. 1.85
The team scored 0(3) + 1(5) + 2(6) + 3(4) + 4(2) = 37 goals in 20 matches, a mean of 37 ÷ 20 = 1.85 (A). B is the mean of the values 0 to 4, ignoring how often each occurred; C is the mean of the frequencies; D divides 37 by the 5 rows of the table.
Question 4
A dance studio recorded the number of students in each of its 20 classes last month.
Class size Number of classes
8 4
10 6
12 5
14 3
18 2
What is the mean class size?
Enter your own answer.
Question 5
The scores of 9 students on a mathematics quiz were 35, 62, 78, 81, 84, 85, 86, 88 and 90. Which statement about the data is true?
- A.
  The mean is greater than the median.
- B.
  The mean and the median are equal.
- C.
  The median exceeds the mean by more than 10 points.
- D.
  The median exceeds the mean by less than 10 points.
Answer: D. The median exceeds the mean by less than 10 points.
The median is the 5th score, 84. The total is 689, so the mean is about 76.6, and the median exceeds it by about 7.4 points (D). The low scores 35 and 62 form a left tail that pulls the mean below the median, so A and B are false; C overstates the gap.

A youth football team recorded the number of goals it scored in each of its 20 matches.
Goals scored	Number of matches
0	3
1	5
2	6
3	4
4	2

A dance studio recorded the number of students in each of its 20 classes last month.
Class size	Number of classes
8	4
10	6
12	5
14	3
18	2

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