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SAT · Math · Problem-Solving and Data Analysis
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.
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?
97/5
47/2
49/2
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
In a physics class, a student hangs a 100-gram mass from each of 40 springs chosen at random from a batch of 800 identical springs. The mean extension of the 40 springs is 4.2 cm, with a margin of error of 0.3 cm. Which of the following is the most appropriate conclusion?
The mean extension of all 800 springs in the batch is likely between 3.9 cm and 4.5 cm.
Every spring in the batch extends between 3.9 cm and 4.5 cm.
The mean extension of springs from any manufacturer is likely between 3.9 cm and 4.5 cm.
The mean extension of all 800 springs in the batch is exactly 4.2 cm.
Answer: A. The mean extension of all 800 springs in the batch is likely between 3.9 cm and 4.5 cm.
The sample mean of 4.2 cm with a margin of error of 0.3 cm supports an interval of 3.9 to 4.5 cm for the mean of the population the sample came from, the 800 springs. B applies the interval to every individual spring; C extends it to springs the sample did not represent; D claims an exact value that a sample cannot give.
Question 3
A weather station in Chennai recorded the rainfall, in millimeters, on 5 days during the monsoon: 12, 18, 7, 25 and x. The mean rainfall for the 5 days was 15 millimeters. What is the value of x?
Enter your own answer.
Answer: 13
A mean of 15 over 5 days means a total of 15 × 5 = 75 millimeters. The four known days add to 62, so x = 75 − 62 = 13. Subtracting 62 from 15, or dividing by 4 instead of 5, skips the step of turning the mean into a total.
Question 4
A factory has 30 day-shift workers who average 9.5 hours of overtime a month and 20 night-shift workers who average 12 hours of overtime a month. What is the mean monthly overtime, in hours, for all 50 workers?
10.5
10.75
11
21.5
Answer: A. 10.5
The total overtime is 30(9.5) + 20(12) = 285 + 240 = 525 hours, and 525 ÷ 50 = 10.5 (A). B averages the two group means as if the groups were equal; C attaches each mean to the other group's size; D adds the means.
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?
The mean is greater than the median.
The mean and the median are equal.
The median exceeds the mean by more than 10 points.
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 few questions show the style. The free scan shows which SAT skills cost you the most points, then builds a plan around them.