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SAT Digital Math · SL AA

Problem-Solving and Data Analysis: One Variable Statistics and Probability

Targeted Practice Sheet 3

Worked Examples: SAT Strategy

Example 1: Easy (Working Backward from the Mean)

Question: The mean of a set of 5 numbers is 24. If four of the numbers are 18, 22, 26, and 30, what is the fifth number?
SAT Strategy: Use the "Sum = Mean $\times$ Count" shortcut. Don't set up a massive fraction equation if you can avoid it.
  1. If the mean of 5 numbers is 24, the total sum of all 5 numbers must be exactly $5 \times 24 = 120$.
  2. Find the sum of the four known numbers: $18 + 22 + 26 + 30 = 96$.
  3. The missing fifth number is the difference between the total sum and the known sum: $120 - 96$.
  4. $120 - 96 = 24$.
  5. Answer: 24.

Example 2: Medium (The Effect of Outliers)

Question: A dataset contains the following values: 12, 14, 15, 16, 18. A new value of 85 is added to the dataset. Which of the following statements best describes the effect of this new value on the mean and median?
SAT Strategy: An outlier is a value that is significantly higher or lower than the rest of the data. Outliers violently drag the mean toward them, but the median (the middle number) barely moves.
  1. Original dataset: 12, 14, 15, 16, 18. The median is 15. The mean is 15.
  2. New dataset: 12, 14, 15, 16, 18, 85. The median shifts slightly to 15.5.
  3. The mean, however, skyrockets because of the massive sum ($160 \div 6 \approx 26.6$).
  4. Answer: The mean will increase significantly more than the median.

Example 3: Hard (Conceptual Standard Deviation)

Question: Data Set A consists of the numbers: 5, 5, 5, 5, 5. Data Set B consists of the numbers: 1, 3, 5, 7, 9. Which data set has the greater standard deviation?
SAT Strategy: Do NOT use the standard deviation formula! Standard deviation is simply a measure of "how far the data points are spread away from the mean."
  1. Look at Data Set A. The mean is 5. Every single number is exactly 5. The data has zero spread. Therefore, the standard deviation is exactly 0.
  2. Look at Data Set B. The mean is 5. The numbers are spread far away from the mean (1 and 9 are 4 units away!).
  3. Because the values in Set B are more spread out from the center, Set B has a greater standard deviation.
  4. Answer: Data Set B has the greater standard deviation.

Practice Questions

Level 1: Easy (Foundational SAT)

  1. The mean of 3 numbers is 40. If two of the numbers are 35 and 45, what is the third number? (Grid-in)
  2. A dataset consists of the following values: 4, 7, 7, 9, 10, 15. What is the median of this dataset?
    • A) 7
    • B) 8
    • C) 9
    • D) 10
  3. A bag contains 4 red marbles, 5 blue marbles, and 3 yellow marbles. If one marble is drawn at random, what is the probability that it is NOT blue?
    • A) $\frac{5}{12}$
    • B) $\frac{7}{12}$
    • C) $\frac{1}{2}$
    • D) $\frac{7}{5}$
  4. The ages of 5 children at a party are 6, 6, 7, 8, and 10. If an 11-year-old child joins the party, what will be the new median age? (Grid-in)
  5. Data Set X has a standard deviation of 0. Which of the following could be Data Set X?
    • A) 1, 2, 3, 4, 5
    • B) -5, -5, 0, 5, 5
    • C) 8, 8, 8, 8, 8
    • D) 0, 10, 100, 1000

Level 2: Medium (Standard SAT)

  1. A student's average (mean) score on 4 tests is an 82. What score must the student earn on the 5th test to raise their average to an 85? (Grid-in)
  2. A frequency table shows the number of pets owned by 15 families. {|c|c|} Number of Pets & Frequency
    0 & 3
    1 & 5
    2 & 4
    3 & 2
    4 & 1
    What is the median number of pets owned by these families? (Grid-in)
  3. A list of numbers contains 10 values. The largest value is removed from the list. Which of the following statistics MUST decrease?
    • A) The mean
    • B) The median
    • C) The standard deviation
    • D) The range
  4. In a class of 30 students, 18 play an instrument and 12 play a sport. If 5 students play both an instrument and a sport, what is the probability that a randomly selected student plays neither?
    • A) $\frac{1}{6}$
    • B) $\frac{1}{5}$
    • C) $\frac{5}{30}$
    • D) $\frac{1}{4}$
  5. Data Set A: 10, 12, 14, 16, 18
    Data Set B: 10, 14, 14, 14, 18
    Which of the following statements comparing the standard deviations of the two data sets is true?
    • A) The standard deviation of A is greater than B.
    • B) The standard deviation of B is greater than A.
    • C) The standard deviations are equal.
    • D) There is not enough information to compare them.

Level 3: Hard (Advanced SAT)

  1. A dataset of 20 integers has a mean of 50 and a median of 48. If the highest number in the dataset is increased by 10, what is the new mean of the dataset? (Grid-in)
  2. A factory produces lightbulbs. The probability that a randomly selected lightbulb is defective is $0.04$. If the factory produces 5,000 lightbulbs in a day, what is the expected number of non-defective lightbulbs? (Grid-in)
  3. The mean of $a, b, c,$ and $d$ is 15. The mean of $a, b, c, d,$ and $e$ is 18. What is the value of $e$? (Grid-in)
  4. Data Set 1: A normally distributed set of 1,000 values with a mean of 50 and a range of 20.
    Data Set 2: The identical set of 1,000 values from Data Set 1, but the number 5 is added to every single value.
    Which of the following statements about the standard deviation of the two datasets is true?
    • A) The standard deviation of Set 2 is 5 greater than Set 1.
    • B) The standard deviation of Set 2 is 25 greater than Set 1.
    • C) The standard deviations of the two sets are exactly the same.
    • D) The standard deviation of Set 2 is 5 times the standard deviation of Set 1.
  5. A dataset of 9 consecutive even integers has a median of 20. What is the mean of this dataset?
    • A) 18
    • B) 20
    • C) 22
    • D) It cannot be determined from the information given.

Level 4: Very Hard (Slightly beyond SAT)

  1. The average (mean) weight of a group of 10 players is 160 pounds. Two new players join the group, and the new average weight of all 12 players becomes 165 pounds. If one of the new players weighs 180 pounds, what is the weight, in pounds, of the other new player? (Grid-in)
  2. A distribution of test scores has a mean of 75 and a standard deviation of 8. If every score in the distribution is multiplied by 1.5, what is the standard deviation of the new distribution? (Grid-in)
  3. The probability of Event A occurring is $0.4$, and the probability of Event B occurring is $0.5$. If Event A and Event B are independent, what is the probability that Event A occurs but Event B does NOT occur?
    • A) $0.20$
    • B) $0.40$
    • C) $0.50$
    • D) $0.90$
  4. A dataset has 15 identical values. If one of the values is changed from 10 to 40, the new mean is 12. What was the original mean of the dataset? (Grid-in)
  5. Two classes take the same exam. Class X has 20 students and a mean score of 80. Class Y has 30 students and a mean score of 90. What is the combined mean score of all 50 students?
    • A) $84$
    • B) $85$
    • C) $86$
    • D) $88$

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