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Center, Shape, & Spread of Data - SAT Math

  1. Two students each recorded 5 commute times (in minutes). Student A: 18, 19, 20, 21, 22. Student B: 10, 18, 20, 22, 30. Which statement correctly compares the medians and ranges of the two data sets?
    Answer: Same median; Student B has larger range. This question compares medians and ranges for two commute time data sets. For Student A (18, 19, 20, 21, 22): the median is 20 (middle value) and range is 22-18=4. For Student B (10, 18, 20, 22, 30): the median is 20 (middle value) and range is 30-10=20. Both students have the same median of 20 minutes, but Student B has a much larger range due to the extreme values of 10 and 30. This illustrates how median is resistant to extreme values while range is sensitive to them. When comparing data sets, calculate each measure separately and be careful not to assume that different distributions must have different centers.
  2. A coach tracked the number of laps each athlete ran in practice: 12, 12, 13, 13, 13, 14, 14, 15. What are the mode and the range of this data set?
    Answer: Mode 13; Range 3 This question asks for the mode and range of lap counts. The mode is the most frequent value: looking at 12, 12, 13, 13, 13, 14, 14, 15, we see 13 appears three times (more than any other value), so the mode is 13. The range is the difference between maximum and minimum values: 15 - 12 = 3. A common error is confusing mode with median or mean, or calculating range as max + min instead of max - min. Remember that mode identifies the most popular value, while range measures the total spread.
  3. The following data set represents the ages of participants in a survey: 22, 25, 27, 22, 30, 26, 31, 29, 22. What is the mode of the data?
    Answer: 22 This question asks for the mode, which is the value that appears most frequently in a dataset. To find the mode, count how many times each value appears in the dataset: 22, 25, 27, 22, 30, 26, 31, 29, 22. Counting the frequencies: 22 appears 3 times, 25 appears 1 time, 27 appears 1 time, 30 appears 1 time, 26 appears 1 time, 31 appears 1 time, and 29 appears 1 time. Since 22 appears most frequently (3 times), it is the mode. A common error is confusing mode with median or mean, so remember that mode specifically refers to the most frequent value.
  4. A box plot shows the following five-number summary: Min = 5, Q1 = 10, Median = 15, Q3 = 20, Max = 30. What is the interquartile range (IQR) of the data?
    Answer: 10 This question asks for the interquartile range (IQR), which measures the spread of the middle 50% of the data. The IQR is calculated as Q3 minus Q1, where Q3 is the third quartile and Q1 is the first quartile. From the five-number summary given: Q3 = 20 and Q1 = 10, so IQR = 20 - 10 = 10. The IQR represents the range of the middle half of the data and is less affected by outliers than the total range. A common error is confusing IQR with the total range (Max - Min) or incorrectly adding the quartiles instead of subtracting. Remember that IQR specifically measures the spread between the 25th and 75th percentiles.
  5. For the data set 4, 7, 9, 12, 15, 15, 18, 21, the value 21 is replaced by 10. By how much does the mean change?
    Answer: -1.375 The total decreases by 11, so the mean changes by -11/8 = -1.375. The distractors arise from forgetting to divide by 8 or dividing by an incorrect number.
  6. A coach recorded the number of successful free throws (out of 20) made by a player in 9 practice sessions: 11, 12, 12, 13, 13, 13, 14, 15, 18. What is the median number of successful free throws for these sessions?
    Answer: 13 This question asks for the median of 9 practice session scores. To find the median, we first verify the data is already ordered: 11, 12, 12, 13, 13, 13, 14, 15, 18. With 9 values (odd number), the median is the middle value, which is the 5th value in the ordered list: 13. A common error would be averaging the two middle values (13 and 13), but this is only necessary when we have an even number of values. When finding the median, always count carefully to identify the exact middle position in odd-numbered data sets.
  7. A list of 9 salaries (in thousands of dollars) at a small company is: 38, 40, 41, 42, 43, 44, 45, 46, 120. Which statement is true about the mean and median of these data?
    Answer: Mean > median because of the outlier This dataset has an outlier salary of 120 thousand. For the median: ordering gives 38,40,41,42,43,44,45,46,120, so median = 43 (5th value). For the mean: sum = 38+40+41+42+43+44+45+46+120 = 459, so mean = 459/9 = 51. The mean (51) is greater than the median (43) because the outlier pulls the mean upward. This illustrates that outliers affect the mean more than the median. When a distribution has a high outlier, expect mean > median; with a low outlier, mean < median.
  8. A student recorded the number of minutes spent studying each day for a week: 25, 30, 30, 35, 40, 45, 45. What is the standard deviation most likely to do if each value increases by 10 minutes (for example, 25 becomes 35)?
    Answer: Stay the same The question asks what happens to standard deviation when all values increase by the same amount (10 minutes). Standard deviation measures the spread of data around the mean. When you add the same constant to every value, all values shift equally, maintaining the same distances from each other and from the new mean. Therefore, the standard deviation stays the same. The original spread is preserved because we're just shifting the entire dataset up by 10. This is different from multiplying each value, which would change the standard deviation. For the SAT, remember that adding a constant to all values doesn't change measures of spread (standard deviation, IQR, range), only measures of center.
  9. Consider the following scores of a test: 15, 18, 22, 18, 25, 28, 30, 15, 18. What is the range of the scores?
    Answer: 15 This question asks for the range of test scores, which is the difference between the maximum and minimum values. First, identify the highest and lowest scores from the dataset: 15, 18, 22, 18, 25, 28, 30, 15, 18. The maximum score is 30 and the minimum score is 15. Calculate the range by subtracting: 30 - 15 = 15. The range measures the spread of the data by showing how far apart the extreme values are. Don't confuse range with other measures of spread like standard deviation or interquartile range.
  10. The box plot (given by its five-number summary) represents the distribution of daily temperatures (°F) over 15 days: min 48, Q1Q_1 52, median 56, Q3Q_3 60, max 75. What is the interquartile range (IQR) of the temperatures?
    Answer: 8 This question asks for the IQR from a box plot's five-number summary. Given: min = 48, Q1 = 52, median = 56, Q3 = 60, max = 75. The interquartile range is calculated as IQR = Q3 - Q1 = 60 - 52 = 8. Don't confuse IQR with the full range (75 - 48 = 27) or use the median in the calculation. The IQR represents the spread of the middle 50% of the data. When reading box plots, remember that IQR only involves the first and third quartiles, not the minimum, maximum, or median.
  11. The numbers of books read by 6 students last month were 3, 5, 2, 4, 6, and 5. What is the range?
    Answer: 4 Range = largest minus smallest = 6 - 2 = 4. The distractors come from subtracting the wrong values or misreading the data.
  12. A data set has mean 50 and standard deviation 0. Which description must be true about the data values? Use the idea that standard deviation measures spread and equals 0 only when there is no variation.
    Answer: All values are 50 Standard deviation measures how far values sit from the mean, and it can equal 0 only when no value deviates at all. With a mean of 50, that forces the data to be 50, 50, 50, and so on, which is the choice saying every value equals 50. The choice saying all values are 0 contradicts the given mean, because a list of zeros averages to 0, not 50. The choice describing values evenly spread around 50 still requires values above and below the mean, and any such gap makes the squared deviations positive, pushing the standard deviation above 0. The choice putting half the values above 50 and half below fails for the same reason: that split guarantees variation, so its standard deviation could not be 0.
  13. A box plot summarizes two classes' test scores. Class 1 has five-number summary (min 55, Q1Q_1 70, median 78, Q3Q_3 85, max 95). Class 2 has (min 50, Q1Q_1 72, median 78, Q3Q_3 90, max 98). Which statement is true about the distributions' centers and spreads?
    Answer: Same median; Class 2 has larger IQR This question compares two classes using their five-number summaries. Class 1: median = 78, IQR = Q3 - Q1 = 85 - 70 = 15. Class 2: median = 78, IQR = Q3 - Q1 = 90 - 72 = 18. Both classes have the same median (78), but Class 2 has a larger IQR (18 > 15), indicating greater spread in the middle 50% of scores. This shows that distributions can have identical centers but different spreads. The IQR is a better measure of spread than range when comparing distributions because it's not affected by extreme values. When interpreting box plots, always calculate IQR = Q3 - Q1 to measure spread.
  14. A box plot summarizes the distribution of quiz scores for a class with five-number summary: minimum 50, Q1=65Q_1=65, median 75, Q3=85Q_3=85, maximum 100. Based on this summary, what is the interquartile range (IQR) of the quiz scores?
    Answer: 2020 The question asks for the interquartile range (IQR) based on the five-number summary of quiz scores. The IQR is calculated as Q3 minus Q1, so 85 - 65 = 20. This measure focuses on the middle 50% of the data, making it resistant to outliers like the minimum or maximum. The five-number summary provides all needed values without requiring the full dataset. A common error is subtracting median from Q1 instead, but IQR specifically uses Q3 and Q1. In box plots, visualize the IQR as the box length to understand spread quickly.
  15. A store tracked the number of customers entering during 10 consecutive hours: 18, 19, 20, 20, 21, 21, 22, 22, 23, 80. Which measure of center is most affected by the outlier value 80: the mean or the median? Assume the rest of the data stays the same.
    Answer: Mean, because it uses all values The question asks which measure of center, the mean or the median, is most affected by the outlier 80 in the customer data for 10 hours. To calculate the mean, sum all values (18+19+20+20+21+21+22+22+23+80=266) and divide by 10, giving 26.6; without the 80, the mean of the first 9 is about 20.7, showing a significant change. For the median, with 10 values ordered, it's the average of the 5th and 6th (21 and 21), remaining 21 even if 80 changes, as it doesn't affect the middle. Thus, the mean is more affected because it uses all values, while the median is robust to outliers. A common error is assuming the median shifts with extreme values, but it only depends on the middle order. Understanding the difference helps choose the right center measure in skewed data. When outliers are present, compare both measures to see their sensitivity.
  16. The weights (in kg) of a group of 10 students are recorded as follows: 45, 48, 52, 50, 49, 48, 51, 47, 46, 48. What is the mode of the data?
    Answer: 48 This question asks for the mode, which is the value that appears most frequently in a data set. Looking at the weights 45, 48, 52, 50, 49, 48, 51, 47, 46, 48, we need to count how often each value appears. The value 48 appears three times, while all other values appear only once. Therefore, 48 is the mode since it has the highest frequency.
  17. The mean of five numbers is 12. When a sixth number, 18, is added, what is the new mean?
    Answer: 13 Original total is 5×12 = 60; new total is 60 + 18 = 78, and 78 ÷ 6 = 13. The other choices reflect leaving the mean unchanged or averaging 12 and 18 incorrectly.
  18. A cafe recorded the number of customers each day for a week: 22, 18, 22, 25, 25, 25, 18. What is the mode of these data?
    Answer: 25 The value 25 occurs most frequently (three times). The other values occur fewer times or not at all.
  19. Seven test scores have a mean of 78. One score, 92, was mistakenly recorded as 29. What is the correct mean?
    Answer: 87 Correct sum is 7×78+(92−29)=6097 \times 78 + (92 - 29) = 609, so the mean is 609÷7=87609 \div 7 = 87. The other choices reflect not correcting the error or adjusting by the wrong amount.
  20. The mean of 12 numbers is 15, and the mean of 8 different numbers is 21. When all 20 numbers are combined, what is the mean?
    Answer: 17.4 Combined mean is (12×15+8×21)/20=(180+168)/20=17.4. The other choices come from simple averaging or incorrect weighting.

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