20 medium SAT One-variable data: center & spread questions
Medium is where most scores are actually won and lost. These questions are not tricky for the sake of it, but every one of them has a wrong answer built to catch a specific shortcut.
Every question below is a real item from the SAT Climb bank, tagged medium by the same difficulty model the app uses to build your practice. Pick an answer before you open the explanation.
Math · Problem-Solving and Data Analysis~2 per testMedium tier
A data set consists of the following 8 values: 12, 15, 15, 18, 18, 18, 21, 45. What is the difference between the mean and the median of this data set?
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Why B is right
The median of the 8 values is the average of the 4th and 5th values when arranged in order: (18 + 18) / 2 = 18. The mean is (12 + 15 + 15 + 18 + 18 + 18 + 21 + 45) / 8 = 162 / 8 = 20.25. The difference is 20.25 - 18 = 2.25.
Why the others are wrong
AThis incorrectly assumes the mean and median are equal, not accounting for the effect of the outlier value 45.
CThis results from computing the difference between the maximum and minimum of the middle values rather than properly finding the median.
DThis results from an error in calculating the median position, such as using only the 4th value (18) instead of averaging the 4th and 5th values.
Question 2Medium
Value Frequency 20 3 25 5 30 8 35 4 The table shows the frequency of values in a data set. What is the median of the data set?
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Why C is right
The total number of data values is 3 + 5 + 8 + 4 = 20. The median is the average of the 10th and 11th values when the data is ordered. The first 3 values are 20, the next 5 values are 25 (cumulative: 8 values), and the next 8 values are 30 (cumulative: 16 values). Both the 10th and 11th values are 30, so the median is 30.
Why the others are wrong
AThis results from computing the mean of the data set instead of the median.
BThis results from incorrectly averaging the middle two frequency positions instead of the actual data values.
DThis results from incorrectly identifying the median position or averaging 30 and 35.
Question 3Medium
Value Frequency 5 2 7 5 9 8 11 4 13 1 The table shows the frequency of values in a data set. If the value 13 is removed from the data set, by how much will the median change?
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Why C is right
The data set has 2+5+8+4+1 = 20 values. The median is the average of the 10th and 11th values. Counting: values 1-2 are 5, values 3-7 are 7, values 8-15 are 9, values 16-19 are 11, and value 20 is 13. The 10th and 11th values are both 9, so the median is 9. After removing 13, there are 19 values, and the median is the 10th value, which is still 9. The median does not change.
Why the others are wrong
AThis incorrectly applies the change in mean to the median. The mean decreases when 13 is removed, but the median is resistant to changes in extreme values.
BThis results from miscounting the position of the median or confusing the effect of removing a value on position versus value.
DThis incorrectly suggests that removing a high value increases the median, which contradicts how medians behave.
Question 4Medium
The box plots below summarize the distributions of annual incomes (in thousands of dollars) for Neighborhood R and Neighborhood S.
[FIGURE: Two box plots. Neighborhood R: minimum = 30, Q1 = 45, median = 65, Q3 = 85, maximum = 100. Neighborhood S: minimum = 35, Q1 = 55, median = 65, Q3 = 75, maximum = 95.]
Which of the following is true about the two data sets?
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Why C is right
Both Neighborhood R and Neighborhood S have a median of 65 thousand dollars. The IQR for Neighborhood R is 85 - 45 = 40 thousand dollars, while the IQR for Neighborhood S is 75 - 55 = 20 thousand dollars. Therefore, Neighborhood R has a greater interquartile range.
Why the others are wrong
AThis option correctly identifies that the medians are equal but reverses which neighborhood has the greater IQR.
BThis option incorrectly states that Neighborhood R has a greater median, possibly confusing the median with the maximum.
DThis option correctly identifies that the medians are equal but incorrectly concludes the IQRs are equal.
Question 5Medium
Data set A: 12, 14, 15, 16, 18
Data set B: 10, 13, 15, 17, 20
Which data set has the greater standard deviation?
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Why B is right
Standard deviation measures the spread of data around the mean. Data set A has values clustered from 12 to 18 (range of 6), while data set B has values spread from 10 to 20 (range of 10). Data set B shows greater variability and therefore has the greater standard deviation.
Why the others are wrong
AThis incorrectly compares the data sets based on a different measure such as mean rather than spread around the mean.
CThis fails to recognize that data set B has noticeably greater spread than data set A.
DStandard deviation can be compared qualitatively by examining the spread of values without explicit computation.
Question 6Medium
A data set consists of the 5 values 3, 7, 9, 11, and 15. A new data set is created by adding 4 to each value in the original data set. How does the range of the new data set compare to the range of the original data set?
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Why C is right
The range of a data set is the difference between the maximum and minimum values. The original range is 15 - 3 = 12. When 4 is added to each value, the new minimum is 7 and the new maximum is 19, giving a range of 19 - 7 = 12. The range remains unchanged because adding a constant to all values shifts the entire distribution without changing the spread.
Why the others are wrong
AThis choice confuses the effect of adding a constant to each value with the effect on range, which actually remains unchanged.
BThis choice incorrectly applies a subtraction of the added constant from the range.
DThis choice incorrectly assumes the range scales multiplicatively when a constant is added to all values.
Question 7Medium
A data set consists of the 7 values 12, 15, 18, 18, 21, 24, and 30. If the value 30 is removed from the data set, which of the following quantities will change the least?
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Why B is right
The median of the original data set is 18 (the middle value when arranged in order). After removing 30, the six remaining values are 12, 15, 18, 18, 21, 24, and the median becomes 18 (the average of the two middle values 18 and 18). The median remains 18, so it changes by 0, which is the least change among all the quantities.
Why the others are wrong
AThe mean changes from approximately 19.7 to 18, a change of about 1.7, which is more than the change in median.
CThe range changes from 18 (30 - 12) to 12 (24 - 12), a change of 6, which is more than the change in median.
DThe maximum changes from 30 to 24, a change of 6, which is more than the change in median.
Question 8Medium
Value Frequency 20 3 25 7 30 12 35 5 40 3 The table shows the frequency of values in a data set. If the value 40 appears one additional time in the data set, by approximately how many standard deviations will the mean increase?
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Why A is right
The original data set has 30 values with mean (20×3 + 25×7 + 30×12 + 35×5 + 40×3)/30 = 895/30 ≈ 29.83. Adding one value of 40 gives new mean 935/31 ≈ 30.16. The increase is approximately 0.33. The standard deviation is approximately 6. The increase in terms of standard deviations is 0.33/6 ≈ 0.05, closest to 0.1.
Why the others are wrong
BThis overestimates the increase in standard deviation units.
CThis significantly overestimates the standardized increase.
DThis grossly overestimates the effect of adding one value.
Question 9Medium
Data value: 8, 9, 10, 11, 12, 13, 14. Frequency: 2, 4, 6, 8, 6, 4, 2. The frequency table summarizes the data values in a data set. What is the median of the data set?
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Why C is right
The total number of data values is 2 + 4 + 6 + 8 + 6 + 4 + 2 = 32. The median is the average of the 16th and 17th values. Counting from the left: values 1-2 are 8, values 3-6 are 9, values 7-12 are 10, values 13-20 are 11. Both the 16th and 17th values are 11, so the median is 11.
Why the others are wrong
AThis results from miscounting the position of the median, such as treating the median as the 15th value instead of the average of the 16th and 17th values.
BThis incorrectly uses the mean of the data set instead of finding the median.
DThis results from incorrectly identifying the middle position without properly accounting for the frequency distribution.
Question 10Medium
Value Frequency 15 4 20 6 25 2 30 8 The table shows the frequency of values in a data set. If the value 10 is added to this data set, which measure will change the most?
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Why C is right
The original data set has 20 values with a minimum of 15 and maximum of 30, giving a range of 15. When 10 is added as the new minimum, the range increases to 20, a change of 5 units. The mean decreases slightly from 23 to 22.38, a change of 0.62. The median remains at 22.5 (the average of the 10th and 11th values). The range changes by the largest absolute amount.
Why the others are wrong
AThis choice underestimates the effect on range, which changes by 5 units compared to the mean changing by less than 1 unit.
BThis choice incorrectly assumes the median changes significantly, when it actually remains nearly constant at the average of the 10th and 11th values.
DThis choice ignores the fact that different measures respond differently to outliers, with range being most sensitive to extreme values.
Question 11Medium
The box plots below summarize the distributions of commute times (in minutes) for Workers in District V and Workers in District W.
[FIGURE: Two box plots. District V: minimum = 10, Q1 = 20, median = 32, Q3 = 44, maximum = 55. District W: minimum = 15, Q1 = 26, median = 32, Q3 = 38, maximum = 50.]
Which of the following is true about the two data sets?
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Why C is right
Both District V and District W have a median commute time of 32 minutes. The IQR for District V is 44 - 20 = 24 minutes, while the IQR for District W is 38 - 26 = 12 minutes. Therefore, District V has a greater interquartile range.
Why the others are wrong
AThis option incorrectly states that District V has a greater median, possibly confusing the median with the range or maximum.
BThis option correctly identifies that the medians are equal but incorrectly concludes the IQRs are equal.
DThis option correctly identifies that the medians are equal but reverses which district has the greater IQR.
Question 12Medium
Data set J: 100, 104, 108, 112, 116
Data set K: 94, 102, 108, 114, 122
Which data set has the greater standard deviation?
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Why A is right
Data set J has values ranging from 100 to 116 (range of 16), while data set K has values ranging from 94 to 122 (range of 28). Data set K shows greater dispersion, with values more spread out from the mean, indicating greater standard deviation.
Why the others are wrong
BThis incorrectly identifies the data set with smaller spread as having greater standard deviation.
CThis fails to recognize that data set K has noticeably wider spread than data set J.
DStandard deviation measures spread around the mean, and can be compared qualitatively without calculating the median.
Question 13Medium
A data set has 9 values with a median of 45 and a range of 20. If the minimum value is removed from this data set, what is the range of the resulting data set with 8 values?
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Why D is right
The range is the difference between the maximum and minimum values. While we know the current range is 20, removing the minimum value means the new minimum is the second-smallest value, which is not specified. Without knowing the distribution of values, we cannot determine the new range.
Why the others are wrong
AThis incorrectly assumes all remaining values are equal, which contradicts the original range of 20.
BThis assumes the range decreases by exactly 1, but the gap between the minimum and second-minimum is unknown.
CThis incorrectly assumes the range stays the same after removing the minimum value.
Question 14Medium
Data value Frequency 45 6 48 4 51 8 54 7 57 5 The frequency table shows the 30 data values in a data set. If the value 45 is removed from the data set once, what is the new median?
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Why C is right
After removing one instance of 45, there are 29 values. The median is the 15th value. The first 5 values are 45 (6-1=5 remaining), the next 4 are 48 (total 9), the next 8 are 51 (total 17). The 15th value falls within the 51 group, so the median is 51.
Why the others are wrong
AThis incorrectly assumes the median shifts to 48 after removing 45, but there are still only 9 values at or below 48, so the 15th value is 51.
BThis is close to the mean of the new data set, not the median. The median is determined by position in the ordered list.
DThis would be the median if there were an even number of values and the 14th and 15th values were different, but with 29 values, the median is simply the 15th value, which is 51.
Question 15Medium
Value Frequency 8 2 10 5 12 8 14 3 18 2 The table shows the frequency of values in a data set. What is the median of this data set?
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Why C is right
The data set has 20 total values (2+5+8+3+2). The median is the average of the 10th and 11th values when arranged in order. Counting through the frequencies: values 1-2 are 8, values 3-7 are 10, values 8-15 are 12. Both the 10th and 11th values are 12, so the median is 12.
Why the others are wrong
AThis choice represents the value just below the median position, likely from miscounting the cumulative frequency.
BThis choice represents the mean of the data set rather than the median.
DThis choice represents a value in the data set but not at the median position.
Question 16Medium
A data set consists of 9 values. The minimum value is 4, the maximum value is 28, and the mean is 16. If the minimum value is removed from the data set, which of the following quantities must increase?
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Why A is right
When the minimum value 4 is removed, the mean must increase because the sum of the remaining 8 values is 140, giving a new mean of 17.5, which is greater than 16. The range stays at 24 because both the old and new ranges are from some value to 28. The median may or may not change depending on the distribution of the other values.
Why the others are wrong
BThis choice incorrectly claims only the range increases, but the range remains unchanged at 24 while the mean does increase.
CThis choice incorrectly claims the median must increase, but the median change depends on the specific values in the data set.
DThis choice incorrectly claims the range increases, but the range remains 24 since the maximum is unchanged.
Question 17Medium
The box plots below summarize the distributions of daily high temperatures (in degrees Fahrenheit) for City A and City B during a 30-day period.
[FIGURE: Two box plots. City A: minimum = 62, Q1 = 68, median = 74, Q3 = 80, maximum = 86. City B: minimum = 58, Q1 = 70, median = 74, Q3 = 78, maximum = 82.]
Which of the following is true about the two data sets?
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Why A is right
The median for both City A and City B is 74 degrees. The IQR for City A is 80 - 68 = 12 degrees, while the IQR for City B is 78 - 70 = 8 degrees. Therefore, City A has a greater interquartile range.
Why the others are wrong
BThis option incorrectly identifies the median by possibly confusing it with the maximum or mean, and incorrectly states the IQRs are equal.
CThis option correctly identifies that the medians are equal but reverses which city has the greater IQR.
DThis option correctly identifies that the medians are equal but incorrectly concludes the IQRs are equal, possibly by comparing a different measure of spread.
Question 18Medium
Data set E: 150, 155, 160, 165, 170
Data set F: 143, 153, 160, 167, 177
Which data set has the greater standard deviation?
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Why D is right
Data set E has values ranging from 150 to 170, while data set F has values ranging from 143 to 177. Data set F exhibits greater variability around its mean, with values more spread out, resulting in greater standard deviation.
Why the others are wrong
AThis fails to recognize that data set F has moderately wider spread than data set E.
BThis incorrectly identifies the data set with less spread as having greater standard deviation.
CStandard deviation can be compared qualitatively by examining the spread of values without explicit calculation.
Question 19Medium
A data set consists of 11 values. The minimum value is 42 and the maximum value is 78. If both the minimum and maximum values are removed from the data set, the range of the remaining 9 values is 24. What was the range of the original 11 values?
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Why B is right
The range of the original data set is the difference between the maximum value (78) and the minimum value (42): 78 - 42 = 36. The information about the range of the remaining 9 values is additional context but does not affect the calculation of the original range.
Why the others are wrong
AThis is the range of the reduced data set after removing the minimum and maximum, not the original range.
CThis might result from adding the new range (24) to some other value incorrectly.
DThis results from an arithmetic error or misunderstanding how to calculate range from given minimum and maximum values.
Question 20Medium
The table shows the frequency of values in a data set. Value: 8, Frequency: 3; Value: 10, Frequency: 5; Value: 12, Frequency: 7; Value: 14, Frequency: 2; Value: 16, Frequency: 3. What is the median of this data set?
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Why C is right
There are 3+5+7+2+3 = 20 data values total. The median is the average of the 10th and 11th values when arranged in order. Counting: values 1-3 are 8, values 4-8 are 10, values 9-15 are 12. Both the 10th and 11th values are 12, so the median is 12.
Why the others are wrong
AThis results from finding the value at position 9 instead of the middle position between 10 and 11.
BThis is the mean of the data set, not the median.
DThis results from incorrectly averaging 12 and 14 without considering frequency.
What to do after medium
Medium is the tier that decides most scores. If these are landing, the hard set is where the remaining points are.
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