SAT One-variable data: center & spread
Mean, median, mode, and comparing spread.
How to score it
- Median: sort first, then find the middle (average the two middles if the count is even).
- Frequency-table mean = Σ(value × frequency) ÷ Σ(frequency).
- You never compute standard deviation — just compare which set is more spread out.
Common traps
- Forgets to sort before taking the median.
- Ignores frequencies and averages the distinct values.
- Confuses spread (SD) with center (mean).
The 6 question types, with real examples
Median / mean / mode of a list
“What is the [median / mean] of the data shown?”
A data set consists of the 8 values 3, 5, 7, 9, 11, 13, 15, 17. What is the mean of this data set?
- A
- B✓
- C
- D
The mean is the sum of all values divided by the number of values. The sum is 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 80. Dividing by 8 gives 80 ÷ 8 = 10.
Mean from a frequency table
“What is the [mean / maximum] of the data set?”
The table shows the frequency of values in a data set. Value Frequency 8 4 10 6 12 5 14 3 What is the median of this data set?
- A✓
- B
- C
- D
There are 4 + 6 + 5 + 3 = 18 total values. The median is the average of the 9th and 10th values when ordered. The first 4 values are 8, the next 6 values are 10, so both the 9th and 10th values are 10. The median is 10.
Compare standard deviation (qualitative)
“Which data set has the greater standard deviation?”
Data set A: 12, 14, 15, 16, 18 Data set B: 10, 13, 15, 17, 20
Which data set has the greater standard deviation?
- AData set A
- BData set B✓
- CThe two data sets have equal standard deviations.
- DThe standard deviations cannot be determined from the given information.
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.
Effect of adding / removing a data point
“[list] with mean [mu]. If [new value v] is added, what is the new [mean / median]?”
A data set consists of the 5 values 8, 12, 14, 16, 20. If the value 8 is removed from the data set, how does the mean change?
- AThe mean increases by 2.
- BThe mean decreases by 2.
- CThe mean increases by 1.5.✓
- DThe mean remains the same.
The original mean is (8 + 12 + 14 + 16 + 20) / 5 = 70 / 5 = 14. After removing 8, the mean is (12 + 14 + 16 + 20) / 4 = 62 / 4 = 15.5. The increase is 15.5 - 14 = 1.5.
Comparison of two distributions
“[FIGURE: two box plots / histograms]. Which of the following is true about the two data sets?”
The box plots below summarize two data sets, Set X and Set Y. [FIGURE: Box plot for Set X shows minimum = 5, Q1 = 15, median = 25, Q3 = 35, maximum = 45. Box plot for Set Y shows minimum = 8, Q1 = 18, median = 25, Q3 = 40, maximum = 52.] Which of the following is true about the two data sets?
- ASet X and Set Y have the same median, and Set X has a smaller interquartile range than Set Y.✓
- BSet X has a smaller median than Set Y, and the two sets have the same interquartile range.
- CSet X and Set Y have the same median and the same interquartile range.
- DSet X and Set Y have the same median, and Set X has a larger interquartile range than Set Y.
Both sets have median = 25. For Set X, IQR = Q3 - Q1 = 35 - 15 = 20. For Set Y, IQR = 40 - 18 = 22. Since 20 < 22, Set X has a smaller IQR than Set Y.
Range / spread of a data set
“[data set]. What is the range of the data set? (inverse: given range + min/max, find the missing value)”
A data set consists of the 6 values 5, 7, 9, 9, 11, 13. What is the range of this data set?
- A
- B
- C✓
- D
The range is the difference between the maximum and minimum values. The maximum is 13 and the minimum is 5, so the range is 13 - 5 = 8.
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