What this SAT skill is testing
One-variable data questions ask you to describe a distribution using its center, spread, and shape. You may work from a list, table, dot plot, histogram, or box plot.
Measures of center and spread
- Mean: add all values and divide by the number of values. It is sensitive to extreme values.
- Median: the middle value after ordering the data. It is resistant to extreme values.
- Range: maximum minus minimum.
- Interquartile range (IQR): (Q_3-Q_1), the spread of the middle 50%.
- Standard deviation: a measure of how far values typically lie from the mean. More spread means a larger standard deviation.
Fast comparison: shifting every value by the same amount changes the mean and median by that amount but does not change the range, IQR, or standard deviation.
Worked SAT-style example
Data set A is (4,5,5,6,20). The median is 5, while the mean is (40/5=8). The high outlier pulls the mean upward but has much less effect on the median.
If the value 20 is replaced by 7, the center and spread both decrease. The new data are more tightly clustered, so the standard deviation is smaller.
How transformations affect a distribution
- If every value increases by 3, the mean and median increase by 3; measures of spread stay the same.
- If every value is multiplied by 2, the mean, median, range, IQR, and standard deviation are all multiplied by 2.
- Adding one new extreme value can change the mean and range substantially.
- Changing the frequency of an existing value can change the mean and median without changing the minimum or maximum.
Reading graphs accurately
- Read the axis labels and scale.
- Locate the center of the distribution.
- Compare overall spread, not just the endpoints.
- Notice clusters, gaps, skew, and possible outliers.
- Use the statistic named in the question; do not substitute a different measure.
Common traps
- Finding the median before putting values in order.
- Dividing by the number of distinct values instead of the total frequency.
- Assuming two data sets with the same mean have the same spread.
- Reading bar height without checking the graph scale.
- Using the range when the question asks about the middle 50%.
Practice strategy
Before calculating, predict the direction of change: should the center go up or down, and should the data become more or less spread out? That prediction catches many calculator and transcription errors.
