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One-Variable Data: Distributions, Center & Spread for the SAT — Complete Study Guide + Free Practice Problems
Everything the SAT tests about one-variable data in one place: mean, median, and mode, range, interquartile range, and standard deviation, distribution shapes, the effect of outliers, and comparing data sets — with step-by-step examples, worked problems, and 2 free practice quizzes.
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Free · No signupWhat is one-variable data?
One-variable data is a data set that records a single measurement or characteristic for each item in a group — test scores for a class, heights of a group of plants, ages of a set of employees. The SAT asks you to summarize and interpret that single list of numbers using two ideas: where the data is centered, and how spread out it is.
Two data sets can have the exact same mean and completely different spreads. Always consider center and spread together — neither one alone fully describes a data set.
Measures of center: mean, median & mode
| Measure | Definition | Sensitive to outliers? |
|---|---|---|
| Mean | Sum of all values divided by the count | Yes — pulled toward extreme values |
| Median | Middle value when data is ordered | No — resistant to outliers |
| Mode | Most frequently occurring value | No, but may not exist or may not be unique |
Find the mean and median of: 4, 7, 7, 9, 15
For an even number of data points, the median is the average of the two middle values, not either one alone. Always sort the data first — the median of an unsorted list is meaningless.
Measures of spread: range, IQR & standard deviation
| Measure | Definition |
|---|---|
| Range | Maximum value minus minimum value |
| Interquartile range (IQR) | Third quartile (Q3) minus first quartile (Q1) — the spread of the middle 50% of the data |
| Standard deviation | A measure of how far, on average, data points are from the mean — larger means more spread out |
Find the range and IQR of: 3, 5, 7, 8, 9, 12, 14
The SAT rarely asks you to calculate standard deviation by hand — it more often asks you to compare which of two data sets has a larger or smaller standard deviation just by looking at how clustered or spread out the values are.
Distribution shapes
Symmetric
Roughly mirror images on both sides of the center. Mean and median are approximately equal.
Right-skewed
A long tail stretches toward higher values. Mean is pulled higher than the median.
Left-skewed
A long tail stretches toward lower values. Mean is pulled lower than the median.
A fast way to identify skew: compare mean and median. If mean > median, the data is right-skewed. If mean < median, it's left-skewed. If they're close, the distribution is roughly symmetric.
The effect of outliers
An outlier is a value far removed from the rest of the data set. Outliers pull the mean toward themselves but leave the median mostly unaffected — which is exactly why the SAT loves testing this comparison.
A data set is 4, 5, 6, 7, 8. Compare the mean and median before and after adding the outlier 50.
When a question asks which measure of center "best represents" a data set with an outlier, the answer is almost always the median, since it isn't distorted by the extreme value the way the mean is.
Comparing data sets
SAT questions often show two data sets, or a data set before and after a change, and ask you to compare their centers and spreads. Approach these by evaluating center and spread as two separate comparisons.
- Same center, different spread: the data set with values more tightly clustered around the center has a smaller standard deviation and IQR.
- Adding a constant to every value: shifts the mean and median by that same constant, but does not change the spread (range, IQR, standard deviation stay the same).
- Multiplying every value by a constant: multiplies both the center and the spread measures by that same constant.
Reading dot plots, histograms & box plots
| Display | What it shows |
|---|---|
| Dot plot | Every individual data value plotted above a number line |
| Histogram | Frequency of values grouped into intervals (bars) |
| Box plot | Five-number summary: minimum, Q1, median, Q3, maximum |
On a box plot, the box itself spans the IQR (Q1 to Q3), and the line inside the box is the median — not the mean. A wider box always means a larger IQR, regardless of where the whiskers extend.
Test day strategy for one-variable data
| Question signal | Fastest approach |
|---|---|
| "Best measure of center" with an outlier present | Choose median — it resists outliers |
| Mean vs. median comparison | Mean > median means right-skewed; mean < median means left-skewed |
| "Which data set has greater spread" | Compare how clustered the values are around the center, not just the range |
| Every value shifted by a constant | Center shifts by that constant; spread measures stay unchanged |
| Every value scaled by a constant | Both center and spread measures scale by that same constant |
| Box plot given | Read the five-number summary directly; the line in the box is the median |
Now put it to work
Two quiz sets, each building on the last — start with Quiz 1 and work through in order, or jump straight to the topic you need.
One-Variable Data 1
Mean, median, mode, and measures of spread.
Start quiz → Quiz 2One-Variable Data 2
Distribution shapes, outliers, and comparing data sets.
Start quiz →