SAT One-Variable and Two-Variable Data Analysis

One-variable and two-variable data questions ask you to read a data display and describe what it shows. One-variable questions give you a single list of values, quiz scores, plant heights, delivery times, and ask for the mean, median, mode, range or standard deviation, or ask you to pull those values off a table, histogram or dot plot. Two-variable questions give you a scatterplot of two related quantities and ask you to describe the trend, fit a line or curve to the data, or interpret slope and correlation in context.

This guide walks through both skills with original worked examples for each question type covered on the digital SAT. It closes with a list of the traps test writers build into wrong answers, and links to free drill sets at three difficulty levels.

Where One-Variable and Two-Variable Data Fit in Problem-Solving and Data Analysis

One-variable and two-variable data are two of seven official skills inside the SAT Math Problem-Solving and Data Analysis domain, one of four Math domains along with Algebra, Advanced Math, and Geometry and Trigonometry. The Math section runs 44 questions across two 35-minute modules, and Problem-Solving and Data Analysis accounts for 5 to 7 of those questions, about 15% of the Math section.

College Board names the two skills covered here One-variable data: distributions and measures of center and spread and Two-variable data: models and scatterplots. For the remaining five Problem-Solving and Data Analysis skills, see our SAT Ratios, Rates and Percentages guide and our SAT Probability and Statistical Inference guide, and see our SAT Math section guide for the full section breakdown. The table below maps every Problem-Solving and Data Analysis skill to where it is covered.

One-Variable Data: Center and Spread

One-variable data questions give you a list of values and ask you to compute or interpret a single summary number. Center describes where the middle of the data sits, and spread describes how far the values stretch from the center.

Mean, Median and Mode

Mean, median and mode each describe the center of a data set in a different way, and the SAT tests whether you pick the correct one for the situation described. The mean is the sum of every value divided by the count of values. The median is the middle value when the data is ordered from least to greatest, or the average of the two middle values when the count is even. The mode is the value which appears most often. A data set has no mode when every value is different, one mode when a single value repeats most, or more than one mode when two or more values tie for the highest count.

Here is an original example: a class records quiz scores of 68, 74, 74, 81, 85, 90 and 93. Find the mean, median and mode.

1. Add every score and divide by the count: 68 plus 74 plus 74 plus 81 plus 85 plus 90 plus 93 equals 565, and 565 divided by 7 equals 80.71, the mean.
2. Order the scores and pick the middle value: the seven scores are already ordered, so the fourth score, 81, is the median.
3. Find the value which repeats most: 74 appears twice and every other score appears once, so 74 is the mode.

Range and Standard Deviation

Range measures the total spread of a data set as the difference between the maximum and minimum values. Standard deviation measures spread by tracking, on average, how far each value sits from the mean. Two data sets with the same mean have a low standard deviation when the values cluster near the mean, and a high standard deviation when the values scatter far from the mean.

Here is an original example: two vending machines each dispense a mean of 12 ounces per cup. Machine A dispenses cups of 11.8, 12.1, 11.9 and 12.2 ounces. Machine B dispenses cups of 9.5, 14.0, 10.5 and 14.0 ounces. Both machines share the same mean, but Machine A's values cluster tightly around 12, so Machine A has the lower standard deviation, and Machine B's values scatter far from 12, so Machine B has the higher standard deviation.

How an Outlier Changes Center and Spread

An outlier is a value which sits far from the rest of a data set, and it pulls the mean toward itself while leaving the median mostly unaffected. Adding an extreme high or low value raises the range and the standard deviation, since the new value sits far from the center.

Here is an original example: five houses on a street sell for 210,000 dollars, 215,000 dollars, 220,000 dollars, 225,000 dollars and 900,000 dollars. The mean price is 354,000 dollars, pulled upward by the 900,000-dollar sale, while the median price stays at 220,000 dollars, the middle value in the ordered list. The median gives a better picture of a typical sale price on this street, since the mean is skewed by one outlier.

Reading Tables, Histograms and Dot Plots

SAT data questions display one-variable data as a frequency table, a histogram or a dot plot, and each format needs a different reading strategy. Reading the axis labels first prevents a slow start on these questions.

Reading a Frequency Table

A frequency table lists each value in one column and the count of data points in another. Multiply each value by its frequency, add the products, then divide by the total count to find the mean of data given in a frequency table.

Here is an original example: the table below shows the number of siblings reported by 20 students. Find the mean and the median number of siblings.

Work the four steps below to get both answers straight from the table.

1. Multiply each value by its frequency: 0 times 4 equals 0, 1 times 8 equals 8, 2 times 5 equals 10, 3 times 3 equals 9.
2. Add the products: 0 plus 8 plus 10 plus 9 equals 27.
3. Divide by the total count: 27 divided by 20 equals 1.35, the mean number of siblings.
4. Find the median: the first 4 students report 0 siblings and the next 8 report 1, covering positions 5 through 12, so the 10th and 11th values both equal 1, and the median is 1 sibling.

Reading a Histogram

A histogram groups values into equal-width intervals along the horizontal axis and shows the count in each interval as a bar height on the vertical axis. Read the bar height for each interval you need, then add or compare counts to answer the question.

Here is an original example: a histogram of race finish times groups runners into 10-minute intervals. The bar for 30 to 40 minutes has a height of 12 runners, and the bar for 40 to 50 minutes has a height of 18 runners. Find the number of runners who finish between 30 and 50 minutes.

1. Read the height of each bar in the range: 12 runners and 18 runners.
2. Add the two counts: 12 plus 18 equals 30.
3. State the answer: 30 runners finish between 30 and 50 minutes.

Reading a Dot Plot

A dot plot stacks one dot for every data point above its value on a number line. The tallest stack marks the mode, and counting every dot gives the sample size. Read the number line scale before you count, because a dot plot still prints values with no dots above them. Find the median by counting dots from the left until you reach the middle position.

Here is an original example: a dot plot shows the number of books read last month by each student in a class. Three dots sit above 1, five dots sit above 2, six dots sit above 3, and two dots sit above 4. Find the mode and the total number of students.

1. Find the tallest stack: six dots sit above 3, so 3 books is the mode.
2. Add every stack to find the sample size: 3 plus 5 plus 6 plus 2 equals 16.
3. State the answer: the mode is 3 books, and 16 students are represented in the plot.

Two-Variable Data: Scatterplots and Lines of Best Fit

Two-variable data questions plot two related quantities as points on a scatterplot, one quantity on each axis. The digital SAT gives you a built-in calculator you are allowed to toggle between scientific and graphing modes at any point in the Math section, including on scatterplot questions. The shape and direction of the point cloud describe the relationship between the two quantities.

Describing a Scatterplot Trend

A scatterplot trend has a direction and a form. Direction is positive when one quantity rises as the other rises, and negative when one quantity falls as the other rises. Points which follow no consistent pattern show no correlation. Form is linear when the points cluster around a straight line, and nonlinear, curved into a quadratic or exponential shape, for example, when they do not.

Here is an original example: a scatterplot plots hours of daily sunlight on the horizontal axis against plant height in centimeters on the vertical axis for 15 plants. The points rise from left to right and cluster tightly around a straight line. Describe the trend. The scatterplot shows a positive, linear trend, since plant height rises as sunlight hours rise, and the points sit close to a straight line.

Fitting a Line or Curve to the Data

A line of best fit is a straight line drawn through a scatterplot to model a linear trend, positioned so it passes as close as possible to every point. Some scatterplots curve instead of following a straight path, and a curve of best fit, often a quadratic or exponential model, fits the data more closely than a straight line in those cases.

Here is an original example: a scatterplot of a company's yearly revenue against years since founding shows points which bend upward more steeply each year. Find the type of model which fits the trend. The steepening upward bend fits an exponential curve of best fit better than a straight line, since a straight line underestimates the later, faster-growing years.

Interpreting Slope and Correlation in Context

Line of best fit questions test whether you connect the slope and intercept of an equation to the real quantities in the problem, not only the algebra. Wrong answers reuse the same two numbers with the units swapped or the direction reversed. Read both axis labels first, then say out loud what one unit on each axis stands for before you pick an interpretation.

Slope as a Rate of Change

The slope of a line of best fit gives the predicted change in the vertical-axis quantity for every one-unit increase in the horizontal-axis quantity. State the slope with both units from the problem to interpret it correctly.

Here is an original example: a line of best fit for hours studied versus test score is y equals 4x plus 62, where x is hours studied and y is predicted test score. Interpret the slope and the y-intercept.

1. Read the slope: 4.
2. Interpret the slope in context: predicted test score rises by 4 points for every additional hour studied.
3. Read the y-intercept: 62.
4. Interpret the y-intercept in context: a student who studies 0 hours is predicted to score 62.

Extending a line of best fit past the range of the collected data is called extrapolation, and predictions made this way carry less confidence than predictions made inside the data range. A model built from study times of 0 to 10 hours supports no reliable prediction for 40 hours of study. Check the smallest and largest horizontal-axis values in the display before you trust a predicted value.

Correlation Direction and Strength

Correlation describes how closely two quantities move together, in a direction, positive or negative, and a strength, from weak to strong. Points sitting close to the line of best fit show strong correlation, and points scattered widely around the line show weak correlation. Correlation describes a pattern between two quantities. It does not prove one quantity causes the other, a distinction covered further in our SAT Probability and Statistical Inference guide.

Here is an original example: a scatterplot of ice cream sales versus daily temperature shows points clustered tightly along an upward line. A second scatterplot of ice cream sales versus the number of pool visits on the same day shows a similarly tight upward line. Explain why the second scatterplot does not prove pool visits cause higher ice cream sales. Both temperature and pool visits rise together on hot days, so the correlation between ice cream sales and pool visits is explained by a third factor, temperature, rather than one causing the other directly.

Common One- and Two-Variable Data Traps

Test writers build wrong answers around specific misreadings of data displays, not random numbers. Naming the trap during practice prevents picking it under time pressure.

• Reading the mean off a data display when the question asks for the median, or the reverse, especially when a data set includes an outlier.
• Reporting the height of a single histogram bar as the total count for a question which spans multiple intervals.
• Counting dots on a dot plot as the mode instead of finding the value with the tallest stack.
• Fitting a straight line of best fit to a scatterplot which curves, producing bad predictions near the ends of the data range.
• Extrapolating a line of best fit far past the collected data range and treating the prediction as certain.
• Reading a strong correlation as proof of cause and effect between the two plotted quantities.

Practice One- and Two-Variable Data With Us

We built free drill sets for one-variable data and for two-variable data and scatterplots, at three difficulty levels. Every set gives instant feedback with a full explanation for each question. No signup is required.

• One-variable data: easy set, medium set, hard set.
• Two-variable data and scatterplots: easy set, medium set, hard set.

Start at the easy set for each topic, then move up once you answer every question without rechecking your steps. For full-length timed practice covering Problem-Solving and Data Analysis alongside the other three Math domains, visit our SAT practice page. Working the drills and a full test together turns center, spread and scatterplot reading into steps you run on sight.

Sources

• College Board: The Math Section Overview, SAT Suite
• College Board: Problem-Solving and Data Analysis, SAT Suite
• College Board: How the SAT Is Structured, SAT Suite
• College Board: Assessment Framework for the Digital SAT Suite
• College Board: SAT Calculator Policy