SAT Problem Solving and Data Analysis questions test how well students use numbers, units, and data in real situations. This cheat sheet gives quick reminders for ratios, rates, percents, statistics, probability, and interpreting graphs. Students need these tools to read word problems carefully, choose efficient calculations, and avoid common traps on the SAT.
It is designed as a fast reference for grades 11-12 SAT prep.
Key Facts
- A ratio compares quantities using division, so the ratio of to is or .
- A unit rate has a denominator of , such as .
- Percent means per hundred, so is equal to .
- Percent change is .
- The mean of values is .
- Probability is when outcomes are equally likely.
- For independent events, .
- A line of best fit can be written as , where is the predicted change in for each -unit increase in .
Vocabulary
- Ratio
- A comparison of two quantities by division, often written as or .
- Unit Rate
- A rate that compares a quantity to exactly unit of another quantity.
- Percent Change
- The relative change from an old value to a new value, calculated by .
- Mean
- The arithmetic average of a data set, found by dividing the sum of the values by the number of values.
- Median
- The middle value of an ordered data set, or the average of the two middle values when there is an even number of values.
- Correlation
- A measure of the direction and strength of the relationship between two quantitative variables.
Common Mistakes to Avoid
- Using the new value as the denominator in percent change is wrong because percent change is measured relative to the original value.
- Confusing percent with decimal form is wrong because equals , not .
- Adding ratios directly without matching units is wrong because each part of a ratio must refer to the same kind of quantity or a clearly stated comparison.
- Assuming correlation proves causation is wrong because two variables can move together without one directly causing the other.
- Using the mean when the median is better is wrong for skewed data because outliers can pull the mean far from a typical value.
Practice Questions
- 1 A car travels miles in hours. What is its unit rate in miles per hour?
- 2 A jacket originally costs and is discounted to . What is the percent decrease?
- 3 The data set is . Find the mean and median.
- 4 A scatterplot shows a strong positive association between hours studied and test score. Explain why this does not prove that studying alone caused every score increase.
Understanding SAT Math Problem Solving and Data Analysis Reference
Many data problems become easier when every number keeps its unit. A rate is not just a number. It describes one quantity for each unit of another quantity.
If a store sells fruit by weight, dollars per pound is useful. Pounds per dollar answers a different question. On the SAT, a table may contain totals, rates, and percentages together.
Write the units beside each value before calculating. This helps prevent multiplying quantities that should be divided or comparing values measured in different ways.
Scale matters too. A ratio can stay the same when both quantities are multiplied by the same factor, but a difference does not behave that way.
Percent problems often test the starting amount more than the arithmetic. An increase of twenty percent followed by a decrease of twenty percent does not return to the original value. The second change uses a new base.
For example, a price of one hundred dollars becomes one hundred twenty dollars, then falls by twenty four dollars. The final price is ninety six dollars. When a problem gives a percent of a group, first identify the whole group.
When it asks for a percent increase, compare the change with the original amount, not the new amount. These ideas appear in sale prices, tax, population reports, test score summaries, and survey results.
Statistics requires attention to the shape and source of data. The mean uses every value, so one unusually large or small value can pull it away from what is typical. The median is the middle value after the data are ordered, making it less sensitive to extreme values.
A data set can have the same mean as another set while having much more spread. Read graphs for clusters, gaps, peaks, and outliers instead of relying on one summary number. For survey questions, ask whether the sample represents the larger population.
A random sample can support a reasonable estimate. A sample chosen from one narrow group may not. An association found in survey data does not prove that one factor causes another.
Scatterplots show how two variables move together. A positive trend means larger values of one variable tend to pair with larger values of the other. A negative trend means one tends to decrease as the other increases.
The slope of a fitted line has meaning only with its units. If it is three dollars per hour, it predicts a three dollar change for each extra hour. The intercept may be meaningful, such as a starting fee, or it may describe an unrealistic situation outside the data.
Use a fitted line mainly for interpolation, which means predicting within the observed range. Predictions far beyond that range are less reliable. In probability, list the possible outcomes carefully and check whether events affect each other.
Replacing an item before a second draw creates a different situation from keeping it out. A clear sketch, table, or tree diagram can make the sample space visible and reduce errors.