In SAT data-analysis questions, the most important step is often identifying the denominator. A percentage of which group? A rate per which unit? A probability under which condition? Write that relationship before calculating, and many seemingly complicated problems become ordinary arithmetic.
College Board includes Problem-Solving and Data Analysis among the Math domains. The following examples are original and focus on reasoning about quantities rather than memorizing one equation for every story. See the official Math overview.
Ratios compare parts; proportions connect equivalent ratios
A paint mixture uses blue and white paint in a 2:5 ratio. That means two parts blue for every five parts white. The total mixture contains seven parts.
If the mixture totals 28 liters, each part is four liters. Blue paint contributes eight liters and white paint contributes twenty liters.
The blue fraction of the whole is 2/7, not 2/5. The ratio 2/5 compares blue with white; the fraction 2/7 compares blue with the total. Decide which relationship the question requests.
For a scale drawing, suppose three centimeters represent twelve meters. A length of eight centimeters represents 8 × 4 = 32 meters. The factor is four meters per centimeter. Labeling the units makes it harder to accidentally reverse the conversion.
Distinguish a percent from a percentage-point change
Suppose participation rises from 40% to 50%. That is an increase of ten percentage points. Relative to the original 40%, it is a 25% increase because (50 − 40)/40 = 0.25.
These descriptions answer different questions. A change in percentages does not automatically equal the percent change.
For a price rising from $80 to $92, the change is $12. Divide by the original $80 to obtain 15%. Dividing by the new price answers a different comparison.
Use the structure:
Percent change = (new − original) / original × 100%
Write the original amount explicitly. Words such as “increased to” and “increased by” are not interchangeable.
Reverse a percentage change with division
A jacket costs $72 after a twenty-percent discount. The sale price is 80% of the original price, so:
0.80P = 72, giving P = 90.
Adding twenty percent of $72 does not reverse the discount. That would add a percentage of the smaller amount, producing $86.40 instead of $90.
Successive changes also use successive bases. A $100 value that increases twenty percent becomes $120. A later twenty-percent decrease produces $96. The changes do not cancel because the second percentage applies to $120.
Before doing arithmetic, write the multiplier: 1.20 for a twenty-percent increase, 0.80 for a twenty-percent decrease.
Use units as a calculation guide
A runner covers 7.5 kilometers in thirty minutes. Thirty minutes is half an hour, so the average speed is 15 kilometers per hour.
You could first calculate 0.25 kilometers per minute and multiply by sixty. Both methods preserve units.
Area and volume conversions require special care. A square that measures two meters per side has area four square meters. Since one meter is one hundred centimeters, each side measures two hundred centimeters and the area is forty thousand square centimeters.
The conversion from square meters to square centimeters is 100², not 100. Similarly, a cubic conversion uses the cube of the linear factor.
When you are unsure, convert the dimensions of a simple example rather than applying a remembered multiplier blindly.
Read a two-way table by choosing the correct group
Here is a fictional survey:
| Transportation | Grade 10 | Grade 11 | Total |
|---|---|---|---|
| Bus | 18 | 12 | 30 |
| Walk | 12 | 18 | 30 |
| Total | 30 | 30 | 60 |
The probability that a randomly selected student walks is 30/60 = 1/2.
The probability that a randomly selected Grade 10 student walks is 12/30 = 2/5. The condition restricts the sample to Grade 10, so the denominator is thirty.
The probability that a randomly selected student both walks and is in Grade 10 is 12/60 = 1/5. “Both” asks for an intersection within the full group; “given Grade 10” asks for a fraction within a restricted group.
Mark the relevant row or column before dividing. Many errors in conditional probability are denominator errors rather than failures of arithmetic.
Do not assume events are independent
If a bag contains three red and two blue tokens, the chance of drawing a red token first is 3/5. Without replacement, a red first draw leaves two red among four tokens. The probability of two red draws is therefore (3/5)(2/4) = 3/10.
With replacement, the bag returns to its original contents. The probability becomes (3/5)(3/5) = 9/25.
The words “with replacement” change the model. A similar issue arises when choices affect what remains available, even when the context does not involve a bag.
For “at least one,” a complement can be simpler. If the probability of no success is known, subtract it from one. Make sure the complement describes every way the requested event could fail.
Compare mean and median deliberately
For the data set 4, 5, 6, 7, 8, both the mean and median are six. Replace eight with twenty-eight. The mean becomes ten, while the median stays six.
The mean uses every value’s magnitude. The median depends on the ordered middle. A large extreme value can therefore change the mean substantially without changing the median.
When two groups are combined, do not average their means unless their sizes are equal. A group of ten students with a mean of eight and a group of thirty students with a mean of twelve have a combined mean of:
(10 × 8 + 30 × 12) / 40 = 11
The group sizes provide the weights.
For an unknown-value problem, convert an average into a total. Five numbers with a mean of fourteen have a sum of seventy. If four sum to fifty-three, the missing number is seventeen.
Interpret spread without confusing it with center
Two data sets can have the same mean and different variability. The sets 9, 10, 11 and 2, 10, 18 both average ten, but the second is more spread out.
Standard deviation measures spread around the mean. You may be able to compare it from the distribution without calculating a formula.
Adding the same constant to every value shifts the center but does not change the distances among the values. Multiplying every value by a positive factor multiplies those distances by that factor.
For example, converting measurements from meters to centimeters multiplies both the mean and standard deviation by one hundred. It does not change which observations are relatively close together.
Separate association from cause
If students who attend a voluntary workshop have higher scores, the observation alone does not establish that the workshop caused the difference. The students may differ in prior preparation, available time, or motivation.
Random sampling concerns whether a sample can represent a population. Random assignment concerns how participants are allocated to conditions in an experiment. They solve different problems.
A larger sample does not automatically remove selection bias. Surveying more volunteers may still leave out people who would never volunteer.
Read the conclusion as carefully as the numbers. An answer that extends results to a different population or turns an association into a causal claim can fail even if every percentage is correct.
Review the reasoning, not only the arithmetic
After a missed question, identify the chosen denominator, unit, population, or statistical measure. Ask which word in the prompt should have changed that choice.
Build a short mixed set with a reverse percentage, a conditional-probability table, a weighted mean, and a study-design conclusion. Explain the model in a sentence before calculating.
The SAT score calculator explains score formats after practice; these topic results are most useful as evidence about specific reasoning habits. Explore SHSPrep’s SAT program for lessons, targeted practice, and two original full-length mock tests with raw performance results.