SAT Math: Problem Solving & Data Analysis — Complete Study Guide

Introduction

The Problem Solving & Data Analysis domain measures how well you can read real-world situations, translate them into math, and make decisions from data. On the Digital SAT, this domain usually contributes about 5 to 7 questions out of 44 Math questions.

The most important structural fact: this domain appears only in Module 2 of the adaptive math section. That means these questions show up after Module 1 performance determines difficulty. Depending on your module path, some questions will be straightforward computations, while others involve layered interpretation, multi-step setup, and careful reading of tables or graphs.

Unlike many Algebra questions, nearly every PSDA question is a word problem with context: business pricing, health studies, survey results, test scores, scientific models, sports statistics, and everyday rates. You are rarely asked to do abstract manipulation without context clues.

This domain rewards four habits:

1. Identify what quantity is actually being asked.
2. Label units before calculating (miles per gallon, dollars per hour, percent of total, etc.).
3. Keep proportions and percent relationships explicit.
4. Interpret results in context, not just as raw numbers.

The built-in calculator is very useful here because arithmetic can be heavier than in pure Algebra questions. Still, setup matters more than button pressing.

Strategy tip: In PSDA, many wrong answers come from solving the wrong quantity correctly. Circle the target quantity in the prompt before doing any math.

1. Ratios, Rates, and Proportions

Ratios and rates are foundational in PSDA.

• A ratio compares two quantities (for example, 3 red marbles to 5 blue marbles).
• A rate is a ratio with different units (for example, 60 miles per hour).
• A unit rate has denominator 1 unit (for example, 30 miles per gallon).

Setting up proportions

A proportion states that two ratios are equal:

A common SAT move is cross-multiplication:

Use this only after confirming the ratios compare corresponding quantities.

Unit conversion and consistency

Always align units before forming a proportion. Examples:

• minutes to hours
• ounces to pounds
• centimeters to meters

If units are mixed, convert first; otherwise, proportion setup is wrong.

Direct and inverse proportion

• Direct proportion: as one quantity increases, the other increases at a constant ratio. $$y=kx$$
• Inverse proportion: as one quantity increases, the other decreases so product stays constant. $$y=\frac{k}{x}$$

Scaling problems

Scaling questions ask how a recipe, map, model, or design changes when dimensions are adjusted.

• Linear dimensions scale by factor $s$.
• Area scales by $s^2$.
• Volume scales by $s^3$.

SAT PSDA may hide this inside words like "enlarged," "reduced," "scaled model," or "same concentration."

Worked Example 1: Proportion with unit conversion

Worked Example 2: Rate problem

Strategy tip: In work problems, always use "job per unit time" rates first. Adding times directly is almost always wrong.

2. Percentages

Percentage questions appear often in PSDA and can be disguised in many forms.

Core formulas

Percent of quantity:

Percent increase/decrease:

Original from final after percent change:

• After increase by $r$: $\text{Final}=\text{Original}(1+r)$
• After decrease by $r$: $\text{Final}=\text{Original}(1-r)$

Successive percent changes

Percent changes in sequence are multiplicative, not additive.

If a price goes up 20% then down 20%:

So final is 96% of original, not 100%.

Tax, tip, and discount structure

• Add tax/tip by multiplying by $(1+r)$.
• Apply discount by multiplying by $(1-r)$.
• If multiple changes happen, multiply all factors in order.

Worked Example 3: Successive percentage changes

Worked Example 4: Percent change

Strategy tip: In percent change questions, denominator is always the original amount unless the problem says otherwise.

3. Probability

Probability in PSDA blends arithmetic and interpretation.

Basic probability

• Probability is between 0 and 1.
• Convert to percent by multiplying by 100.

Two-way tables

Two-way tables organize counts by two categories (for example, grade level and breakfast habit). Use row totals, column totals, and grand total carefully.

Conditional probability

Probability of event $A$ given $B$:

In table terms, denominator is the "given" group.

At least one

Use complement:

This is often easier than summing multiple outcomes directly.

Independent vs dependent events

• Independent: one event does not affect the other. $P(A\cap B)=P(A)P(B)$
• Dependent: probabilities change after first event.

SAT wording clues:

• "with replacement" suggests independence.
• "without replacement" suggests dependence.

Worked Example 5: Two-way table probability

Strategy tip: For conditional probability, draw a box around the "given" category and treat that box total as the new denominator.

4. Statistics: Center and Spread

SAT statistics questions are mostly conceptual with moderate computation.

Measures of center

• Mean: arithmetic average.
• Median: middle value when data are ordered.
• Mode: most frequent value.

Measures of spread

• Range: max minus min.
• Standard deviation (concept only on SAT): typical distance from mean.
  • Smaller SD: values clustered near mean.
  • Larger SD: values more spread out.

Outliers and center choice

Outliers affect mean more than median. If distribution has extreme values, median is often better representative center.

Comparing distributions

Use three ideas:

1. Shape (symmetric/skewed)
2. Center (typical value)
3. Spread (variability)

Box plots and dot plots

• Box plot highlights median, quartiles, spread, and possible outliers.
• Dot plot shows individual values and clusters directly.

Margin of error and confidence intervals (conceptual)

A poll estimate like 52% with margin of error ±3% means plausible true value is roughly 49% to 55%. Smaller margin of error generally means more precision, often from larger random samples.

Worked Example 6: Mean vs. median

Worked Example 7: Interpreting standard deviation

Strategy tip: If two groups have equal means, the group with smaller standard deviation is the more consistent one.

5. Data Interpretation: Scatterplots & Line of Best Fit

Scatterplots appear frequently in PSDA Module 2.

Reading scatterplots

Each point represents an ordered pair $(x,y)$ from context.

Look for:

• Direction (positive, negative, none)
• Strength (tight cluster vs wide cloud)
• Outliers
• Whether a linear model is reasonable

Correlation types

• Positive correlation: as $x$ increases, $y$ tends to increase.
• Negative correlation: as $x$ increases, $y$ tends to decrease.
• No correlation: no clear trend.

Line of best fit

A linear model approximates trend:

• Slope $m$: predicted change in $y$ for 1-unit increase in $x$.
• Intercept $b$: predicted $y$ when $x=0$ (interpret only if context makes sense).

Predictions

Substitute an $x$-value into model to predict $y$. Be cautious with extrapolation far outside data range.

Residuals

• Positive residual: actual above model.
• Negative residual: actual below model.
• Large residual magnitude may indicate outlier or weak fit.

Worked Example 8: Scatterplot interpretation

Strategy tip: In context interpretation, include units in your sentence (points per hour, dollars per month, etc.).

6. Evaluating Statistical Claims

PSDA includes reasoning about study design and claim quality, not just calculations.

Random sampling and bias

A sample should represent the target population. Bias occurs when selection method over/underrepresents groups.

Examples of bias:

• Online poll from one fan forum to infer national opinion.
• Surveying only morning students to estimate all-student sleep patterns.

Generalizability

Can results from sample apply to larger population?

• Large, random, representative samples support generalization.
• Narrow or self-selected samples weaken generalization.

Observational vs experimental studies

• Observational: researcher observes without assigning treatments.
• Experiment: researcher assigns treatments/interventions, ideally with random assignment.

Experiments can support stronger causal claims than observational studies.

Correlation vs causation

Correlation means two variables move together; it does not prove one causes the other. Confounding variables may drive both.

Margin of error interpretation

A poll result with margin of error ±$m$ defines a plausible interval around estimate. If intervals overlap strongly, differences may not be statistically meaningful.

Worked Example 9: Study design

Strategy tip: If a question asks "can conclude cause?" and evidence is just trend or association, the safe answer is usually no.

Quick Reference, Digital SAT Tips, and 5 Practice Problems

Quick Reference

Digital SAT Tips for PSDA

1. Use the calculator for arithmetic, but set up equations yourself.
2. Build mini-tables in scratch work for percent steps and multi-stage rates.
3. In long prompts, mark three things: given numbers, units, target quantity.
4. For table probability questions, physically highlight numerator and denominator groups.
5. On scatterplot questions, interpret slope in words with units before selecting answer.
6. Be cautious with causation claims; "correlated" rarely means "caused by."
7. If answer choices are close, estimate first to eliminate unreasonable options quickly.

Practice Problems

Problem 1 (ratio/proportion)

A recipe uses 5 cups of flour for 8 batches of muffins. At the same rate, how many cups of flour are needed for 20 batches?

A) 10.5
B) 12.0
C) 12.5
D) 13.0

Problem 2 (successive percentages)

A laptop price is increased by 15% and then discounted by 20%. What is the overall percent change from original price?

A) 5% increase
B) 5% decrease
C) 8% decrease
D) 12% decrease

Problem 3 (probability, conditional)

In a club, 60 students play chess, 40 play debate, and 25 play both. If a student is selected from the 60 chess players, what is the probability that the student also plays debate?

A) $\frac{25}{100}$
B) $\frac{25}{60}$
C) $\frac{40}{60}$
D) $\frac{25}{40}$

Problem 4 (statistics interpretation)

Two stores have the same mean daily sales. Store X has smaller standard deviation than Store Y. Which statement is true?

A) Store X always has higher sales than Store Y
B) Store X sales are more consistent day to day
C) Store Y has a higher mean than Store X
D) Standard deviation gives no useful information

Problem 5 (line of best fit)

A line of best fit for monthly advertising spend $x$ (in hundreds of dollars) and monthly sales $y$ (in thousands of dollars) is:

What is the predicted sales value when $x=10$?

A) 12.8 thousand dollars
B) 20 thousand dollars
C) 8 thousand dollars
D) 22 thousand dollars

Problem Solving & Data Analysis is highly trainable when you focus on translation and interpretation. The arithmetic is manageable once setup is clean. If you can consistently define quantities, align units, and interpret results in context, this Module 2 domain becomes a strong source of points.