SAT Problem-Solving & Data Analysis

What You Need to Know

Problem-Solving & Data Analysis (PSDA) questions test whether you can turn real-world info (tables, graphs, surveys, experiments, units) into correct math, then interpret the result. It’s less about fancy algebra and more about setting up the right relationships (ratios, rates, percents, averages, probabilities) and reading data carefully.

What shows up most

• Ratios, rates, proportions, and unit conversions (including multi-step conversions)
• Percents (percent change, discounts, tax, percent error)
• Averages (mean/median, weighted mean, average speed)
• Data interpretation (tables, scatterplots, line graphs, histograms)
• Statistics concepts (sampling, bias, margin of error, center/spread, standard deviation conceptually)
• Two-way tables & probability (joint/marginal/conditional, independence)
• Linear models (line of best fit, slope meaning, residuals, outliers)

PSDA is where careful reading + units + “what is the question actually asking?” matters more than complicated computation.

Core idea (the “rule” behind the whole topic)

Almost every PSDA problem is:
1) define variables clearly,
2) write the relationship (often a ratio/proportion or probability),
3) compute,
4) interpret with correct units/context.

Step-by-Step Breakdown

A. Universal PSDA setup (works for most questions)

1. Underline what you’re solving for. Is it a rate, a percent, a probability, a predicted value, or a comparison?
2. List given quantities with units. Write units explicitly (minutes, dollars, miles, people).
3. Pick the relationship type:
   • “per” or “each” ⟶ unit rate
   • “out of” ⟶ probability / fraction
   • “is/of” or “percent” ⟶ percent equation
   • “compared to” ⟶ ratio
   • table with categories ⟶ two-way table probabilities
   • scatterplot/line of best fit ⟶ linear model
4. Set up the math cleanly (proportion, equation, or counting).
5. Solve + sanity check:
   • Does the answer have the right units?
   • Is the value reasonable (probabilities between $0$ and $1$, percents between $0\%$ and $100\%$ unless a change)?

B. Percent problems (fast method)

1. Identify Part, Whole, and Percent.
2. Use:
   $\text{part} = \text{percent} \times \text{whole}$
3. Convert percent to decimal: $35\% = 0.35$.

Mini-example: “$18$ is what percent of $60$?”

• $\text{percent} = \frac{18}{60} = 0.3 = 30\%$

C. Proportions & unit conversions

1. Write a ratio with matching units.
2. Multiply by conversion factors that equal $1$.

Mini-example: Convert $90\,\text{miles/hour}$ to $\text{miles/minute}$.

• Multiply by $\frac{1\,\text{hour}}{60\,\text{minutes}}$:
  $90 \times \frac{1}{60} = 1.5\,\text{miles/minute}$

D. Two-way tables (probability) procedure

1. Decide what events mean (example: $A$ = “female”, $B$ = “plays a sport”).
2. Find counts in the table.
3. Choose probability type:
   • Joint: $P(A \cap B)$ ⟶ cell ÷ total
   • Marginal: $P(A)$ ⟶ row/column total ÷ total
   • Conditional: $P(A \mid B)$ ⟶ cell ÷ (the “given” group total)
4. Reduce the fraction or convert to decimal if needed.

E. Scatterplots, lines, and residuals

1. If given a line, use $y = mx + b$ to predict.
2. Compute residual:
   $\text{residual} = \text{actual} - \text{predicted}$
3. Interpret:
   • residual $> 0$: model underpredicted
   • residual $< 0$: model overpredicted
4. Watch for outliers and nonlinear patterns.

Key Formulas, Rules & Facts

Ratios, rates, proportions, and averages

Percents

Probability & two-way tables

Statistics & study design (SAT-level essentials)

• Mean vs median: mean is sensitive to outliers; median is resistant.
• Effect of transformations (very common):
  • Add constant $c$ to every value ⟶ mean and median increase by $c$; range and standard deviation unchanged.
  • Multiply every value by $k$ ⟶ mean and median multiply by $k$; range and standard deviation multiply by $|k|$.
• Standard deviation (conceptual): bigger standard deviation means data are more spread out from the mean.
• Sampling:
  • Random sample supports generalizing to the population.
  • Random assignment (in experiments) supports cause-and-effect.
  • Bias happens with leading questions, undercoverage, nonresponse, convenience samples.
• Correlation is not causation: a relationship in observational data does not prove one variable causes the other.

Examples & Applications

Example 1: Percent change + interpretation

A jacket’s price increased from $80$ dollars to $92$ dollars. What is the percent increase?

Setup:
$\frac{92 - 80}{80} \times 100\%$
Compute:
$\frac{12}{80} = 0.15$ so percent increase is $15\%$.

Key insight: the denominator is the original value.

Example 2: Weighted mean (groups)

A class has $12$ freshmen with an average score of $78$ and $18$ sophomores with an average score of $86$. What is the overall average?

Setup weighted mean:
$\bar{x} = \frac{12 \times 78 + 18 \times 86}{12 + 18}$
Compute numerator:
$12 \times 78 = 936$
$18 \times 86 = 1548$
Total:
$\bar{x} = \frac{936 + 1548}{30} = \frac{2484}{30} = 82.8$

Key insight: you can’t average $78$ and $86$ directly because groups aren’t equal sizes.

Example 3: Two-way table conditional probability

A survey of $200$ students:

Find $P(\text{Junior} \mid \text{Plays a sport})$.

“Given plays a sport” ⟶ denominator is total who play a sport: $110$.

$P(\text{Junior} \mid \text{Sport}) = \frac{50}{110} = \frac{5}{11}$

Key insight: conditional probability denominator is the given group total, not the grand total.

Example 4: Line of best fit + residual

A line of best fit is $y = 2.5x + 40$, where $x$ is hours studied and $y$ is test score. A student studied $8$ hours and scored $55$.

Predicted:
$2.5(8) + 40 = 20 + 40 = 60$
Residual:
$55 - 60 = -5$

Interpretation: residual is negative, so the model overpredicted by $5$ points.

Common Mistakes & Traps

1. Using the wrong denominator in percent change

   • Wrong: divide by new value.
   • Right: percent change is always relative to old value: $\frac{\text{new} - \text{old}}{\text{old}}$.
   • Fix: circle “original” in the problem.

2. Averaging averages without weights

   • Wrong: $\frac{78 + 86}{2}$ when group sizes differ.
   • Right: use weighted mean $\frac{\sum (w_i x_i)}{\sum w_i}$.
   • Fix: if you see different group counts, think “weights.”

3. Mixing units (especially in rates)

   • Wrong: adding minutes to hours, or using $\text{miles/hour}$ with time in minutes.
   • Right: convert units first or keep units symbolic.
   • Fix: write units next to every number while solving.

4. Conditional probability with the wrong “given” group

   • Wrong: dividing by grand total when asked $P(A \mid B)$.
   • Right: denominator is total outcomes where $B$ occurred.
   • Fix: rewrite in words: “Out of all $B$, what fraction are $A$?”

5. Confusing marginal vs joint vs conditional

   • Joint: both happen (a single cell).
   • Marginal: totals along the edge.
   • Conditional: restricted to a subgroup.
   • Fix: point to what part of the table you’re using before computing.

6. Assuming correlation implies causation

   • Wrong: “More ice cream causes crime” from a correlation.
   • Right: a lurking variable (like temperature) can drive both.
   • Fix: ask: was there random assignment? If not, no causal claim.

7. Treating a line of best fit as exact

   • Wrong: expecting every point to lie on the line.
   • Right: it’s a model; use residuals and understand prediction is approximate.
   • Fix: if asked “best estimate,” use the line; if asked “exact from data,” use the data point.

8. Misreading graphs (especially axes and scales)

   • Wrong: ignoring that the $y$-axis starts at $50$, not $0$.
   • Right: read axis labels, tick marks, and units first.
   • Fix: spend $5$ seconds checking scale before doing any math.

Memory Aids & Quick Tricks

Quick Review Checklist

• You can set up and solve ratios/proportions with units aligned.
• You use $\text{part} = \text{percent} \times \text{whole}$ and keep “old on bottom” for percent change.
• You compute weighted means when group sizes differ.
• You handle unit conversions with conversion factors equal to $1$.
• You read tables/graphs by checking axes, scales, and labels first.
• You can compute joint, marginal, and conditional probabilities from a two-way table.
• You know independence means $P(A \mid B) = P(A)$.
• You interpret a line of best fit and compute residuals using $\text{actual} - \text{predicted}$.
• You can spot bias, and you know: random sample ⟶ generalize; random assignment ⟶ causation.

One last pass: keep your units visible and your denominator choices intentional—you’ve got this.