Digital SAT Problem-Solving and Data Analysis: a complete guide to ratios, percentages, statistics and probability

What Problem-Solving and Data Analysis demands

Problem-Solving and Data Analysis (PSDA) is about 15% of the Digital SAT Math section, and it is the most context-heavy domain: ratios, percentages, statistics and probability wrapped in real-world word problems. The math is not hard, but the reading is, so careful setup is everything. This guide ties together the matching dot-point pages, each with its own practice: ratios, rates and proportional relationships, percentages, one-variable data and statistics, two-variable data and scatterplots, and probability and conditional probability.

Ratios, rates and proportions

A proportion sets two ratios equal, $\frac{a}{b} = \frac{c}{d}$, and is solved by cross-multiplying. A unit rate is the amount per one unit (divide to find it), and in a proportional relationship $y = kx$ that unit rate is the constant of proportionality and the slope through the origin. Convert units by multiplying by factors arranged so the unwanted units cancel.

Percentages

To find $p\%$ of a number, multiply by $\frac{p}{100}$. An increase of $p\%$ multiplies by $1 + \frac{p}{100}$; a decrease by $1 - \frac{p}{100}$. Percent change is $\frac{\text{new} - \text{old}}{\text{old}} \times 100\%$. Successive percentages multiply their factors (a $10\%$ rise then a $10\%$ fall is $0.99$, a net loss). For a reverse percent, divide the result by the multiplier.

One-variable statistics

The mean is the average, the median the middle value, the mode the most frequent, and the range max minus min. Standard deviation measures spread, which you compare rather than compute on the SAT. The median resists outliers while the mean is pulled toward them, so right-skewed data has mean $>$ median.

Inference extends this: a sample estimates a population, a larger random sample shrinks the margin of error, and a statistical claim is only valid if the sample is random and representative (and only a randomised experiment supports causation).

Two-variable data

A scatterplot shows paired data; its shape may be linear, quadratic, or exponential. A line of best fit $y = mx + b$ summarises a linear trend: the slope is the average rate of change (with units and direction), and the intercept is the predicted value at $x = 0$. Predict by substituting, preferring interpolation (within the data) over extrapolation (beyond it).

Probability

Simple probability is $\frac{\text{favorable}}{\text{total}}$. From a two-way frequency table, conditional probability ("given that...") restricts the group, so the subgroup total becomes the denominator. The complement rule $P(\text{not } A) = 1 - P(A)$ helps with "at least one" questions.

How PSDA is examined

• Ratios and rates. Solve proportions, find unit rates, convert units.
• Percentages. Percent of, percent change, successive and reverse percentages.
• One-variable data. Center and spread, outliers and skew, sampling and margin of error, statistical claims.
• Two-variable data. Read scatterplots, interpret a line of best fit, predict.
• Probability. Simple and conditional probability from tables.

Check your knowledge

Work these under timed conditions, then read the solutions.

1. A map scale is 2 cm to 5 km. How many km does 7 cm represent? (2 marks)
2. A $120 item is marked up 15%. What is the new price? (2 marks)
3. For the data 6, 8, 8, 10, 28, is the mean greater or less than the median? (2 marks)
4. A line of best fit is $y = -3x + 50$. What does the slope mean? (2 marks)
5. Of 80 surveyed adults, 50 own a car; of those 50, 20 also own a bike. What is the probability a car owner also owns a bike? (2 marks)