SAT Math — Problem-Solving & Data Analysis: Learn-It-From-Scratch Notes

Ratios, Rates, Proportional Relationships, and Units

A lot of “real-world math” on the SAT is about comparing quantities. The test often hides straightforward arithmetic inside contexts like recipes, maps, travel, pricing, or science measurements. The big idea is that when you compare quantities correctly—and keep units consistent—the problem usually becomes simple.

Ratios: comparing quantities (with or without units)

A ratio compares two quantities by division. If a class has 12 juniors and 18 seniors, the ratio of juniors to seniors is \frac{12}{18}, which simplifies to \frac{2}{3}. Ratios can be written in multiple forms:

• \frac{a}{b}
• a:b
• “a to b”

What matters is the order. “Juniors to seniors” is not the same as “seniors to juniors.” A common SAT trap is giving you one ratio and asking for the inverse.

Ratios sometimes compare parts to parts (juniors to seniors) and sometimes parts to whole (juniors to total students). If total students are 12+18=30, then juniors to total is \frac{12}{30}=\frac{2}{5}.

Rates: ratios with “per” and units

A rate is a ratio that includes units, often using “per.” Examples:

• Speed: miles per hour
• Price: dollars per pound
• Density: grams per cubic centimeter

A unit rate is a rate with 1 in the denominator (like \$2.50 per 1 pound). Unit rates let you compare options quickly.

Example (unit rate comparison):
A 12-ounce bottle costs \$1.80 and a 20-ounce bottle costs \$2.60. Which is cheaper per ounce?

Compute unit prices:

• 12 oz: \frac{1.80}{12}=0.15 dollars per ounce
• 20 oz: \frac{2.60}{20}=0.13 dollars per ounce

So the 20-ounce bottle is cheaper per ounce.

A common mistake here is dividing the wrong way (ounces per dollar instead of dollars per ounce). Either can work if you interpret it correctly, but the question usually signals which direction it wants.

Proportional relationships: constant multiplicative change

Two quantities are in a proportional relationship if one is always a constant multiple of the other. If y is proportional to x, then:

y = kx

where k is the constant of proportionality.

How you recognize proportionality:

• In a table, \frac{y}{x} is constant (for nonzero x).
• In a graph, it’s a straight line through the origin.
• In words, “at a constant rate,” “per,” or “proportional to.”

Example (find k):
A car travels 180 miles in 3 hours at a constant speed. Let y be miles and x be hours.

k = \frac{y}{x} = \frac{180}{3} = 60

So y = 60x, and the speed is 60 miles per hour.

A frequent SAT error is assuming any straight line is proportional. A line like y = 60x + 10 is linear but not proportional because it doesn’t go through the origin.

Unit conversion and dimensional analysis

PSDA problems often test whether you can convert units without getting lost. The safest method is to multiply by “1” in a clever form.

Key idea: a conversion factor like “12 inches = 1 foot” can be written as either:

\frac{12 \text{ inches}}{1 \text{ foot}}

or

\frac{1 \text{ foot}}{12 \text{ inches}}

You choose the version that cancels units.

Example (convert speed):
Convert 90 miles per hour to feet per second.

Use 1 \text{ mile} = 5280 \text{ feet} and 1 \text{ hour} = 3600 \text{ seconds}.

90 \frac{\text{miles}}{\text{hour}} \cdot \frac{5280 \text{ feet}}{1 \text{ mile}} \cdot \frac{1 \text{ hour}}{3600 \text{ seconds}}

Now cancel miles and hours:

= 90 \cdot \frac{5280}{3600} \frac{\text{feet}}{\text{second}}

= 90 \cdot \frac{22}{15} = 132 \frac{\text{feet}}{\text{second}}

Common mistake: converting only the numerator unit and forgetting to convert the denominator unit.

Exam Focus

• Typical question patterns:
  • Compare two options using unit rates (best buy, speed, density, productivity).
  • Identify whether a relationship is proportional from a table/graph/equation.
  • Multi-step unit conversions embedded in context.
• Common mistakes:
  • Swapping ratio order (part-to-part vs part-to-whole).
  • Treating any linear equation as proportional (forgetting “through the origin”).
  • Dropping or mismatching units during conversions.

Percentages

Percent problems are everywhere on the SAT because they measure whether you can interpret “out of 100” flexibly. The key is to translate percent language into multiplication and to keep track of what the “whole” is.

What a percent means

A percent is a fraction with denominator 100. So 35\% means:

\frac{35}{100} = 0.35

To find p\% of a quantity A, multiply:

\text{part} = \frac{p}{100} \cdot A

Percent increase/decrease (multipliers)

Percent change is easiest when you use multipliers.

• Increase by r\%: multiply by 1 + \frac{r}{100}
• Decrease by r\%: multiply by 1 - \frac{r}{100}

Example (decrease):
A jacket costs \$80 and is discounted by 25\%.

80 \cdot (1 - 0.25) = 80 \cdot 0.75 = 60

So the sale price is \$60.

Reverse percent problems

Sometimes you’re given the final amount after a percent change and asked for the original. The strategy is still multipliers—just divide instead of multiply.

Example (original price):
After a 20\% discount, a shirt costs \$48. What was the original price P?

A 20\% discount means final is 0.80P:

0.80P = 48

P = \frac{48}{0.80} = 60

So the original price was \$60.

A common mistake is subtracting 20 from 48 or dividing by 20—those confuse “percent” with “dollars.”

Percent of what? (choosing the correct base)

Many percent errors happen because the “whole” changes.

Example (percent change relative to the original):
A population increases from 50,000 to 60,000. What is the percent increase?

Change is 10,000. Base is original 50,000:

\frac{10000}{50000} = 0.20 = 20\%

If you mistakenly divide by 60,000, you’d be using the new value as the base.

Percent vs percentage points

SAT questions sometimes describe survey results changing from, say, 40\% to 55\%.

• Percentage point change: 55 - 40 = 15 percentage points.
• Percent increase relative to original: \frac{15}{40} = 0.375 = 37.5\%.

They are not the same, and the wording tells you which is wanted.

Exam Focus

• Typical question patterns:
  • Compute percent of a quantity, or find the whole given the part and percent.
  • Percent increase/decrease using a multiplier (discounts, tax, growth).
  • Interpret “percentage points” vs “percent increase.”
• Common mistakes:
  • Using the wrong base (dividing by the new value instead of the original).
  • Treating a percent like a number of units (mixing up dollars and percent).
  • Doing additive changes instead of multiplicative changes for repeated percent changes.

One-variable data: distributions and measures of center and spread

One-variable statistics is about describing a single list of numbers—test scores, incomes, heights, wait times. The SAT wants you to interpret what data “looks like” and to summarize it using center (typical value) and spread (variability).

Distributions: shape, clusters, gaps, and outliers

A distribution describes how often values occur. You might see it as:

• A dot plot
• A histogram
• A box plot
• A table of values

When you read a distribution, notice:

• Center: where the data tends to be
• Spread: how far values vary
• Shape: symmetric, skewed left, skewed right
• Outliers: unusually high or low values

Skew matters because it affects which measure of center is most “typical.” In a right-skewed distribution (long tail to the right), the mean is often pulled right by large values.

Measures of center: mean and median

The mean is the arithmetic average:

\text{mean} = \frac{\text{sum of data}}{\text{number of data values}}

If the data values are x1, x2, \dots, x_n, then:

\bar{x} = \frac{x1 + x2 + \cdots + x_n}{n}

The median is the middle value when data are ordered. If there are two middle values (even number of points), the median is their average.

Why two centers? Because:

• The mean uses every value, so it’s sensitive to outliers.
• The median is resistant to outliers, so it often represents “typical” better in skewed data.

Example (outlier effect):
Data: 2, 3, 3, 4, 20

Mean:

\bar{x} = \frac{2+3+3+4+20}{5} = \frac{32}{5} = 6.4

Median is 3.

The outlier 20 pulls the mean upward, but the median stays near the bulk of the data.

Measures of spread: range and interquartile range

The range is:

\text{range} = \text{max} - \text{min}

But range can be distorted by a single outlier. A more robust spread measure is the interquartile range (IQR).

To find IQR:

1. Order the data.
2. Find Q_1 (the median of the lower half).
3. Find Q_3 (the median of the upper half).
4. Compute:

\text{IQR} = Q3 - Q1

Example (IQR):
Data: 4, 6, 7, 9, 10, 12, 13, 20

Lower half: 4, 6, 7, 9 so Q_1 = \frac{6+7}{2} = 6.5

Upper half: 10, 12, 13, 20 so Q_3 = \frac{12+13}{2} = 12.5

\text{IQR} = 12.5 - 6.5 = 6

Standard deviation (often conceptual on the SAT)

Standard deviation measures typical distance from the mean. Larger standard deviation means data are more spread out.

You are often asked to compare standard deviations after a change:

• Adding the same constant to every data value shifts the mean but does not change spread.
• Multiplying every data value by a constant scales the spread.

If each value becomes x+c, standard deviation stays the same.

If each value becomes kx, standard deviation becomes |k| times as large.

The SAT may provide a formula when needed; conceptually, remember it measures spread around the mean.

Box plots: visual summaries

A box plot shows the five-number summary:

• Minimum
• Q_1
• Median
• Q_3
• Maximum

The “box” length represents the IQR. Box plots are great for comparing two groups quickly (which has greater median? greater variability?).

Common mistake: thinking the box plot shows how many data points are in each segment by length. It doesn’t; it shows quartiles (each contains 25% of the data), not equal distances.

Exam Focus

• Typical question patterns:
  • Compute or interpret mean, median, range, IQR from a list, histogram, or box plot.
  • Decide which measure of center/spread is more appropriate given skew or outliers.
  • Predict how mean/median/spread changes after adding or multiplying all values.
• Common mistakes:
  • Finding median without ordering data first.
  • Mixing up part-to-whole ideas in box plots (segment lengths vs 25% chunks).
  • Using range when IQR is the more appropriate “typical spread” under outliers.

Two-variable data: models and scatterplots

Two-variable statistics asks whether two quantities are related—like study hours and score, advertising and sales, temperature and electricity use. The SAT focuses on reading scatterplots, understanding trend lines, and interpreting model parameters.

Scatterplots and association

A scatterplot graphs paired data (x,y). You look for:

• Direction: positive association (upward trend) or negative association (downward trend)
• Form: linear vs curved
• Strength: how tightly points cluster around a trend
• Outliers: points far from the pattern

Be careful: association does not automatically mean causation (you return to this idea in study design).

Linear models and lines of best fit

A common model is linear:

y = mx + b

• m is the slope: change in y for a 1-unit increase in x
• b is the y-intercept: predicted y when x=0 (only meaningful if x=0 makes sense in context)

SAT questions often ask you to interpret slope in context. If m=2.5 where x is hours and y is dollars earned, then “each additional hour corresponds to about \$2.50 more earnings.”

Example (interpret slope):
A best-fit line for data relating number of books x to points earned y is:

y = 15x + 20

Slope 15 means each additional book is associated with about 15 more points.

Intercept 20 means predicted score is 20 when x=0 books (which might represent a starting bonus).

Predictions, interpolation, and extrapolation

Using a model to estimate y for an x value within the data’s range is interpolation and is usually safer. Predicting outside the range is extrapolation and is riskier because the relationship may change.

The SAT may ask which prediction is more reliable: typically the one that stays closer to observed x values.

Residuals: how far off the model is

A residual is:

\text{residual} = \text{actual } y - \text{predicted } y

• Positive residual: point is above the line.
• Negative residual: point is below the line.

Residuals help you judge fit. If residuals show a curved pattern, a linear model might not be appropriate.

Example (residual computation):
Line: y = 3x + 2. At x=4, predicted y=14. If actual y=12:

\text{residual} = 12 - 14 = -2

Nonlinear models (as patterns, not heavy algebra)

Sometimes the pattern is clearly not linear (for example, rapid growth). The SAT may use exponential language like “grows by a constant percent each year.” The key skill is recognizing that “constant percent change” suggests exponential behavior, while “constant amount change” suggests linear.

Exam Focus

• Typical question patterns:
  • Interpret slope and intercept of a fitted line in context.
  • Compute a predicted value and/or a residual.
  • Decide whether linear modeling is reasonable, and whether interpolation or extrapolation is appropriate.
• Common mistakes:
  • Interpreting b (the intercept) as meaningful when x=0 is outside the data’s context.
  • Mixing up residual sign (using predicted minus actual instead of actual minus predicted).
  • Assuming correlation implies causation.

Probability and conditional probability

Probability is about uncertainty: how likely an event is. The SAT emphasizes translating words into events, using tables, and understanding “given that” (conditional probability).

Basic probability and sample spaces

For an event A, probability is:

P(A) = \frac{\text{number of favorable outcomes}}{\text{number of total equally likely outcomes}}

Probabilities range from 0 to 1.

A big hidden phrase is “equally likely.” If outcomes are not equally likely, you can’t just count outcomes—you must use the given distribution.

Complement rule

Often it’s easier to compute the probability something does NOT happen.

If A^c is the complement of A, then:

P(A^c) = 1 - P(A)

Example (at least one):
Flip a fair coin twice. Probability of at least one head.

It’s easier to compute “no heads” (both tails):

P(\text{TT}) = \frac{1}{4}

So:

P(\text{at least one head}) = 1 - \frac{1}{4} = \frac{3}{4}

Conditional probability

Conditional probability is probability given that some condition is true.

P(A\mid B) = \frac{P(A \cap B)}{P(B)}

Interpretation: you restrict your “world” to outcomes where B happens, then ask how often A happens within that restricted set.

The SAT often gives a two-way table so you can compute conditionals by dividing within a row or column.

Example (two-way table):
A survey of 100 students:

Find P(\text{plays sport} \mid \text{has job}).

Given “has job,” focus on that row: 40 students. Of those, 18 play a sport.

P(\text{sport} \mid \text{job}) = \frac{18}{40} = 0.45

Common mistake: dividing by 100 (the overall total) instead of by the conditional total (40).

Independence

Events A and B are independent if learning that one happened doesn’t change the probability of the other:

P(A\mid B) = P(A)

Equivalently:

P(A \cap B) = P(A)P(B)

On SAT problems, independence is often described in words (like “randomly selected,” “with replacement,” or “unrelated traits”). Be cautious: “mutually exclusive” is different from independent.

• Mutually exclusive means they cannot happen together, so P(A \cap B)=0.
• Independent events can happen together; they just don’t affect each other’s probabilities.

Exam Focus

• Typical question patterns:
  • Compute probability from a sample space, table, or described process.
  • Conditional probability using two-way tables (“given that”).
  • Decide whether events are independent based on probabilities.
• Common mistakes:
  • Using the overall total instead of the conditional total for P(A\mid B).
  • Confusing independence with “mutually exclusive.”
  • Assuming outcomes are equally likely when they aren’t.

Inference from sample statistics and margin of error

Inference is about using a sample to learn about a population. The SAT doesn’t require advanced formulas here; it tests whether you understand the logic and can interpret statements about random sampling, sample statistics, and margin of error.

Population vs sample, parameter vs statistic

• A population is the full group you want to know about.
• A sample is the subset you actually collect data from.

A parameter is a true population value (usually unknown), like the true proportion of voters who support a policy.

A statistic is a value computed from the sample, like the sample proportion.

The goal is to use a statistic to estimate a parameter.

Why random sampling matters

A random sample gives each member of the population a chance to be chosen. Random sampling reduces bias and makes your sample more representative.

If the sample is not random (for example, only people from one neighborhood, or only volunteers), then your estimate may be systematically off.

Margin of error and confidence intervals (interpretation)

A common SAT output is something like:

“52\% of respondents support the proposal, with a margin of error of \pm 3\%.”

The margin of error tells you a range of plausible values for the population parameter (under the sampling method used). You interpret it as an interval:

52\% - 3\% = 49\%

52\% + 3\% = 55\%

So an interval of 49\% to 55\%.

Important: this does not guarantee the true value is in the interval. It’s about the reliability of the method over many samples (often phrased informally on the SAT).

How sample size affects margin of error (conceptually)

In general, larger samples tend to produce smaller margins of error because they reduce random sampling variability. The SAT may ask which of two studies is likely to have a smaller margin of error—usually the one with the larger random sample (assuming similar methods).

A common misconception is that a larger sample fixes bias. It doesn’t. A huge biased sample can still give a biased estimate.

Example (interpreting a claim with margin of error):
Study A: 60\% \pm 4\% support.

Is it reasonable to claim “a majority supports the policy”? The lowest plausible value is 56\%, still above 50\%, so “majority” is supported.

Is it reasonable to claim “about 60% support”? Yes, because 60% is the point estimate, but you should acknowledge uncertainty.

Exam Focus

• Typical question patterns:
  • Interpret a reported statistic with margin of error as an interval.
  • Decide whether two results are meaningfully different by checking interval overlap.
  • Compare studies based on sample size and sampling method.
• Common mistakes:
  • Treating margin of error as covering individual data values rather than the population parameter.
  • Thinking larger sample size removes sampling bias.
  • Ignoring the margin of error when judging whether a claim like “majority” is supported.

Evaluating statistical claims: observational studies and experiments

The SAT often tests whether you can critique a study and decide what conclusions are justified. This is less about computation and more about reasoning: what was measured, how data were collected, and whether causation can be claimed.

Observational studies: measuring without imposing treatment

An observational study records data as they naturally occur, without assigning treatments. Example: surveying people about how much they sleep and their GPA.

Observational studies can show association (a relationship), but they generally cannot prove causation, because other variables may be responsible.

A key term here is confounding variable: a factor related to both the explanatory variable and the response variable that can distort the apparent relationship.

Example: If coffee drinking is associated with higher stress, a confounder might be workload—people with heavier workloads might both drink more coffee and have more stress.

Experiments: testing cause with treatment assignment

An experiment imposes a treatment to see its effect. The gold standard for causal conclusions is a randomized controlled experiment:

• Participants are randomly assigned to treatment groups.
• There is often a control group for comparison.

Random assignment helps balance confounders between groups, making causation claims more credible.

Placebos and blinding (reducing bias)

• A placebo is an inactive treatment used so that expectations don’t drive results.
• Blinding means participants and/or researchers don’t know who received which treatment.

These design choices reduce bias in measuring outcomes.

What conclusions are justified?

A very common SAT question is essentially: “Can this study show that X causes Y?”

Use this rule of thumb:

• Random sampling helps you generalize to the population.
• Random assignment helps you claim causation.

You can have one without the other:

• A randomized experiment using volunteers may support causation for that volunteer group but may not generalize well.
• A random sample survey can generalize associations to the population but cannot establish causation.

Example (survey claim):
A school surveys 200 randomly selected students and finds that students who play sports report higher happiness.

• You can reasonably say: “In this school, there is an association between playing sports and reported happiness.”
• You cannot say: “Playing sports causes higher happiness,” because students weren’t randomly assigned to play sports.

Example (experiment claim):
Researchers randomly assign students to either a new study program or a standard one and compare test score gains.

• If the experiment is well-controlled, you can claim the program caused the difference in outcomes (for the participants).

Common mistake: mixing up random sampling and random assignment—they solve different problems.

Exam Focus

• Typical question patterns:
  • Decide whether a study is observational or experimental from a description.
  • Identify design flaws: biased sampling, lack of control group, confounding.
  • Judge whether a conclusion is about association, causation, or generalization.
• Common mistakes:
  • Claiming causation from an observational study.
  • Assuming random sampling and random assignment are interchangeable.
  • Ignoring plausible confounders when evaluating a claimed relationship.