Review and reference

Worksheets and ledgers are optional reference records. Write on paper or keep your own notes; these records do not save progress on this website.

On this page

The practice question numbers below are the stable Guide question numbers shown on each question, even when the bank shuffles their order.

Review Section 1

Check the reasoning, not only the result. Use the linked lesson or practice question for a targeted repair.

Open a question, attempt it, then use Check answer or Show model answer to review the explanation.

Ratios, rates, proportions, and units

Section 1 exit-check practice

Review Section 2

Check the reasoning, not only the result. Use the linked lesson or practice question for a targeted repair.

Open a question, attempt it, then use Check answer or Show model answer to review the explanation.

Percentages

Section 2 exit-check practice

Review Section 3

Check the reasoning, not only the result. Use the linked lesson or practice question for a targeted repair.

Open a question, attempt it, then use Check answer or Show model answer to review the explanation.

One-variable data: distributions, center, and spread

Section 3 exit-check practice

Review Section 4

Check the reasoning, not only the result. Use the linked lesson or practice question for a targeted repair.

Open a question, attempt it, then use Check answer or Show model answer to review the explanation.

Two-variable data: models and scatterplots

Section 4 exit-check practice

Review Section 5

Check the reasoning, not only the result. Use the linked lesson or practice question for a targeted repair.

Open a question, attempt it, then use Check answer or Show model answer to review the explanation.

Probability and conditional probability

Section 5 exit-check practice

Review Section 6

Check the reasoning, not only the result. Use the linked lesson or practice question for a targeted repair.

Open a question, attempt it, then use Check answer or Show model answer to review the explanation.

Sample-based inference and margin of error

Section 6 exit-check practice

Review Section 7

Check the reasoning, not only the result. Use the linked lesson or practice question for a targeted repair.

Open a question, attempt it, then use Check answer or Show model answer to review the explanation.

Evaluating statistical claims

Section 7 exit-check practice

Rates, percentages and summaries

Relationships to know or reconstruct

The test’s standard reference sheet focuses on geometry; it does not supply a general statistics or percentage formula list.^[3] Learn the meaning of the relationships below. Conversion factors and specialized models explicitly supplied in a question are part of that question’s information.

Task	Relationship	Condition or check
Ratio $A ∶ B = a ∶ b$	$A = ak,$ $B = bk;$ $share of combined whole = a/(a + b)$	A and B must be the only parts of that whole.
Direct proportion	$y = kx$	A constant ratio is not merely a constant slope.
Rate	$r = q/t;$ $q = rt$	Keep numerator and denominator units visible.
Average speed	$Total distance / total elapsed time$	Use total time, including stated stops; do not average speeds without the right weights.
Unit conversion	Multiply by equivalent-unit ratios	Cancel old units. Square or cube a length factor for area or volume.
Percent of	$part = (p/100)base$	Identify the amount representing 100%.
Percent change	$100(new - old)/old$	The original value is the base; for the usual formula it is nonzero.
Successive changes	$final = initial \prod(1 + r i)$	Signed rates $r i$ are decimals; factors apply to each changing base.
Reverse change	$initial = final/factor$	Undo multiplication with division, not an opposite percentage.
Weighted rate	$(n 1 p 1 + n 2 p 2)/(n 1 + n 2)$	Weights represent the appropriate group sizes.
Mean and total	$\bar{x}$ $\bar{x}$	Frequency tables require $value \times frequency.$
Combined mean	${\bar{x}}_{1}$	Do not average group means unless the group sizes match.

Supplied or supporting models, not a new memorization list

When the question supplies simple interest, $I = Prt,$ interpret the rate and time consistently. When it supplies $\sqrt{n}$ use its stated assumptions to compare sample sizes. The long standard-deviation formula is a way to justify comparisons, not the default calculation strategy. The official focus is interpreting spread and uncertainty.

Models, probability and inference

Task	Relationship	Condition or check
Median	Middle ordered value, or average of two middle values	Sort and use positions; frequency counts determine positions.
Range and IQR	$Range = max - min;$ $IQR = Q 3 - Q 1$	Check whisker conventions on a box plot.
Transform all data	For $y = ax + b:$ $\bar{y}$ $SD(y) = \|a\|SD(x)$	Translation changes center; scaling changes spread by an absolute factor.
Linear prediction	$\hat{y}$	The slope’s units are output per input.
Residual	$Observed - predicted:$ $\hat{y}$	A negative residual means the model overpredicts.
Exponential interval	$y = Ab x/h$	Factor b applies every h input units.
Equally likely outcomes	$P(A) = #A/#eligible$	Check that the selected elementary outcomes are equally likely.
Complement	$P(not A) = 1 - P(A)$	“At least one” is often easier as 1 minus “none.”
Inclusive union	$P(A or B) = P(A) + P(B) - P(A and B)$	Subtract the overlap once.
Conditional probability	$P(A ∣ B) = P(A and B)/P(B)$	Requires $P(B) > 0;$ restrict the denominator to B.
Two-stage events	$P(A then B) = P(A)P(B ∣ A)$	Update the second pool after a draw without replacement.
Independence	$P(A and B) = P(A)P(B)$	Do not assume it from separate event names.
Estimate a count or total	$\hat{p}$ or $\bar{x}$	A sample-based estimate is not an exact population fact.
Symmetric interval	Estimate $\pmE;$ width $2E$	Keep percentage points, percentages, and units distinct.

Choose the right relationship

When you see...	Ask...
“Of those” or “given”	Which subgroup is now the full eligible pool?
“Greater than” or “less than”	Which quantity is the comparison base? Does the question ask for the new value or the change?
A histogram	Are exact values shown, or only bins? Can I identify a range without an exact mean or median?
A changed data set	Did every value change, or only one? Did the count also change?
A line on a scatterplot	Am I using points on the model line or observed data points?
A fitted equation	What is the prediction, and is the requested input inside the observed range?
A confidence interval	Which population quantity does it estimate? Is the margin a half-width?
A larger sample	Was the sampling method improved, or only the number of responses?
A randomized study	Were people randomly sampled, randomly assigned to treatments, or both?
A difference in outcomes	Is the result convincing beyond chance, and does the design support a causal interpretation?

False shortcuts worth rejecting immediately

“Equal ranges imply equal standard deviations” ignores the distribution between the endpoints. “A random sample proves causation” confuses selection with assignment. “Overlapping intervals prove equality” overstates what separate uncertainty summaries establish. “More responses eliminate bias” confuses precision with design quality. “The average of the rates is the overall rate” ignores weights.

Four kinds of answer

An exact result follows from the given mathematical quantities. A prediction is what a model returns. An estimate uses sample information about an unknown population quantity. A causal conclusion requires an appropriate comparison design and adequate evidence. Correct calculations do not make these four categories interchangeable.

Calculator workflow

Use the tool for arithmetic, not for deciding the question

Calculators are permitted throughout SAT Math. Bluebook includes Desmos scientific and graphing options. Current College Board policy prohibits handheld calculators with CAS functionality; check the linked policy before your test because requirements can change.^[5] Practice with the test preview so the available interface is familiar.

Fast numerical checks

A single carefully parenthesized expression can preserve the model. To reverse a 30% discount, type $84/0.70,$ not $84+0.30*84.$ To combine group means, enter $(12*78+18*88)/(12+18).$ For average speed, $120/(60/30+60/60)$ preserves total distance divided by total time.

A decimal is not self-interpreting. A displayed result of 0.35 could be a probability, a rate, or a dollar amount. It becomes 35 only when the requested answer is a percentage.

Lists and summary statistics

Desmos accepts numerical data in lists and tables. Its descriptive functions include mean and median; standard deviation has separate sample and population versions.^[12] For the list [4, 6, 6, 8, 11], the mean is 7 and the median is 6. With a small list, finding the median by hand may be faster than typing.

For a frequency table, do not average the distinct labels without repeating or weighting them. Values 0, 1, 2, 3 occurring 2, 4, 3, 1 times have mean

$\frac{0(2) + 1(4) + 2(3) + 3(1)}{2 + 4 + 3 + 1}$

not the unweighted label mean 1.5. A correct calculator operation on the wrong list still gives the wrong statistic.

A reproducible regression workflow

Create a table and enter the paired values into columns $x 1$ and $y 1 .$ ^[6] For Example 4.09, the inputs are 0, 1, 2, 3, 4 and the outputs are 2, 5, 5, 8, 10. Enter a custom regression using a tilde rather than an equals sign:

$y 1 \sim mx 1 + b.$

Desmos fits the free parameters and stores their values for later expressions.^[7] Here the fitted line is $\hat{y}$ At $x = 3,$ it predicts 7.9, even though the observed response there is 8. The residual is $8 - 7.9 = 0.1.$

Models, precision and entry

Use $y 1 \sim ax 12 + bx 1 + c$ for a quadratic fit or $y 1 \sim ab x 1$ for an exponential fit when that family is justified. Do not fit a complicated curve simply because it passes through a small number of observations. Preserve column roles and input units; changing years to decades changes the interpretation of the exponent.

If the question supplies a model, use that model rather than recalculating a different regression from nearby plotted points. If it supplies a drawn line, read coordinates on the line. If it asks for a model from a table, regression may be useful. These are three different tasks.

Graph‐window and precision checks

An unseen intersection may lie outside the viewing window. A plotted line may look steep or flat because of axis scaling. A fitted parameter rounded too early can change a later prediction. Keep the full stored value when possible, then round the final requested quantity.

A calculator cannot infer an exact median from a histogram that only supplies bins. It cannot turn volunteers into a random sample. It cannot establish that a numerical difference was caused by a treatment merely by finding two means.

Student‐produced response format

The official directions allow up to five characters for a positive answer and six, including the minus sign, for a negative answer. Use an improper fraction or a decimal instead of a mixed number. Omit commas, units, currency signs, and percent signs. When an exact fraction fits, it avoids unnecessary rounding.^[3]

Mathematical answer	Suitable entry	Why
Probability $3/8$	$3/8$ or .375	Both are exact representations.
“What percent?” with result 35%	35	The question supplies the percent interpretation.
Count 1,600	1600	No grouping comma.
Value $\frac{1}{2}$	$7/2$ or 3.5	Not a mixed-number entry.
Probability $2/3$	$2/3$ or .6667	Use sufficient precision, not .67.

Follow the on-screen truncation or rounding directions when the decimal expansion does not fit. The decimal point and a fraction slash count toward the available space. If a question specifically requests rounding, obey that instruction before formatting the entry. The guide’s answer keys often include units to explain meaning; those units are not typed into a numeric-response field.

This website’s guide activities check exact numerical answers. Use an exact fraction when a decimal would require rounding; the official test’s entry directions are explained above.

The last five seconds

Re-read only the requested quantity. Check the base or condition, the units, and whether the answer should be a percent or a proportion. Then check the actual characters entered. This final check is often more valuable than repeating correct arithmetic.

Skills map: Sections 1–4

Quantities and data summaries

Use these links to isolate a weakness. A section is not mastered merely because one representation feels familiar: move between words, tables, graphs, and equations.

Section	Skill cluster	Worked examples
1	Parts versus wholes; proportional scaling	1.01–1.04
1	Unit rates and conversion chains	1.02, 1.05–1.11
1	Combined work and average speed	1.12, 1.13
1	Changing ratios and production factors	1.14, 1.15
2	Percent bases; relative change; percentage points	2.01–2.06
2	Successive and reverse comparisons	2.07–2.10
2	Nested groups and weighted rates	2.11, 2.12
2	Interest, concentration, and revenue models	2.13–2.15
3	Means, medians, and frequency tables	3.01–3.03
3	Dot plots, histograms, and box plots	3.04–3.06
3	Weighted means and unknown observations	3.07, 3.08, 3.14
3	Outliers and transformations	3.09–3.11
3	Standard-deviation comparisons and constraints	3.12, 3.13, 3.15
4	Read points and association; interpret parameters	4.01–4.06
4	Residuals and regression	4.07–4.09
4	Linear, exponential, and quadratic patterns	4.10–4.12
4	Extrapolation, axes, and competing models	4.13–4.15

Select the right kind of repair

A concept error needs a return to the lesson and a new explanation in your own words. A representation error needs a labeled diagram or table. An execution error needs a check on arithmetic, parentheses, or entered values. A scope error needs a statement of what the design actually supports. Do not assign the same generic “study more” remedy to all four.

Skills map: Sections 5–7 and independent practice

Section	Skill cluster	Worked examples
5	Single-category and complement probabilities	5.01, 5.02
5	Marginal, joint, and conditional denominators	5.03–5.06
5	Overlaps, unknown counts, independence	5.07–5.09
5	Conditional area and two-stage selection	5.10–5.13
5	Weighted and reverse conditional probabilities	5.14, 5.15
6	Statistics, parameters, and population estimates	6.01–6.03
6	Intervals, units, and compatible values	6.04–6.09
6	Sample size and precision	6.10–6.12
6	Bias, population weights, and interval comparison	6.13–6.15
7	Observation, assignment, and population scope	7.01–7.06
7	Confounding, nonrandomized experiments, controls	7.07–7.10
7	Undercoverage, nonresponse, question wording	7.11–7.13
7	Within-group randomization and evidence limits	7.14, 7.15

Independent‐practice map

Every section has five exit questions and six mixed questions. Mixed questions are deliberately unlabeled on their question pages; use this map after completing a set.

Section	Exit questions	Mixed questions
1	1A–1E	1, 8, 15, 22, 29, 36
2	2A–2E	2, 9, 16, 23, 30, 37
3	3A–3E	3, 10, 17, 24, 31, 38
4	4A–4E	4, 11, 18, 25, 32, 39
5	5A–5E	5, 12, 19, 26, 33, 40
6	6A–6E	6, 13, 20, 27, 34, 41
7	7A–7E	7, 14, 21, 28, 35, 42

Review and retrieval

Turn an error into an observable change

After each independent set, record every incorrect answer, correct guess, and solution that depended on a hint. Before reading the explanation, identify the earliest step where the reasoning went off track. The first wrong assumption is more useful than the last wrong arithmetic result.

Rewrite the question’s structure in one sentence: “I need the bus total because the selection is conditioned on bus riders,” or “I must divide by the retained price factor to recover the original.” Then solve again from a clean start. Do not simply copy the printed solution.

A practical review sequence

First pass: build the model. Work untimed until you can state why the setup is correct. Use the Foundation examples to repair prerequisites and the Standard examples to move among representations.

Second pass: retrieve the method. Revisit a section after a delay. Start with its exit questions, not its lesson. Read only the specific explanation needed for an identified gap.

Third pass: remove topic cues. Complete a mixed set without the skills map. Record elapsed time and confidence, but prioritize correct setup before speed. These 14-question domain sets are not official modules and have no score conversion.

Fourth pass: transfer the skill. Change one feature of a missed problem: reverse the conditional probability, use a new base, ask for a count instead of a percentage, or replace random assignment with random sampling. Explain how the solution or conclusion changes.

Track the kind of mistake

Code	What to repair
B: Base or denominator	Name the whole or conditioned pool before dividing.
U: Units	Write a cancellation chain and verify the final dimension.
M: Model	Choose the correct relationship before calculating.
D: Display	Re-read axes, frequencies, bins, or the model line.
S: Statistical scope	Separate sample facts, population estimates, association, and causation.
E: Execution or entry	Check arithmetic, rounding, calculator input, and response format.

A workable mastery rule

Treat a skill as secure after you solve two different questions without hints on separate occasions and can explain a plausible wrong answer. This is a study convention, not a validated score threshold. Reopen any skill that fails during mixed practice.

Reusable error log

Use one entry per important mistake. A specific repair is more valuable than a large number of copied solutions.

Question and first attempt

Question ID: __________

Date: __________

My answer: __________

Correct answer: __________

Confidence before checking: low / medium / high

Error code: B / U / M / D / S / E

What was the first incorrect decision? __________

Correct reasoning

The requested quantity or claim is: __________

The correct base, units, model, or study-design distinction is: __________

One sentence explaining why my original approach failed: __________

A quick check that would have caught the mistake: __________

Delayed retrieval

Related worked example: __________

First no-hint retry date and result: __________

Different transfer question and result: __________

The new habit I will use next time: __________

The goal is not to remember this answer. It is to recognize the next structure.

Sources and editorial notes

The linked sources below support content alignment, current test policies, calculator use, and selected methodological explanations. Sources were checked on September 10, 2026. The numerical examples and practice data are original rather than adapted official test questions.

Official test and calculator references

[1] College Board. Problem-Solving and Data Analysis.

The seven official skill areas used to organize this volume.

[2] College Board. Math Specifications.

Domain weighting and content descriptions, including linear, quadratic, and exponential models.

[3] College Board. Assessment Framework for the Digital SAT Suite, version 3.01 (August 2024).

Detailed domain skills, Math directions and the reference sheet.

[4] College Board. How the SAT Is Structured.

Question counts, standard module timing, and the adaptive structure.

[5] College Board. SAT Suite of Assessments Calculator Policy.

Current handheld-calculator restrictions and the graphing/scientific options embedded in Bluebook.

[6] Desmos. Tables.

Entering and organizing paired numerical data.

[7] Desmos. Regressions.

Custom regression with a tilde relation and reusing fitted parameter values.

Statistics and methodology

[8] NIST/SEMATECH. e-Handbook of Statistical Methods: Confidence Limits for the Mean.

Confidence-interval meaning, precision, and the role of sample size and variability.

[9] U.S. Census Bureau. Statistical Testing Tool: explanatory guidance.

Why comparing survey estimates requires uncertainty information, not only ranking point estimates.

[10] NIST/SEMATECH. e-Handbook of Statistical Methods: Completely Randomized Designs.

Random allocation of experimental units to treatment conditions.

[11] NIST/SEMATECH. e-Handbook of Statistical Methods: Randomized Block Designs.

Comparing treatments within groups defined by an important background factor.

[12] Desmos. Statistics.

Descriptive summaries, lists, and the distinction between sample and population standard deviation functions.

[13] Statistics Canada. Statistics Canada Quality Guidelines: Sample Design (archived edition).

Probability sampling, design quality, and why sample size often matters more for precision than the sampling fraction. Used for stable methodological principles, not current program rules.

Editorial boundaries

The seven-section structure follows College Board’s domain categories, while chapter names, teaching order, examples, and difficulty labels are this guide’s instructional design. A supporting method is not a claim that every named technique is a separately tested skill. All study claims in exercises are evaluated only from the stated assumptions; invented findings should not be interpreted as real research.

This guide is intended for study and review. Official policies and test directions control on test day. Revisit the linked calculator policy and practice in Bluebook rather than relying on an older printed rule.

End of guide

105 worked examples • 35 section-exit questions • 42 mixed questions