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SAT Problem-Solving and Data Analysis
Build your understanding of ratios, percentages, probability, and statistics with free SAT math lessons.
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Open interactive lessons →Ratios, rates, proportional relationships, and units
Ratios, rates, and proportions
Scale a relationship and distinguish part-to-part from part-to-whole.
Equivalent ratios
Multiply both quantities by the same factor to preserve a ratio. If red:blue is 2:3, the whole has 5 equal parts, so red is 2/5 of the total.
Unit rates
Divide to find the amount per one unit. A proportional relationship has the form y = kx and passes through the origin.
A ratio compares quantities; a rate includes units
The ratio 2:3 means one quantity has two equal-sized parts while the other has three. It does not tell their actual sizes until you know a scale. A rate such as 60 miles per hour compares different units. Dividing by the denominator quantity gives the amount per one denominator unit.
Proportional relationships preserve a multiplicative link
If y = kx, doubling x doubles y and y/x stays constant for nonzero x. Adding a fixed charge changes the relationship to y = kx + b, which is linear but not proportional when b ≠ 0. Decide whether the situation includes a fixed starting amount before using a proportion.
Worked example
A mix uses oats and nuts in a 3:2 ratio. How many cups of nuts are in 20 cups of mix?
- There are 5 ratio parts total.
- Each part is 20/5 = 4 cups.
- Nuts use 2 parts.
8 cups of nuts
Watch for this: Do not divide a part-to-part ratio by just one part when the question asks for a fraction of the whole.
Check your understanding
Five notebooks cost $15 at a constant unit price. What do eight cost?
- $18
- $24
- $30
- $40
Show answer and explanation
Answer: B. $15/5 = $3 per notebook; 8 × $3 = $24.
Percentages
Percent change and successive changes
Choose the correct base and convert a percent to a multiplier.
Percent of and percent change
p% of A is (p/100)A. Percent change is (new − old)/old × 100%; the original value is the denominator.
Successive changes
Apply multipliers one at a time. A 20% increase followed by a 20% decrease multiplies by 1.2 × 0.8 = 0.96, so it does not return to the original.
Percent literally means per hundred
A percentage is a fraction with denominator 100. To take 18% of an amount, multiply by 18/100 = 0.18. To find what percent one amount is of another, divide part by reference amount and multiply by 100. The reference amount is essential: the same difference can represent different percentages.
Change is measured against the original base
If a value rises from 50 to 60, the increase is 10/50 = 20%. Reversing the change gives a decrease of 10/60, about 16.7%. Successive changes use the updated base each time, so combine multiplicative factors rather than simply adding or canceling percentages.
Worked example
A $50 item is discounted 20%, then taxed 10%.
- Discount: 50 × 0.80 = 40.
- Tax applies to $40: 40 × 1.10 = 44.
Final price: $44
Watch for this: A rise from 40% to 50% is 10 percentage points, or a 25% relative increase.
Check your understanding
A value rises from 80 to 100. What is the percent increase?
- 20%
- 80%
- 25%
- 125%
Show answer and explanation
Answer: C. (100 − 80)/80 × 100% = 25%.
Ratios, rates, proportional relationships, and units
Convert units and compound rates
Use conversion factors so unwanted units cancel.
Cancel units
Multiply by a ratio equal to 1, such as 60 minutes/1 hour. Arrange it to cancel the unit you are replacing.
Squared and cubed units
Area conversions square the length factor; volume conversions cube it. If 1 m = 100 cm, then 1 m² = 10,000 cm².
A conversion factor is another way to write one
If 1 hour = 60 minutes, then 60 minutes/1 hour equals 1. Multiplying by it changes the units used to express a quantity without changing the quantity itself. Arrange the factor so the unwanted unit cancels, just as a common nonzero factor cancels in a fraction.
Compound dimensions require every relevant conversion
A speed has both distance and time units, so converting mph to feet per second changes numerator and denominator. An area is length multiplied by length, so a length conversion factor is squared. A volume uses three length factors. Write the unit powers to prevent a factor-of-100 or factor-of-1,000 error.
Worked example
Convert 72 kilometers per hour to meters per second.
- Multiply by 1,000 m/km and by 1 hour/3,600 seconds.
- Compute 72 × 1,000 / 3,600.
20 meters per second
Watch for this: Write the units during the calculation; a plausible number with the wrong unit is still wrong.
Check your understanding
How many square centimeters are in 3 square meters?
- 300
- 3,000
- 300,000
- 30,000
Show answer and explanation
Answer: D. 3 m² × (100 cm/m)² = 30,000 cm².
One-variable data: Distributions and measures of center and spread
Mean, median, and weighted averages
Find a center and understand how each measure responds to change.
Mean and median
Mean = sum/count. For a median, sort the values and take the middle value, or the mean of the two middle values. Outliers often affect the mean more than the median.
Weighted averages
To combine groups, add their total values and divide by total count. Do not simply average group means unless the groups have equal size.
Mean balances the total; median locates the middle
The mean is the value every observation would have if the total were evenly redistributed. The median is the midpoint of the ordered observations. An unusually large observation adds strongly to the total, moving the mean, while it may leave the middle position and median unchanged.
Frequencies and group sizes act as weights
A frequency of 4 means a value appears four times, so it contributes four copies to the total and four observations to the count. For combined groups, multiply each mean by its group size to recover each group’s total, then divide the combined total by the combined size.
Worked example
Group A has 4 scores averaging 70; group B has 6 averaging 80. Find the combined mean.
- The totals are 4 × 70 = 280 and 6 × 80 = 480.
- Divide the combined total 760 by the combined count 10.
Combined mean: 76
Watch for this: A frequency table requires counting repeated values, not averaging just the distinct labels.
Check your understanding
What is the median of 3, 8, 5, 12, 7?
- 7
- 5
- 8
- 35
Show answer and explanation
Answer: A. Sorted values are 3, 5, 7, 8, 12; the middle is 7.
One-variable data: Distributions and measures of center and spread
Spread, outliers, and distribution shape
Compare variability without confusing it with center.
Measures of spread
Range = maximum − minimum. Interquartile range is Q3 − Q1 and describes the middle half. Standard deviation measures typical distance from the mean.
Shifts and scaling
Adding the same constant to every value shifts the mean and median but leaves range and standard deviation unchanged. Multiplying all values by c multiplies standard deviation by |c|.
Center and variability answer different questions
A center describes a typical location. Spread describes how far values extend or fluctuate around that location. Standard deviation uses distances from the mean, so values concentrated near the mean produce a smaller standard deviation than values scattered far away, even when both sets have the same mean.
Shifts preserve distances; scaling changes distances
Adding 5 to every value also adds 5 to the mean, leaving every value’s difference from the mean unchanged. Multiplying by 3 triples those distances and the standard deviation. Range responds to extremes; interquartile range focuses on the middle half and is less sensitive to a single extreme observation.
Worked example
Every score in a data set is increased by 5. What changes?
- Every distance from the mean remains the same.
- The center moves up by 5, while the spread is preserved.
Mean increases by 5; standard deviation is unchanged.
Watch for this: Two distributions can have the same mean and very different spread.
Check your understanding
Data set A is 9, 10, 11. Data set B is 1, 10, 19. Which statement is true?
- B has a smaller mean.
- A has a larger range.
- Both have mean 10, but B has more spread.
- Both have the same standard deviation.
Show answer and explanation
Answer: C. Both average 10, but B’s values lie farther from that center.
Probability and conditional probability
Probability and frequency tables
Identify the sample space and count favorable outcomes.
Basic probability
For equally likely outcomes, probability = favorable outcomes / total outcomes. The complement rule is P(not A) = 1 − P(A).
Adding events
For mutually exclusive events, add their probabilities. If events overlap, subtract their intersection once: P(A or B) = P(A) + P(B) − P(A and B).
Probability compares favorable outcomes with the sample space
The sample space is the complete set of possible outcomes. When outcomes are equally likely, count the favorable ones and divide by the total. A probability must lie from 0 through 1. In a frequency table, the table’s counts determine the relative frequencies; the number of category labels is not the sample size.
Avoid double counting overlapping groups
“A or B” includes outcomes in either group, including outcomes in both. Adding the two group counts counts their intersection twice, so subtract it once. The complement contains all outcomes outside a chosen event, making 1 − P(event) an efficient way to compute “not.”
Worked example
A bag holds 3 red, 5 blue, and 2 green tokens. Find the probability of not selecting blue.
- There are 10 tokens total and 5 are not blue.
- Equivalently, subtract P(blue) = 5/10 from 1.
P(not blue) = 1/2
Watch for this: “Or” is inclusive unless the problem says otherwise; check for overlapping outcomes.
Check your understanding
A fair six-sided die is rolled. What is the probability of rolling greater than 4?
- 1/6
- 1/3
- 1/2
- 2/3
Show answer and explanation
Answer: B. Two of six equally likely outcomes qualify, giving 2/6 = 1/3.
Probability and conditional probability
Conditional probability and independence
Restrict the denominator to the group named by the condition.
Given means restrict
P(A given B) counts outcomes in both A and B divided by the number in B. In a two-way table, start with the specified row or column.
Independence
Independent events satisfy P(A given B) = P(A), and P(A and B) = P(A)P(B). Do not assume independence merely because two categories are different.
A condition creates a smaller sample space
“Given B” means that B is already known to have occurred. You therefore compare the favorable outcomes inside B with all outcomes inside B, not with the entire original group. In a two-way table, circle the relevant row or column before choosing the numerator and denominator.
Independence means the condition does not change the probability
If learning that B occurred leaves the chance of A unchanged, A and B are independent. Independence is a mathematical property, not a guess based on whether categories sound unrelated. Without replacement, selections often become dependent because the remaining contents change after each draw.
Worked example
Of 30 club members, 12 are seniors. Of those seniors, 9 volunteer. Given that a selected member is a senior, find the probability they volunteer.
- Restrict attention to the 12 seniors.
- Nine of these 12 volunteer.
9/12 = 3/4
Watch for this: P(A given B) usually differs from P(B given A). Read the condition carefully.
Check your understanding
Among 20 students who take art, 8 also take music. Given that a student takes art, what is the probability they also take music?
- 8/28
- 20/8
- 12/20
- 8/20
Show answer and explanation
Answer: D. Restrict to the 20 art students; 8 qualify, giving 8/20 = 0.4.
Inference from sample statistics and margin of error
Samples, estimates, and margin of error
Connect a representative sample to a population without overclaiming precision.
Population and sample
The population is the full group of interest. A random sample helps make an estimate representative of that population. A biased sampling method is not fixed by merely collecting more responses.
Margin of error
An estimate ± margin of error gives an interval of plausible population values at a stated confidence level. With comparable random sampling and confidence, larger samples generally reduce the margin of error.
A sample estimates a population characteristic
A population parameter describes the whole group, such as its true mean or proportion. A sample statistic estimates that parameter from a subset. Random sampling reduces systematic selection bias, but different random samples still produce different statistics; this sampling variability is why estimates have uncertainty.
Margin of error describes uncertainty, not individual variation
An estimate with margin of error gives a range of plausible values for the population parameter at the stated confidence level. It does not say every individual falls in that range. With comparable random sampling and confidence level, a larger sample generally narrows the margin, but it does not automatically repair a biased sampling method.
Worked example
A survey estimates support at 54% with a margin of error of 4 percentage points.
- Subtract and add 4 to the estimate.
- The interval includes 50%, so it does not establish that support is above half at that confidence level.
Interval: 50% to 58%.
Watch for this: A confidence interval is not a guarantee about every individual or every possible sample.
Check your understanding
An estimate is 62% with a margin of error of 3 percentage points. What interval does this give?
- 59%–65%
- 3%–62%
- 61%–63%
- 62%–65%
Show answer and explanation
Answer: A. Subtract and add 3: 62 − 3 = 59 and 62 + 3 = 65.
Evaluating statistical claims: Observational studies and experiments
Study design and statistical claims
Distinguish generalization from cause-and-effect evidence.
Random sampling
Random selection from a population supports generalizing to that population. Volunteers or one convenient class may not represent a larger group.
Random assignment
Randomly assigning treatments in a well-designed experiment helps isolate a causal effect. Observing an association alone does not establish causation; another variable may explain it.
Sampling controls who the study represents
Random sampling selects people or objects from a defined population, supporting generalization to that population. A random sample of one school can support statements about that school; it does not automatically represent every school in the country. Always name the group the sample was drawn from.
Assignment controls how treatments are compared
Random assignment distributes participants among treatment conditions to reduce systematic differences between groups. A well-designed randomized experiment can support a causal conclusion. In an observational study, people’s existing choices or characteristics can create confounding: a third factor may influence both the suspected cause and the outcome.
Worked example
Students who choose a music club have higher average grades. Can we conclude the club caused higher grades?
- Students selected the club themselves.
- Motivation or other differences could affect both club participation and grades.
The observational association alone does not establish causation.
Watch for this: Random sampling and random assignment answer different questions; one does not automatically imply the other.
Check your understanding
Which design best supports a claim that a study method causes an improvement in this study group?
- Compare volunteers who already use different methods.
- Randomly assign participants to the new method or a comparison method.
- Ask students which method they like.
- Survey only the highest scorers.
Show answer and explanation
Answer: B. Random assignment helps balance other factors so the treatment effect can be compared.
Two-variable data: Models and scatterplots
Scatterplots, models, and residuals
Use a model to predict and measure its error.
Read a trend
Describe direction, form, and strength. A line may fit an approximately linear relationship; curved patterns may need another model. Avoid extending a model far outside the observed input range.
Residual
Residual = observed output − predicted output. A positive residual means the actual point is above the model. A fitted association alone does not prove causation.
Each point pairs two measurements
A scatterplot places one observation at (input, output), allowing you to see direction, shape, strength, and unusual points. A fitted line summarizes an approximately linear pattern. Its prediction is the height of the line at a chosen x-value; it need not pass through each observed point.
Residuals distinguish an observation from a prediction
Residual = observed y − predicted y. A positive residual lies above the line; a negative residual lies below. Predicting within the observed input range is interpolation. Extending beyond it is extrapolation and may be unreliable because the pattern could change outside the data.
Worked example
A model predicts 24 sales when the observed count is 29. Find the residual.
- Use observed minus predicted.
- Compute 29 − 24.
Residual: +5 sales
Watch for this: Regression coefficients describe the fitted model; interpret their units and the data range.
Check your understanding
A model predicts a height of 150 cm; the observed height is 146 cm. What is the residual?
- 4 cm
- 296 cm
- −4 cm
- −296 cm
Show answer and explanation
Answer: C. Observed − predicted = 146 − 150 = −4 cm.
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