SAT Data Analysis: Every Question Type and Method

The digital SAT's Math section has four content domains, and Problem-Solving and Data Analysis is about 15% of it: 5 to 7 of its 40 scored questions. These are the questions with stories, tables and charts, and the College Board splits them into seven skills:

• Ratios, rates, proportional relationships and units: recipes, speeds and conversions.
• Percentages: a part of a whole, a percent change, and the price before a discount.
• One-variable data: means, medians, spread and the displays that show them.
• Two-variable data: scatterplots and lines of best fit.
• Probability and conditional probability: usually from a two-way table.
• Inference from sample statistics: what a sample and its margin of error say about everyone.
• Evaluating statistical claims: which conclusions a survey or an experiment supports.

Aniko's free SAT data analysis guide covers every question type in seven groups: one-variable data splits into center and spread, and the last two skills share a group, since both ask what a study can conclude. The table below gives each one's method, and each section after it teaches the method, says when Desmos is faster, and gives you an SAT-style question to try. On most of these questions, the arithmetic is the easy part: you earn the points by choosing the right whole, the total a percent is of, the group a probability picks from, the population a sample speaks for.

How to answer an SAT data analysis question

The same four moves work on every question in this guide:

1. Read the last line first. Is it asking for a count, a percent, a probability, a mean, or which conclusion holds?
2. Name the whole: the original before a change, the group a pick comes from, the population a sample was drawn from.
3. Pick the faster route. Reasoning and short arithmetic go by hand. Long lists, a line of best fit for a table, awkward percents and long unit chains go faster in Desmos, built into the digital SAT for the whole Math section.
4. Answer the question, then check it against the table or the story.

Each section below turns these moves into the steps for its question type. The Desmos course teaches the calculator itself.

If data analysis is new to you, start with lesson 1.

Ratio, rate and unit questions give amounts in one form and ask for another: a ratio scaled to a total, an amount at a steady rate, a speed in new units. The trap is rarely the arithmetic. It’s which number belongs to which group, and which way a fraction goes. Put a unit on every number, and the units tell you whether to multiply or divide.

How to answer ratio, rate and unit questions

1. Name the target: the group, the unit or the total the question asks for. Read each ratio in its own order: in $A:B=3:5$, the $3$ goes with $A$.
2. Build the chain: start from the amount you’re given, with its unit, and multiply by rates and conversion factors, each turned so the unit you don’t want cancels, until only the target’s unit is left.
3. Pick the route. When the factor is easy to see, scale by hand. When the chain has several factors or awkward decimals, type it into Desmos as one expression, so nothing gets rounded early: see ratio, rate and unit chains in Desmos.

Ratios and proportions

Give every part of a ratio the same scale factor: for $A:B=a:b$, the amounts are $ak$ and $bk$, and the whole is $(a+b)k$. With students to educators at $3:8$ and $154$ in all, $11$ parts make $154$, so one part is $14$ and there are $8(14)=112$ educators. A part-to-part ratio isn’t a fraction of the whole: seniors to juniors at $3:5$ makes seniors $\frac38$ of the club. See SAT ratios and proportions.

Rates and unit rates

A rate compares two different units, and a unit rate is the amount for one. Multiply a rate by time to get an amount, and divide an amount by a rate to get a time: $165$ km at $55$ km/h takes $3$ hours. An average rate is the total over the total: $60$ miles at $40$ mph, a $30$-minute rest, then $60$ miles back at $60$ mph average $\frac{120}{1.5+0.5+1}=40$ mph. See SAT rates and unit rates.

Unit conversions

A conversion factor equals $1$, so turn it whichever way cancels the unit you don’t want. Convert both parts of a rate: $18$ m/s is $18\cdot\frac{1\text{ km}}{1000\text{ m}}\cdot\frac{3600\text{ s}}{1\text{ h}}=64.8$ km/h. For area and volume, square or cube the whole factor: a square yard is $9$ square feet. See SAT unit conversions.

Mistakes to avoid with ratios, rates and units

The second seed is $5$ of $7+5=12$ parts, so $\frac{5}{12}(288)=120$ g. An average speed is total distance over total time: $60$ miles each way take $1.5+1$ hours, so $\frac{120}{2.5}=48$ mph, and a rest adds to the time. And a square meter is $3.28^2\approx10.8$ square feet, so square the factor.

Percent questions ask for a part of a whole, how much something changed, or what it was before a discount or a chain of changes. Every percent is a percent of something, and the question turns on which amount is the $100\%$.

How to answer percent questions

1. Name the whole: the amount that counts as $100\%$. “Of” usually points to it, and in a change it’s the original.
2. Turn each change into a multiplier: up $18\%$ is $\times1.18$, down $18\%$ is $\times0.82$, and $20\%$ off leaves $80\%$.
3. Multiply the factors for changes in a row, never add the percents. To go back to an original, make it the unknown: $0.80x=184$.
4. Pick the route. Friendly percents are quick by hand. For awkward ones or a chain, type the sentence into Desmos as it reads, like $80\%\text{ of }x=184$, and read the solution: see solve percent problems in Desmos.

Part and whole

Part equals percent times whole, so any two give the third: if $35\%$ of a total is $84$, the total is $240$. When the leftover is given, use the leftover percent: $35\%$ took a tour and $390$ didn’t, so $65\%$ of the total is $390$, and the total is $600$. See SAT percent part whole problems.

Percent change

Percent change is the change over the original: $160$ to $212$ is $\frac{52}{160}=32.5\%$. A multiplier above $1$ is an increase, so $1.18$ is up $18\%$, and one below $1$ is what’s left, so $0.35$ is down $65\%$. “$150\%$ of” a value is $1.5$ times it, but “$150\%$ greater than” is $2.5$ times. See SAT percent increase and decrease.

Reverse and successive changes

Equal changes up and down don’t cancel: up $20\%$ then down $20\%$ is $(1.20)(0.80)=0.96$, a $4\%$ decrease, because the drop works on the larger amount. To find an original, divide by the multiplier: a crowd that grew $15\%$ to $414$ was $\frac{414}{1.15}=360$, not $414$ minus $15\%$. See SAT successive percent changes.

Categories that change differently

When parts of a total change by different percents, call the starting total $1$, multiply each part’s share by its own factor, and add: $30\%$ of members up $20\%$ and $70\%$ down $10\%$ give $0.36+0.63=0.99$, a $1\%$ decrease. See SAT category percent change.

Mistakes to avoid with percentages

Divide by the original: $\frac{20}{80}=25\%$. Multiply the factors: $(1.35)(0.80)=1.08$, up $8\%$. And the discount was $20\%$ of the original, so $0.80x=184$ and $x=230$, not $220.80$.

Center questions give data as a list, a frequency table, or a display (a dot plot, histogram or box plot) and ask for the mean or median, a comparison of the two, or a missing value. First read what the display counts: a frequency says how many, and the median is a position in the ordered data.

How to answer center questions

1. Read the display: what each axis, row or bar counts, and the scale.
2. Pick the route. Count a short list or a frequency table by hand. Type a long, unordered list into Desmos as $L=[\dots]$ and use $\operatorname{mean}(L)$ and $\operatorname{median}(L)$, or a long frequency table as values in $x_1$ and counts in $y_1$, then $\operatorname{total}(x_1 y_1)/\operatorname{total}(y_1)$ for the mean: see mean and median in Desmos.
3. Rebuild the list or the total: put the values in order, or find the total as mean times count, or as each value times its frequency.
4. Find the center: the median sits at position $\frac{n+1}{2}$, or halfway between the two middle values when $n$ is even; the mean is the total over the count.

Data displays

A dot plot’s stack sits at the value, and its height is the count. A histogram bar counts every value in its interval, so “at least $20$ and less than $40$” adds the bars for $20\le x<40$. A box plot marks the minimum, the quartiles, the median and the maximum, and each box holds the middle half of its own group. If a display doesn’t show it, you can’t claim it: a box plot shows no mean and no count. See SAT data displays.

Mean and median

Order the values before you look for the middle: $13,7,11,9,9,15,20$ becomes $7,9,9,11,13,15,20$, so the median is $11$. Every value pulls on the mean, but the median only cares what’s in the middle: $10,11,12,13,34$ has a mean of $16$ and a median of $12$. See SAT mean and median.

Weighted means and missing values

Rebuild the total, then divide. Six values with a mean of $18.5$ total $111$, so if five of them add to $92$, the sixth is $19$. Groups of different sizes combine through their totals: $18$ volunteers averaging $42$ and $12$ averaging $51$ average $\frac{18(42)+12(51)}{30}=45.6$, not $46.5$. See SAT weighted mean.

Mistakes to avoid with center

Order first, then count. A frequency table’s mean weights each value by its count: $\frac{4(2)+7(5)+10(3)}{10}=7.3$. And a dot plot’s value is the number under the stack; the height says how many times it appears.

Spread questions ask which data set is more spread out, what happens when every value follows one rule, or how adding, removing or fixing a value moves the mean, median, range and standard deviation. On the SAT you compare standard deviations rather than calculate them. Standard deviation measures how far the values sit from their own mean, so equal means say nothing about spread.

How to answer spread questions

1. Name the statistics each choice claims: mean, median, range or standard deviation.
2. Test each one on its own. Mean: did the total or the count change? Median: what’s in the middle of the new, ordered list? Range: did the smallest or largest value change? Standard deviation: did values move toward their mean or away from it?
3. Compare distances, not values: find each set’s mean, and how far its values sit from it.
4. Pick the route. Most of these take a few seconds of reasoning by hand. Only for two long, awkward lists with a similar spread, type both into Desmos and compare $\operatorname{stdevp}(A)$ and $\operatorname{stdevp}(B)$: see Desmos box plots and standard deviation.

Range and standard deviation

Range is the largest value minus the smallest. Standard deviation compares distances from the mean: $14,17,20,23,26$ and $8,14,20,26,32$ both have a mean of $20$, but the second set’s distances are twice the first’s, so its standard deviation is twice as large. When one rule $y=ax+b$ changes every value, the mean follows the whole rule, while range and standard deviation are multiplied by $|a|$ and ignore $b$: under $y=-1.5x+6$, a mean of $14$ becomes $-15$ and a standard deviation of $3$ becomes $4.5$. See SAT standard deviation.

Changed data and outliers

When only some values change, test each statistic on its own. Replacing $45$ with $24$ in $11,13,14,15,16,18,45$ lowers the total by $21$, so the mean drops by $3$, while the median stays $15$. The median can move too: replace $30$ with $5$ in $4,6,6,8,9,11,30$, and it falls from $8$ to $6$. A swap that keeps the total but pulls values toward the mean, like $2$ and $18$ becoming $7$ and $13$ around a mean of $10$, keeps the mean and shrinks the standard deviation. See SAT effects of outliers.

Build a list under conditions

Some questions fix the mean and median and ask for the least possible range or the greatest possible median. Turn the mean into a total, lock the slots the median fixes, push the other values as low or high as the rules allow, then write out a full list that meets every condition. Desmos can add up a finished list, but it can’t choose the slots. See SAT ordered data constraints.

Mistakes to avoid with spread

Compare distances from each mean, not the means. Adding $6$ moves every value the same way, so only the multiplier reaches the spread: $1.5(3)=4.5$. And a new value changes the count: six values have a median of $\frac{12+13}{2}=12.5$.

Two-variable data questions show a scatterplot, a line or curve of best fit, or a table of pairs. They ask for the overall trend, a prediction, what a slope means, a residual, or which equation fits the data. The dots are what happened, and the line is what the model predicts.

How to answer scatterplot and model questions

1. Name the variables: what $x$ and $y$ measure, with units.
2. Decide what you need: the trend (direction, form and strength), one observed point, or the model’s prediction.
3. Pick the route. When the line or its equation is given, read it or plug in by hand. When you’re given a table of data to fit, Desmos regression is faster and safer: put $x$ in $x_1$ and $y$ in $y_1$, then type $y_1\sim mx_1+b$. See Desmos regression.
4. Check it in context: the slope’s sign matches the trend, a slope is the predicted change in $y$ for each one-unit increase in $x$, and a residual is actual minus predicted.

Scatterplots

Each dot is one pair. Describe the cloud by its direction (as $x$ increases, does $y$ tend to rise or fall?), its form (straight or curved) and its strength: how closely the points follow the pattern, not how steep it is. An outlier sits far from the overall pattern, so describe most of the cloud first. And a trend alone doesn’t show that one variable causes the other. See SAT scatterplots.

Lines of best fit

A line of best fit gives predictions. In $\hat d=42t+3$, $42$ is the predicted change in $d$ per hour and $3$ is the prediction at $t=0$. A residual is actual minus predicted: positive above the line, negative below it. With equal steps in $x$, constant differences in $y$ mean linear, constant second differences mean quadratic, and a constant ratio means exponential. See SAT lines of best fit.

Rewriting a model

When a variable is redefined, write the old input in terms of the new one, substitute, then apply the new output’s definition. If $x$ counts hours in $y=14x+80$ and $t=60x$ counts minutes, then $x=\frac{t}{60}$ and $y=\frac{7}{30}t+80$. See rewrite line of best fit SAT.

Mistakes to avoid with scatterplots and models

Actual minus predicted: $32-34.8=-2.8$, a point below the line. A prediction comes from the line, not the nearest dot. And the model holds the old input, so solve for it first: $x=\frac{t}{60}$.

Probability questions usually come with a two-way table and ask the chance that someone picked at random fits a description. Some pick from everyone. Others tuck a condition into a short phrase, “given that”, “among those who” or “selected from”, and that phrase shrinks who can be picked. Decide the whole before you count the part.

How to answer probability questions

1. Name the group the pick comes from: everyone, or only the people a condition names.
2. Find that group’s total: the grand total, or one row or column total. That’s the denominator.
3. Count the target inside that group: that’s the numerator. “And” is one cell, and “not” is the rest of the group.
4. Simplify and check: a probability lies between $0$ and $1$. These are quicker by hand: Desmos can divide, but it can’t choose the denominator.

Probability and relative frequency

Probability is the favorable outcomes over all the outcomes the pick could land on, and a relative frequency is the same fraction for data. If $38$ of $70$ attendees chose the afternoon session, the probability a random attendee did is $\frac{38}{70}=\frac{19}{35}$. “Not” is a complement: $1$ minus the probability. See SAT probability and relative frequency.

Conditional probability

$P(A\mid B)$ is the count in both $A$ and $B$ over the count in $B$. Of $90$ students, $36$ ride the bus and $22$ of them aren’t in the club, so given bus, the probability of not in the club is $\frac{22}{36}=\frac{11}{18}$. Flip the condition and the denominator changes. See SAT conditional probability.

Mistakes to avoid with probability

The condition picks the row, so divide by the bus riders: $\frac{22}{36}$. The group you already know about goes on the bottom. And a probability over $1$ means the fraction is upside down or the whole is too small.

These are reading questions with numbers in them: an estimate from a sample with its margin of error, a survey, or a study, and the conclusion it supports. The SAT always gives you the margin; your job is to read it, and to notice two details: where the sample came from, and who put people in each group. Random sampling is about who a result covers; random assignment is about cause.

How to answer sample and study questions

1. Build the interval when there’s a margin: the estimate minus the margin to the estimate plus it. $62\%$ with a margin of $4$ percentage points runs from $58\%$ to $66\%$.
2. Name what it describes: the mean or percent of the population the sample was drawn from at random, not each member, and no group beyond that list.
3. Who picked the groups? If chance did, by random assignment, the result can say “caused”. If people chose, or the researchers only observed, stop at an association.
4. Test each choice for overreach: “plausible” fits, while “exactly”, “every”, “proves”, “impossible”, a claim about individuals or a group bigger than the one sampled doesn’t. Desmos can’t tell who picked the groups; use it only to turn a percent estimate into a count, typed as it reads, like $48\%\text{ of }7200$: see solve percent problems in Desmos.

Sample estimates and margin of error

A sample mean estimates the population mean: $80$ bulbs lasting $1{,}240$ hours on average doesn’t mean every bulb lasts $1{,}240$ hours. Plausible means reasonable, not certain, and a value outside the interval isn’t impossible. A margin in percentage points is added and subtracted: $38\%$ with $5$ points runs from $33\%$ to $43\%$. A larger sample generally gives a smaller margin, a narrower interval, but it doesn’t push the estimate up or down. To estimate a count, take the percent of the population: $48\%$ of $7{,}200$ is about $3{,}456$. To compare two groups, compare their intervals, not their estimates. See SAT margin of error.

Sampling and generalization

A random sample speaks for the list it was drawn from, and no further: $240$ people picked at random from the county’s bus-pass holders speak for bus-pass holders, not every resident. A convenience sample or a voluntary poll describes only the people who answered, and a bigger sample from the same tilted source repeats the same tilt. See SAT sampling and generalization.

Experiments and causation

In an experiment, the researchers decide who gets the treatment; in an observational study, they only record. Students who sign up for a Saturday workshop may be more motivated, so their higher scores show an association, not an effect. A coin flip spreads those differences across both groups, so random assignment supports a causal conclusion, for the people in the study. See SAT experiments and causation, and practice with the evaluating statistical claims questions.

Mistakes to avoid with samples and studies

The interval is for the mean commute, not each one. Random selection works only inside the list it picked from. And when people chose their own group, other differences could explain the gap: stop at an association.