Data & Statistics — 700-800

Data & Statistics: The 700-800 Patterns

Part 1 of 3 — The Archetypes Hard-Tier Items Are Built From

This topic is reading two-way tables and drawing rates out of them. The arithmetic is division. The difficulty is entirely in choosing the denominator, and every hard item is engineered so that the wrong denominators are also available as answer choices.

Archetype 1: The Denominator Decides

Any single cell in a two-way table supports three completely different percentages. Take the cell 84 in a table whose row total is 280, column total is 336, and grand total is 700:

Phrasing	Denominator	Value
"of all respondents"	grand total 700	12.0%
"of respondents in the 30-44 group"	row total 280	30.0%
"of respondents who opposed the plan"	column total 336	25.0%

Same numerator, three answers. Hard items exploit this by pairing a correct number with the wrong phrasing, or by asking two of the three in one question and shuffling them.

The rule: the phrase immediately after "of" names the group you have been restricted to, and that group's total is your denominator. Circle it before you divide.

Archetype 2: Rate Comparison Across Rows

"For each club, the attendance rate is the number of members in that club who attended divided by the number of members in that club. By how many percentage points does the chess club exceed the robotics club?"

Each group's rate uses its own row total — never a shared denominator. That is the whole point: comparing $\frac{63}{400}$ to $\frac{45}{400}$ is not a comparison of rates, it is a comparison of counts wearing a percent sign.

Watch for two variants:

• "Percentage points" vs. "times." A difference of rates is measured in percentage points; a ratio of rates is "how many times." Compute the one asked.
• Adjusted denominators. "The pass rate among students who were not absent" means you must subtract the absent column from each row total first. The unadjusted rate is always an answer choice, and it always ranks the groups differently.

Archetype 3: Missing Cells

Roughly a third of hard items hide one or two entries behind a question mark. Recovering them is mechanical, but the order matters: use whichever margin has only one unknown in it. Then check the recovered value against the other margin — a table that does not reconcile means you made an arithmetic slip, and you would rather find it in five seconds than after computing a percent from it.

Archetype 4: Scaling a Sample to a Population

"The 600 people surveyed are representative of the town's 15,000 residents. Estimate the number of residents ages 50 and over who prefer the evening market."

The one rule that decides these:

The rate's denominator must match the population you are scaling.

• Estimating over the whole population? The rate must come from the whole sample: $\frac{\text{cell}}{\text{grand total}} \times \text{population}$.
• Estimating within a subgroup ("the 320 teachers surveyed were drawn from 5,600 teachers in the county")? The rate must be computed inside that row: $\frac{\text{cell}}{\text{row total}} \times \text{subgroup population}$.

Mixing them is the signature error of this archetype, and the item is built so that the mismatch produces a plausible-looking number. If you scale a within-row rate by the entire population, you overestimate wildly; if you scale a whole-sample rate by a subgroup, you underestimate.

Two further variants:

• Apply a stated rate to a new group. "All 120 Gold members will move to the Silver tier; if the resulting 480 renew at the current Silver rate..." — use the Silver row's rate, not the Gold row's, and not the overall rate. The whole item is a test of reading which group's rate was specified.
• Growth between two years. Scale the same cell rate to both population sizes and subtract. Both intermediate counts will be on the menu.

Archetype 5: "Which Statement Is Supported?"

Four claims, each of which requires a different denominator or a count-versus-rate judgment. Do not evaluate all four fully. Scan for the fastest disqualifier:

• A claim that reads a raw count as a rate ("more students improved with Y, so Y worked better") is wrong whenever the groups differ in size.
• A claim that reverses a conditional ("of the frequent visitors, most were teens" vs. "most teens visit frequently") is wrong unless you check the right margin.
• A claim with a numeric comparison ("more than three times") is often true of the raw counts and false of the rates. That is exactly why it is there.

Data & Statistics: Traps & Speed

Part 2 of 3 — The Distractor Species

Every wrong answer in this topic is a correct calculation of the wrong quantity. That is what makes them dangerous: you never feel like you made a mistake. Learn to recognize them by name.

Species 1: The Reversed Conditional

"Of the jazz fans, what percent are under 25?" and "Of the members under 25, what percent prefer jazz?" share a numerator and differ in denominator — column total versus row total. Both values appear in the choices, usually a few rows apart.

Defense: the group named right after "of" is the denominator. Write it down before dividing.

Species 2: The Grand-Total Denominator

Dividing by the grand total answers "what share of everyone is in this cell." It is correct only when the question actually says "of all 540 employees." When a question restricts to a row or a column and you divide by the grand total anyway, the value is always too small — and always on the menu.

Species 3: The Complement

$27/45$ instead of $18/45$. The arithmetic is flawless and the answer is the percentage of exactly the group you were not asked about. When your answer and one other choice sum to about 100 percent, re-read which side of the split the question wanted.

Species 4: Count Instead of Rate

"150 participants improved with Program B and only 132 with Program A, so B is better." B also had 80 more participants. Raw counts are comparable only when the group totals are equal, and hard items never make them equal. Whenever a claim rests on which number is bigger, convert both to rates before believing it.

The same species appears as ratio of counts instead of ratio of rates: $30/12 = 2.5$ when the true ratio of defect rates is $3.5$, because the two lines inspected different numbers of units.

Species 5: The Mismatched Scaling

The population you are estimating and the rate you multiply by must describe the same group.

Population being estimated	Correct rate
All 15,000 residents	cell ÷ grand total
The 5,600 teachers in the county	cell ÷ teachers row total
The 8,000 North branch members	cell ÷ North branch row total

Scaling a row rate by the whole population, or a whole-sample rate by one subgroup, is the built-in trap on every one of these items.

Species 6: The Intermediate Value

This year's estimate when the question asked for the increase. The total number of passers when the question asked how many passed on a retake. The first of two rates when the question asked for the difference. Re-read the final clause before bubbling.

Speed Technique 1: Label the Margin Before You Divide

Spend three seconds writing "denom = column, 82" next to the problem. Nearly every miss in this topic is a denominator chosen from muscle memory rather than from the sentence.

Speed Technique 2: Reconcile the Table Once

When a cell is missing, recover it from the margin that has a single unknown, then check it against the perpendicular margin. Two missing cells are usually solvable in a chain, and often only one of them is needed — recover the one your question touches and check the grand total.

Speed Technique 3: Percentage Points Are Subtraction, "Times" Is Division

• "By how many percentage points?" → subtract the two rates.
• "Is how many times?" → divide the two rates.
• "By what percent did it increase?" → divide the change by the original.

Three different operations on the same two numbers, and at least two of the three results will appear as choices.

Speed Technique 4: Adjusted Denominators First

If the question excludes a category ("among those who were not absent," "with income of 30 thousand or more"), build the restricted total before doing anything else — usually by adding two columns or by subtracting one column from the row total. Then keep the numerator inside that same restricted set. A numerator drawn from two columns and a denominator drawn from one is the most common structural error on these items.

Speed Technique 5: Estimate to Eliminate

Rates in these tables are usually clean-ish fractions. $87/320$ is a hair over one quarter; $52/290$ is under one fifth. That is enough to rule out two choices instantly and to catch a denominator error before you commit — if your "rate" comes out above 100 percent or absurdly small, you grabbed the wrong margin.

Data & Statistics: Timed Drill

Part 3 of 3 — 5 Questions, About 90 Seconds Each

Run this checklist on every item before you divide anything:

1. What group am I restricted to? The words right after "of" name it. Its total is the denominator.
2. Is the denominator adjusted? "Not absent," "30 thousand or more," "either e-book or audio" — build the restricted total first, and keep the numerator inside the same set.
3. Any missing cells? Recover from the margin with one unknown, then reconcile against the other margin.
4. Rate or count? If a claim compares raw numbers across groups of different sizes, it is almost certainly the trap.
5. Difference, ratio, or level? Percentage points subtract; "how many times" divides; and the intermediate value is always on the menu.
6. Scaling? Match the rate's denominator to the population you are multiplying.