Math · Checkpoint 20 of 30 · Problem-Solving and Data Analysis · ~13 min
Mean vs. median, scatterplots, probability from tables, and what a study can actually claim.
The 30-second version
the mean chases outliers; the median doesn't — one billionaire in the room wrecks the average income but barely moves the middle. Probability from a table is (cell you want)/(row or column you're restricted to). And only a randomized experiment can support a cause-and-effect claim; observational studies support association only, and random sampling is what lets you generalize to the population.
Mean = total ÷ count. Median = middle value when sorted. Skewed data or an outlier drags the mean toward the extreme while the median stays put — so "which is greater?" questions are really "which way does the data lean?" Also useful: if the mean of 5 tests is 84, the total is 420. Many SAT mean questions are secretly total questions ("what must she score on the sixth test…").
The line of best fit is a linear model: its slope is a rate ("each additional hour of practice predicts 4 more points"), its intercept a baseline. A point above the line means the actual value beat the prediction; the vertical gap is the size of the miss. Questions asking "the y-value predicted for x = 30" want the line's height there, not a data point's.
"Given that a randomly chosen participant is a junior, what's the probability they chose robotics?" — the "given" restricts you to the junior row: answer = robotics-juniors ÷ all juniors. Unrestricted probability divides by the grand total. Read which universe the question locks you into before dividing.
A poll's "38% ± 3%" means the population value is plausibly between 35% and 41% — larger random samples shrink the margin. And the claims ladder: random assignment to treatments → may conclude causation; random sampling → may generalize; neither → the study describes only the people in it. The SAT asks "what's the strongest supported conclusion?" — pick the rung the design actually earns.
Nine houses on a street sell for about $300,000 each; a tenth sells for $3 million. Which statement is true?
The median stays near $300,000 (the middle house is ordinary), but the $3M outlier inflates the mean to roughly $570,000. Outliers drag the mean toward themselves and barely touch the median — the single most-tested statistics fact.
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