PSAT Data — centers, chances, and what a study is allowed to say

Problem-Solving and Data Analysis is the domain where the least arithmetic is required and the most points are lost, and two ideas just beyond it are taught almost never: what a margin of error permits, and why an observed difference is not a cause. College Board tests both on the SAT rather than on the PSAT/NMSQT, and they are included here as preparation for it. This material gives each of them a question of its own, and builds the other ten around the statistics that surround them.

Three are about the center of a set. A median of eight sunflower heights where one plant is nearly twice the height of the next tallest. A week of temperatures where a single reading of twenty-eight degrees drags the mean to seven while the median sits at four, so the question becomes which of the two describes the week. And a set of nine commuting times containing a ninety-six, where removing that one reading moves the mean by more than eight minutes and the median by half of one — which is the clearest way to see what each statistic is actually made of.

Two are probability read off counts stated in sentences rather than set in a table. One of them is conditional, and the explanation names the group the fraction is taken from, because taking it from the whole hundred rather than from the forty who prefer digital books is the single error this question type is built to catch.

Three concern a line fitted to data: what its slope means with units attached, what its intercept means, and what it predicts at a stated input — where forgetting to add the intercept is the standard slip.

Two ask what a sample supports. Five hundred students at one university tell you about that university; two hundred employees at one company give an estimate for that company. The wrong options each stretch the conclusion to a population nobody sampled, which is how a sound survey gets misreported.

The last two are the pair named at the start. A margin of error marks out a band around an estimate — thirty-eight percent plus or minus three points means thirty-five to forty-one — and it is not a chance of being wrong. And an observational study, however large its difference, cannot rule out the confounding variable that produced it, because nothing was assigned at random.

Every situation here was invented for the exercise, and nothing is reproduced from any published test.

• Find a median in an ordered list, including a set with an even number of values
• Say which of the mean and the median describes a set better when one extreme value is present
• Predict what removing an outlier does to each statistic, and by roughly how much
• Read a conditional probability off counts by taking the fraction from the subgroup the condition names
• Say what the slope and the intercept of a fitted line mean with their units, and use the line to predict
• Limit a conclusion to the population the sample was drawn from
• State what a margin of error is: a band around an estimate, not a probability of being wrong
• Explain why an observational study cannot establish a cause, however large the difference it reports

Every data set here is small enough to hold in the head and is listed in words inside its own question; there are no tables, and the only drawings are a tree diagram of the library patrons and a graph of the repair-cost line in the quiz, and a blank two-stage tree on one flashcard. Among them: eight sunflower heights of 120, 122, 125, 125, 128, 130, 135 and 250 centimeters; a week of temperatures at 2, 3, 3, 4, 5, 6 and 28 degrees; nine commuting times of 14, 15, 15, 17, 18, 20, 21, 22 and 96 minutes; a hundred library patrons of whom forty are students and forty prefer digital books; two hundred commuters, eighty of them on the train; the fitted lines $y = 50x + 100$ for repair costs and $y = 2.5x + 10$ for a seedling's height; five hundred students surveyed at one university; two hundred employees of five thousand; a county bus survey reporting thirty-eight percent plus or minus three points; and an observational study of tablets and test scores. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the PSAT/NMSQT, and it is not a test center.