Statistics, Spread, and Reading Data

Statistics questions on these tests look easier than they are, because they rarely ask for heavy computation. Instead they ask what a number means. Given two classes with identical averages, which one had more consistent results? Given a survey of 400 students, what can you honestly claim about the 3,000 students who were not surveyed? Given a line drawn through a cloud of dots, what does its slope say in the units of the problem? These are reasoning questions dressed as arithmetic.

On the SAT this material lives in the Problem-Solving and Data Analysis domain, which is a large share of the math section and is deliberately built around tables, scatterplots, and short survey descriptions. The ACT spreads the same content across its statistics and probability questions and adds more direct calculation, such as a probability from a table. Both tests reward students who slow down for one second and identify what the denominator is supposed to be, or who was actually sampled.

The good news is that the standard deviation is never something you have to calculate on these tests. You only ever have to compare spreads qualitatively, and you can do that by eye. Likewise, no one asks you to derive a margin of error; you only need to know what it describes. The previous lesson covered the mechanics of mean, median, and how an outlier moves them. This lesson takes the next step: what a sample lets you say about a population, what a fitted line lets you predict, and how to read a table without picking the wrong total.

Sampling determines what you are allowed to conclude. A sample selected at random from a population is representative of that population, so results from the sample can be generalized to it, but only to it. A random sample of students at one school supports a claim about that school and says nothing reliable about students statewide. A sample that is not random, such as people who volunteer to answer, the first 50 shoppers to enter a store, or every student in one class, is biased toward whatever kind of person was easy to reach, and no conclusion about the wider population is justified no matter how many people were surveyed. When a question describes how a sample was chosen, the first thing to identify is the population that every sampled individual was drawn from, because that is the largest group the result can describe.

A margin of error attached to a survey result gives a plausible interval around the estimate. If 62 percent of a random sample of 400 students support a later start time with a margin of error of 4 percentage points, then the true proportion of all students at that school plausibly lies between 58 and 66 percent. The margin does not mean that exactly 62 percent support it, and it is not a statement about the 400 students in the sample, whose answers are already known. Larger random samples produce smaller margins of error, because more data pin the estimate down more tightly; quadrupling the sample roughly halves the margin. You will never be asked to compute a margin of error on these tests, only to interpret one and to know which direction it moves.

Whether a study can support a cause-and-effect claim depends on a different question: were the treatments assigned at random? In an observational study, the subjects already belong to their groups, so some other difference between the groups may explain the outcome; a survey showing that people who drink coffee sleep less cannot say that coffee causes the lost sleep. In an experiment, the researcher randomly assigns subjects to treatments, which spreads every other difference evenly across the groups, so a difference in outcomes can be attributed to the treatment. The two questions are independent. A random sample with random assignment supports both generalization and causation; a random sample without random assignment supports generalization only; random assignment among volunteers supports causation for subjects like those volunteers but not generalization to a population.

A scatterplot shows paired data, one dot per individual, with the explanatory variable on the horizontal axis and the response on the vertical axis. If the dots trend upward the association is positive, if downward it is negative, and if the dots hug a straight path the association is strong. The line of best fit is the single line that comes closest to the whole cloud of points, and its slope is the predicted change in the response for each one-unit increase in the explanatory variable, carrying the units of both axes. Its y-intercept is the predicted response at an input of zero, which is meaningful only if zero is inside or near the range of the data. For any data point, the residual is the actual y-value minus the y-value the line predicts, so a point above the line has a positive residual and a point below it a negative one. Two cautions are tested constantly: predicting far outside the observed range of x is extrapolation and is unreliable, and a strong association from observational data does not establish that one variable causes the other.

A two-way table sorts individuals by two categorical variables at once, with the row totals, the column totals, and the grand total along the edges. Every probability from such a table is a count divided by a total, and the only real skill is picking the right total. An unconditional probability, such as the probability that a randomly chosen student passed, uses the grand total as the denominator. A conditional probability, signaled by the words given that, among, or of those who, restricts attention to one row or one column, so the denominator becomes that row or column total. Reversing the condition changes the answer: the probability that a student passed given that she studied in a group and the probability that a student studied in a group given that she passed use the same numerator but different denominators, and confusing them is one of the most common errors on the whole test.

Spread describes how far the values of a distribution sit from its center, and it is a separate fact from the center itself. Two data sets can share a mean, a median, a range, and a count and still differ in standard deviation, so an average alone never describes a distribution. These tests never ask you to compute a standard deviation, only to compare two of them, and the previous lesson on mean, median, and standard deviation covers how to make that comparison by looking at how far the values sit from the mean.

Using the grand total for a conditional probability. When a question says given that, among, or of those who, the denominator is a row total or a column total, not the total number of individuals. Before you write the fraction, underline the conditioning phrase and find its total in the table margin, then count the numerator only within that row or column.

Swapping the direction of a conditional. The probability of passing given group study and the probability of group study given passing are different numbers built from the same cell. Read the condition first, fix the denominator, and only then look for the numerator; doing it in that order makes the swap almost impossible.

Assuming equal means imply equal distributions. Two data sets with the same mean can have wildly different standard deviations, and a question that gives you both means and asks about consistency, reliability, or variability is asking about spread, not center. Look at how tightly the values cluster around the mean rather than at the mean itself, and do not let a shared range convince you the spreads match.

Treating a strong association as proof of causation. A scatterplot showing that towns with more libraries have higher test scores does not show that libraries raise scores; the data are observational and some third factor, such as town wealth, may drive both. Only a randomized experiment supports a causal claim, and only a random sample supports generalization to a population.

Trusting a sample that chose itself. Volunteers who answer an online poll, the first people through a door, and everyone in one class are all easy to reach and all unrepresentative, and a large count of them does not repair the bias. When a question describes the sampling method, decide whether every member of the population could have been selected before you accept any generalization.

Overstating what a margin of error says. A result of 62 percent plus or minus 4 points does not mean exactly 62 percent, and it does not mean between 58 and 66 percent of the sample. It means the true value for the population from which the sample was randomly drawn plausibly lies between 58 and 66 percent. Also remember the direction of the sample-size effect: increasing the sample size shrinks the margin of error, it does not grow it.

Predicting outside the data. A line of best fit fit to 1 to 8 hours of study has nothing to say about 40 hours, and a question that asks for such a prediction is usually testing whether you recognize extrapolation. Compare the requested input to the range of x-values in the data before you plug it in.