Statistics & Probability

• Calculate mean, median, mode, and range, and solve weighted average problems
• Calculate probability: favorable over total when outcomes are equally likely, the complement rule P(not A) = 1 − P(A), and P(A and B) = P(A) × P(B) for independent events
• Read a probability off a two-way frequency table (the row/column count divided by the relevant total), including a conditional probability ('given that' means divide by that row or column's total), and use a random sample to estimate a population (a survey proportion, tag and recapture) and spot a biased sample
• Reason about standard deviation without arithmetic, decide whether order matters before counting, and compute an expected value
• Finish on one ACT-hard question and say what made it hard

The mean (average) is the sum divided by the count. The median is the middle value when data is ordered; for an even count, average the two middle values. The mode is the most frequent value, and the range is the largest value minus the smallest. A weighted average multiplies each value by its weight before adding. Probability is always between 0 and 1. The formula probability = favorable outcomes / total outcomes applies only when every outcome is equally likely; when outcomes have unequal probabilities, add the probabilities of the favorable outcomes. For a person selected uniformly from a frequency table, divide the relevant count by the eligible total; observed relative frequencies estimate probabilities from data. Two rules the ACT tests often: the complement rule, P(not A) = 1 - P(A), which is the fast way to handle 'at least one' questions, and the multiplication rule for independent events, P(A and B) = P(A) x P(B). The ACT also asks you to read a probability directly off a two-way frequency table: divide the count in the relevant cell (or row/column) by the appropriate total. Classify before you compute. For spread, decide whether a change shifts the data or stretches it: adding the same number to every value moves the mean but leaves the standard deviation unchanged, multiplying every value by 3 triples the standard deviation, and moving values farther from the mean (or adding an outlier) increases it, so no arithmetic is needed. For counting, first ask whether order matters: arranging 3 of 7 books on a shelf gives 7 × 6 × 5 = 210 arrangements because order matters, while choosing 3 of 7 books to pack gives 210 ÷ 6 = 35 groups because each group of 3 was counted 3 × 2 × 1 = 6 times. Counting gets its own lesson, Standard Deviation and Counting. Expected value is a probability-weighted average: multiply each payoff by its probability and add, then subtract any cost to play; the expected payoff must lie between the smallest and largest possible payoff, which catches many errors. Statistics and probability are a steady part of the Math section. Conditional probability is the top-score trap: 'given that' or 'of those who' shrinks the denominator to that group only. If 60 students take chemistry and 24 of them also take physics, the probability that a randomly chosen chemistry student takes physics is 24/60, not 24 divided by all the students surveyed. For the mean of combined groups, work with totals: each group's total is its mean times its count, then divide the combined total by the combined count. An outlier pulls the mean much more than the median. Mutually exclusive events cannot happen together, so P(A and B) = 0 and P(A or B) = P(A) + P(B). When B can only happen if A happens (a player who plays tennis only on sunny days), P(A and B) = P(B), which is more than 0 if B ever happens and less than P(A) if A sometimes happens without B. Samples: a random sample's proportion estimates the population's, so if 20 of 80 randomly chosen students walk to school, about 1/4 of all 1,200 students, 300, walk; the mean of a random sample's frequency table (each value times its count, added, divided by the sample size) estimates the population mean. Tag and recapture: if 60 fish are tagged and later 10 of 50 caught fish are tagged, the tagged share of the lake is about 10/50, so 60/N = 10/50 and N is about 300. A sample is biased when the way it is chosen is tied to the answer (surveying people leaving the library about library use, or volunteers); a bigger biased sample is still biased. For evidence that a coin is unfair, compare how unlikely each result is for a fair coin: 70 heads in 100 tosses is far stronger evidence than 9 heads in 10, because long runs of a fair coin stay close to half. The hardest questions, at the end of the section: expected value with unequal probabilities and a cost to play, standard deviation questions where one choice shifts the data, one stretches it, and one pulls values toward the mean, a conditional probability read from a table with a missing count, and a tag-and-recapture estimate.

• For median, always arrange the numbers in order first. A common ACT trap is providing unsorted data.
• If a question says 'at least one,' it's often easier to calculate 1 minus the probability of 'none.'
• Weighted averages appear in problems like 'your quiz average is 80% of your grade and the final is 20%.'