ACT Math Statistics and Probability: mean and median, probability, counting, data displays, weighted averages and expected value

What the Statistics and Probability area covers

Statistics and Probability, about 8 to 12 percent of the ACT Math test, covers summary statistics, probability, counting and data interpretation. This guide ties together the dot points: mean, median, mode and range, probability of events, counting, permutations and combinations, data displays and interpretation, and weighted averages and expected value.

Mean, median, mode and range

The mean is the sum over the count; the median is the middle of the sorted list (average the two middle for an even count); the mode is the most frequent value; the range is max minus min. Use total $=$ mean $\times$ count for missing-value questions. An outlier pulls the mean but barely moves the median, so the median better describes skewed data.

Probability of events

Probability is favorable over total outcomes, between 0 and 1. The complement is $1 - P(A)$. For independent events, multiply ("and"); for mutually exclusive events, add ("or"). For "at least one", use $1 - P(\text{none})$. Drawing without replacement makes events dependent.

Counting, permutations and combinations

The fundamental counting principle multiplies the choices at each stage. A permutation counts ordered selections (order matters); a combination counts unordered selections (order does not matter). The test: if rearranging the same items is a different outcome, it is a permutation.

Data displays and interpretation

Read tables, bar and line graphs, histograms, pie charts and box plots carefully, checking axis labels, units and scale. A pie slice is a percentage of the whole; a box plot shows the five-number summary. Compute the statistic asked (mean, total, fraction, probability) after reading the values.

Weighted averages and expected value

A weighted average is $\frac{\sum (\text{value}\times \text{weight})}{\sum \text{weight}}$, used when items count unequally (group sizes, grade categories). Expected value is the same calculation with probabilities as weights: $E = \sum (\text{outcome}\times \text{probability})$, the long-run average. Subtract any cost to play when judging a game.

Check your knowledge

Try these, then read the solutions.

1. Find the median of 7, 2, 9, 4, 6. [1 point]
2. A bag has 5 red and 3 blue marbles. What is the probability of drawing red? [1 point]
3. In how many ways can 3 books be arranged on a shelf? [1 point]
4. A pie chart shows 30% of a $2,000 budget is rent. How much is rent? [1 point]
5. A game pays $8 with probability 0.25 and$0 otherwise. What is the expected payout? [2 points]