ACT Math Probability, Counting and Statistics

Statistics and Probability is one of five subcategories inside the Preparing for Higher Math reporting category on the ACT Math test. It covers counting techniques, probability calculations, and reading and summarizing data. The subcategory accounts for 12 to 15 percent of the questions on the Mathematics test. The same skills also appear inside multi-step problems from other categories, so the payoff reaches past the questions labeled with this name.

This guide breaks down every skill in the category: the counting principle, single and compound event probability, mean, median, mode, range, bivariate data, and how to read charts and tables accurately. Each section gives you the rule first, then a short original example worked from start to finish. Review the rules here, then move to the drills at the end for timed practice. Every drill and practice test on 10exam is free with no signup required.

Where This Category Fits on the ACT Math Test

The ACT Mathematics test has 45 questions with 41 scored and 4 unscored field-test items, and you get 50 minutes to complete it. Preparing for Higher Math makes up 80 percent of the test and splits into five subcategories. Statistics and Probability carries the second smallest weight of the five. The table below shows how the 80 percent divides.

The remaining 20 percent of Math questions fall under Integrating Essential Skills, which measures how you apply skills from multiple categories together, including counting and basic statistics folded into multi-step problems. Every Math question counts toward your Math scale score from 1 to 36. Your Math score is one of three scores averaged into your Composite score alongside English and Reading. Strength in this category lifts both numbers.

The official ACT guide names the core skills tested in this subcategory. Each one maps to a section of this guide. Read the four below before you start drilling, since the official wording tells you what the test writers are measuring.

• Describe the center and spread of distributions.
• Apply and analyze data collection methods.
• Understand and model relationships in bivariate data.
• Calculate probabilities by recognizing related sample spaces.

The Counting Principle and Basic Combinations

Counting questions ask how many ways an event or sequence of events happens. The core tool is the counting principle: if one choice has m options and a second, independent choice has n options, the two together have m times n total outcomes. The rule extends to any number of stages, so three stages multiply three numbers. Break the problem into stages and the arithmetic stays simple.

• List each independent choice or stage in the sequence separately.
• Multiply the number of options at each stage to get the total count.
• Watch for restrictions, such as no repeats allowed, which reduce the options at later stages.

Worked Example

A student picks one shirt from 4 shirts and one pair of pants from 3 pairs. The number of outfits is 4 times 3, which equals 12. If the student also picks socks from 2 colors, the total becomes 4 times 3 times 2, which equals 24. Adding a stage multiplies the count instead of adding to it.

Permutations and Combinations

Permutations count arrangements where order matters, such as ranking three runners in first, second, and third place. Combinations count selections where order does not matter, such as choosing 3 students out of 10 for a committee. Read each question carefully to decide whether swapping two items creates a new outcome. If it does, order matters and you are counting a permutation. If it does not, you are counting a combination.

Probability of Single and Compound Events

Probability measures how likely an event is, expressed as a fraction or decimal between 0 and 1, or as a percent between 0 and 100 percent. The basic formula is the number of favorable outcomes divided by the total number of possible outcomes in the sample space. An impossible event has a probability of 0 and a certain event has a probability of 1. Most lost points here come from miscounting the sample space, not from the division.

Single Events

To find the probability of one event, define the full sample space first. Rolling a standard six sided die gives a sample space of 6 outcomes. The probability of rolling a 4 is 1 divided by 6. The probability of rolling an even number is 3 divided by 6, which reduces to 1/2.

Compound Events

Compound events combine two or more single events. The wording tells you which rule applies, so read for the words and, or, and without replacement. The three rules below handle nearly every compound probability question on the test.

• For independent events (one outcome does not affect the other), multiply the individual probabilities together.
• For mutually exclusive events (both cannot happen at once), add the individual probabilities to find the chance of either one happening.
• For dependent events, adjust the second probability based on what already happened in the first event.

Worked Example

A bag holds 3 red marbles and 5 blue marbles, 8 total. The probability of drawing a red marble is 3 divided by 8. If you replace the marble and draw again, the probability of red both times is 3/8 times 3/8, which equals 9/64, because the two draws are independent. Without replacement, the second draw uses 2 red marbles out of 7 remaining, giving 3/8 times 2/7, which equals 6/56 or 3/28.

Mean, Median, Mode, and Range

These four measures describe the center and spread of a data set, matching the official skill of describing center and spread of distributions. Mean and median report the center of the data, while range reports the spread. Mode reports the value appearing most often. Learn all four definitions cold, since one problem often asks you to compare two of them.

Worked Example

Given the data set 4, 7, 7, 9, 13, the mean is (4+7+7+9+13) divided by 5, which equals 8. The median is 7, the middle value in the ordered list. The mode is 7, since it appears twice. The range is 13 minus 4, which equals 9.

Weighted Averages

Some questions weight certain values more heavily, such as test scores worth different point totals. Multiply each value by its weight, add the products, then divide by the total weight rather than by the plain count of values. A score of 90 on a 30 point assignment and 70 on a 70 point assignment gives (90 times 30 plus 70 times 70) divided by 100, which equals 76. Dividing by 2 instead would give 80 and cost you the question.

Bivariate Data and Data Collection Methods

Bivariate data compares two variables to look for a relationship, such as hours studied against test score. A scatterplot displays this relationship, and a trend line or line of best fit summarizes whether the relationship is positive, negative, or has no clear pattern. Questions ask you to describe the direction of the trend, estimate a value from the trend line, or judge the strength of the association. Read the axis labels first, since the variables are often reversed from what you expect.

The ACT also tests how well you apply and analyze data collection methods. Know the difference between a sample and a population, and recognize when a sampling method introduces bias, such as surveying only one age group and generalizing the result to everyone. A conclusion is only as good as the sample behind it. When a question asks whether a result generalizes, check who was surveyed and how they were chosen.

Reading Data Displays and Interpreting Statistics

Many questions in this category give you a table, bar graph, line graph, circle graph, or scatterplot. You are then asked to read a value, compare categories, or calculate a statistic from the displayed data. The math is usually light, so the difficulty sits in reading the display correctly. Use the checks below on every graph question.

• Read the title, axis labels, and units before answering, since axes are not always in the same scale as the answer choices.
• Check whether the question asks for a raw count, a percent, or a rate, since data displays often show one and the question asks for another.
• For circle graphs, remember all sectors add up to 100 percent or the full total shown in the key.
• For scatterplots, look at the overall trend of points rather than any single outlier point.

Worked Example

A bar graph shows 20 students prefer math, 15 prefer science, 10 prefer history, and 5 prefer art, for 50 students total. The percent who prefer math is 20 divided by 50, which equals 40 percent. If the question instead asks how many more students prefer math than history, the answer is 20 minus 10, which equals 10. Same graph, different question, different arithmetic.

Two free drills on 10exam target every skill in this guide. The Counting & Probability drill covers the counting principle, permutations, combinations, and single and compound event probability. The Statistics & Data Interpretation drill covers mean, median, mode, range, and reading data displays. Both drills are free and untimed, and they give instant explanations with no signup required.

For full length, timed Math practice under real test conditions, take one of our free practice tests on the ACT exam page. Work through the drills first to build the rules into reflexes, then use a full test to check your pacing across all 45 questions. See our ACT Math section guide for the full test format, calculator rules, and content breakdown across every category. For the geometry side of Preparing for Higher Math, our ACT Plane Geometry guide covers angle rules, triangle properties, and area formulas.

Sources

• Preparing for the ACT Test | ACT