Probability and Statistics

Section 1

The Story Behind the Numbers

Long before anyone sat down for a standardized test, people needed ways to make sense of uncertain events — from gambling to predicting harvests. The twin disciplines of probability and statistics grew out of that very human desire to measure the unknown. Understanding where these ideas came from helps you appreciate why the ACT tests them so heavily: they are the mathematical backbone of data-driven decision-making.

1654

Blaise Pascal and Pierre de Fermat exchange a series of letters about gambling problems, launching the formal study of probability theory. Their work on the "problem of points" showed that chance could be analyzed logically.

1713

Jacob Bernoulli's Ars Conjectandi is published posthumously, introducing the Law of Large Numbers — the idea that running an experiment many times causes the observed results to approach the theoretical probability.

1809

Carl Friedrich Gauss describes the normal distribution (the "bell curve") while studying astronomical measurement errors. This distribution would become the most important shape in statistics.

1900s

Karl Pearson and Ronald Fisher develop modern statistical methods such as standard deviation, correlation, and hypothesis testing, making statistics a rigorous science used in medicine, economics, and beyond.

Today, every ACT Math section includes questions about averages, data interpretation, counting, and probability. These aren't abstract museum pieces — they are the same ideas that power everything from weather forecasts to sports analytics. Mastering them means mastering one of the most practical corners of mathematics.

Section 2

Core Principles & Definitions

Before diving into formulas, you need a rock-solid understanding of the vocabulary the ACT uses. These four concepts form the foundation of every statistics and probability question you will encounter.

Mean (Average)

Add all values in a data set, then divide by the count of values. The mean is sensitive to extreme values (outliers) and represents the "balance point" of the data.

Median

The middle value when all data points are sorted in order. If the count is even, it's the average of the two middle values. The median resists outliers better than the mean.

Mode & Range

The mode is the value that appears most often; a set can have multiple modes or none. The range is simply the largest value minus the smallest, measuring overall spread.

Probability

The likelihood of an event occurring, expressed as a fraction, decimal, or percentage between 0 (impossible) and 1 (certain). P(event) = favorable outcomes ÷ total outcomes.

✦ KEY TAKEAWAY

Key Takeaway

Section 3

Visualizing Data: Mean, Median & Distribution

A picture can reveal patterns that raw numbers hide. The diagram below shows a small data set plotted on a number line, with the mean and median marked so you can see how they relate to the spread of values. Notice that the mean gets "pulled" toward higher values while the median stays anchored near the center of the sorted data.

Number line showing data set values 2, 3, 4, 5, 5, 6, 7, 12 with mean at 5.5 and median at 5 marked

In the data set {2, 3, 4, 5, 5, 6, 7, 12}, the median is 5 — the average of the 4th and 5th values when sorted, which are both 5. The mean is 5.5, computed as (2 + 3 + 4 + 5 + 5 + 6 + 7 + 12) ÷ 8 = 44 ÷ 8 = 5.5. Notice that the outlier value of 12 pulls the mean above the median. This is exactly the kind of insight the ACT tests — an unusually large or small value shifts the mean more than the median.

Section 4

Essential Formulas for the ACT

The ACT doesn't give you a formula sheet, so these must be committed to memory. The good news is that there are only a handful of key formulas for statistics and probability, and each one is straightforward once you understand the logic behind it.

Mean (Arithmetic Average)

Mean = (x₁ + x₂ + … + xₙ) / n

Sum every value in the set, then divide by the total number of values, n.

An extremely useful rearrangement of this formula is Sum = Mean × n. The ACT loves questions where you know the mean and the count, and you need to find the total sum — or where a new value is added and you must calculate the new mean. Being comfortable going back and forth between sum, mean, and count is the single most valuable statistics skill on the test.

Probability of a Single Event

P(A) = (favorable outcomes) / (total outcomes)

Always count carefully: favorable outcomes are the specific results you want; total outcomes are all possible results.

Probability of Two Independent Events Both Occurring

P(A and B) = P(A) × P(B)

Events are independent when the outcome of one does not affect the other (e.g., flipping a coin twice).

Probability of Either Event Occurring (OR)

P(A or B) = P(A) + P(B) − P(A and B)

Subtract the overlap to avoid double-counting outcomes that satisfy both events.

✦ KEY TAKEAWAY

Key Takeaway

Section 5

Probability in Depth: Counting & Combinations

Many ACT probability questions require you to correctly count the total number of possible outcomes. The two most common counting tools are the Fundamental Counting Principle and combinations. The diagram below illustrates the counting principle with a classic example: choosing an outfit from a set of shirts and pants.

Tree diagram showing 3 shirts times 2 pants equaling 6 total outfit combinations

The Fundamental Counting Principle states that if you have m choices for one decision and n choices for a second decision, the total number of combined outcomes is m × n. This extends to any number of decisions: just multiply all the counts together. If you also had 4 pairs of shoes, the total outfits would be 3 × 2 × 4 = 24.

When the ACT asks you to choose a group where order doesn't matter — like picking 3 students from a club of 10 — you use a combination. The formula is:

Combinations

C(n, r) = n! / (r! × (n − r)!)

n = total items, r = items chosen. Remember: n! means n × (n−1) × (n−2) × … × 1.

For example, choosing 3 students from 10: C(10, 3) = 10! / (3! × 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120. The trick to making this manageable is canceling the (n − r)! from the numerator — you only need to multiply the top r numbers of n!, then divide by r!.

Probability Scale

Impossible

Equally Likely

Certain

0.5 — Equally Likely

0 — Impossible1 — Certain

Section 6

Worked Example: A Complete ACT-Style Problem

Let's walk through a multi-step problem that combines statistics and probability — exactly the kind of question that appears in the middle-to-late portion of the ACT Math section.

Problem StatementA student has test scores of 78, 85, 90, 72, and 95. She wants her mean score after 6 tests to be exactly 85. What score must she earn on the 6th test? If she then randomly selects 2 of her 6 test papers from a folder, what is the probability that both papers she selects have scores above 85?

Step 1 — Find the current sumSum of 5 tests = 78 + 85 + 90 + 72 + 95 = 420

Step 2 — Find the required total sumShe needs a mean of 85 over 6 tests, so the required total sum is 85 × 6 = 510

Step 3 — Find the needed scoreNeeded score = 510 − 420 = 90

Step 4 — List the final 6 scoresThe six scores are: 72, 78, 85, 90, 90, 95. Scores above 85 are: 90, 90, and 95 — that's 3 scores out of 6.

Step 5 — Calculate the probabilityShe picks 2 papers without replacement. The number of ways to choose 2 from 6 is C(6, 2) = 15. The number of ways to choose 2 from the 3 papers above 85 is C(3, 2) = 3. So the probability = 3/15 = 1/5.

Final AnswerShe needs a score of 90 on the 6th test, and the probability of selecting two papers above 85 is 1/5 (or 0.2).

Section 7

Mean vs. Median: When to Use Which

On the ACT, you might be asked which measure of center "best represents" a data set. The answer depends on the shape of the data and whether outliers are present. The table below compares the two measures across several scenarios.

Feature	Mean	Median
Calculation	Sum ÷ count	Middle value of sorted data
Sensitive to outliers?	Yes — pulled toward extreme values	No — resistant to extremes
Symmetric data	Mean ≈ Median	Mean ≈ Median
Right-skewed data (e.g., incomes)	Mean > Median	Median better represents "typical" value
Left-skewed data	Mean < Median	Median still more stable
Best used when…	Data is roughly symmetric, no extreme outliers	Data is skewed or has outliers

✦ KEY TAKEAWAY

Key Takeaway

Section 8

Beyond the ACT: Connecting to Advanced Topics

The probability and statistics skills you're learning now are the gateway to more powerful tools you'll encounter in AP Statistics or college courses. The table below shows how each ACT-level concept extends into more sophisticated territory.

ACT Concept	Advanced Extension	Real-World Use
Mean & median	Weighted means, trimmed means, standard deviation	GPA calculations, economic indicators
Basic probability	Conditional probability, Bayes' theorem	Medical testing accuracy, spam filters
Counting principle	Permutations, combinations with repetition	Password security, tournament brackets
Data interpretation	Regression analysis, correlation coefficients	Predicting sales trends, climate modeling
Range	Interquartile range (IQR), variance	Quality control in manufacturing

One particularly important extension is the concept of standard deviation, which measures how spread out data is around the mean. While the ACT rarely asks you to calculate standard deviation from scratch, it sometimes asks you to interpret what a "larger" or "smaller" standard deviation means. A data set where every value is close to the mean has a small standard deviation; a set with values far from the mean has a large one. Knowing this distinction can earn you a quick point on test day.

Section 9

Practice Problems

Test your understanding with these five ACT-style questions, arranged from conceptual to challenging. Try each one before revealing the answer.

A data set contains 5 values. The mean of the set is 50 and the median is 50. A sixth value of 200 is added to the data set. Which of the following must be true?

A student's quiz scores are: 14, 17, 15, 19, 15, 16. What is the mean of these scores?

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If you randomly draw one marble, replace it, then randomly draw a second marble, what is the probability that the first marble is red and the second marble is blue?

The mean score of 20 students on a math test is 82. The mean score of 30 students on the same test is 76. What is the mean score of all 50 students combined?

A jar contains 4 red chips and 6 blue chips. Two chips are drawn at random without replacement. What is the probability that at least one of the two chips drawn is red?