Statistics and Probability

Learning Objectives

• Calculate mean, median, mode, and range from raw data and from data displays
• Identify the effect of outliers and distribution changes on each measure of central tendency
• Interpret box plots, histograms, and frequency tables to answer statistical questions
• Read and interpret scatterplots, describe correlation direction and strength, and estimate from lines of best fit
• Calculate basic probability using favorable outcomes / total outcomes
• Apply the complement rule: P(not A) = 1 - P(A)
• Calculate compound probability for independent events: P(A and B) = P(A) x P(B)
• Apply the addition rule for mutually exclusive and overlapping events
• Compute conditional probability and distinguish it from joint probability
• Use the counting principle, permutations, and combinations to count arrangements and selections
• Distinguish correlation from causation in data interpretation contexts
• Calculate expected value as the sum of (outcome x probability) for each outcome

High-Yield Concepts

Measures of Central Tendency and Spread

The mean, median, and mode each describe the "center" of a data set differently:

A common ACT question type: "A student adds a new score to a data set. How does this affect the mean?" Set up the mean equation with n+1 terms, substitute the known mean and new score, and solve for the new mean.

Probability Foundations

The fundamental probability formula: P(event) = favorable outcomes / total possible outcomes. Probability always falls between 0 (impossible) and 1 (certain).

Key rules:

Counting Methods and Data Interpretation

Counting methods determine how many ways events can occur:

Counting Principle: if event A can happen in m ways and event B in n ways, both can happen together in m x n ways.

Permutations: ordered arrangements. P(n,r) = n! / (n-r)! counts how many ways to arrange r items from n options when order matters.

Combinations: unordered selections. C(n,r) = n! / (r! x (n-r)!) counts how many ways to choose r items from n when order does not matter.

Scatterplots describe relationships between two quantitative variables. A positive correlation means both variables increase together; negative correlation means one increases as the other decreases. A line of best fit (trend line) allows predictions. Interpolation (estimating within the data range) is more reliable than extrapolation (predicting outside the data range). Correlation never implies causation -- this is a distinct ACT concept tested in both Math and Science.

Study Strategy

Start with mean, median, mode, and range since these are the highest-frequency statistics topics and span the easiest question types. Drill missing-value problems using the algebraic setup (sum = mean x count).

Study box plots, histograms, and frequency tables as a cluster. These are data display topics that test your ability to read and extract information. Practice identifying median (second quartile), Q1, Q3, and IQR from a box plot.

Then study probability basics (fundamental formula, complement rule, independent events) as an integrated set. These three rules handle the majority of probability questions. Add dependent events, mutually exclusive events, and overlapping events after the basics are solid.

Study counting methods (counting principle, permutations, combinations) as a structured cluster. The key decision is always: does order matter? If yes, use permutations. If no, use combinations.

Cover scatterplots, line of best fit, and correlation as a reading-and-interpretation cluster. These tend to be medium-difficulty questions that reward careful reading of graphs.

Study conditional probability, expected value, and weighted average last. These are the highest-difficulty statistics topics and appear infrequently.

Common Mistakes

• Using the wrong measure of central tendency for the given data set (e.g., using mean when median is more appropriate for skewed data)
• Forgetting to sort data before finding the median, especially for even-count data sets (median = average of two middle values)
• Adding probabilities when they should be multiplied (adding for OR/mutually exclusive events, multiplying for AND/independent events)
• Using the permutation formula when combination is required (not recognizing that order is irrelevant)
• Dividing by the wrong denominator in conditional probability (using the full sample space instead of the conditioned subset)
• Confusing standard deviation (measured in original units) with variance (squared units)
• Reading the wrong quartile from a box plot (confusing Q1 and Q3, or mistaking the median for the mean)
• Interpreting a strong correlation as causation
• Forgetting to subtract the overlap in the addition rule for non-mutually-exclusive events
• Applying independent event probability rules to dependent events (drawing without replacement)

Exam Tips

• For mean problems, convert to totals immediately: if the mean of n numbers is m, their sum is m x n
• For median with an even count, always average the two middle values after sorting
• Use the complement rule whenever a problem includes "at least one," "at least two," or similar phrasing
• For counting problems, ask "does order matter?" -- if you are counting arrangements, use permutations; if selecting a group, use combinations
• For scatterplot line of best fit problems, identify the slope direction (positive or negative correlation) first, then estimate the y-value for a given x
• Expected value questions: multiply each outcome by its probability and sum all products
• For conditional probability: P(A given B) = P(A and B) / P(B); the denominator shifts to the conditioned event
• On data display questions, identify what the axes or rows/columns represent before answering any question