Statistics & Probability

Strategy/Review

Measures of Central Tendency

Mean (average): Sum all values, divide by count. μ = (Σxᵢ) / n

The mean is sensitive to outliers. If Elon Musk walks into a room of 50 people, the mean net worth skyrockets, but nobody else got richer.

Median: Middle value when data is sorted. If even count, average the two middle values. The median is resistant to outliers — it's the "typical" value when data is skewed.

Mode: Most frequently occurring value. A dataset can have no mode, one mode (unimodal), or multiple modes (bimodal, multimodal).

When to Use Which:

• Symmetric data with no outliers → mean is fine.
• Skewed data or data with outliers → median is more representative.
• Categorical data → mode is the only option.

Example: Test scores: 72, 78, 81, 83, 85, 88, 92 Mean = (72+78+81+83+85+88+92)/7 = 579/7 ≈ 82.7 Median = 83 (4th value in sorted list) Mode = none (all unique)

Now add an outlier: 72, 78, 81, 83, 85, 88, 92, 23 Mean = 602/8 = 75.25 (dropped significantly) Median = (81+83)/2 = 82 (barely changed)

Range and Standard Deviation

Range: Max − Min. Simple but only uses two data points.

Standard deviation: Measures average distance from the mean. You don't need to compute it by hand on the DAT (the calculator is basic), but you MUST understand what it means:

• Larger SD = more spread out data.
• Smaller SD = data clustered tightly around the mean.
• SD is always ≥ 0; zero means all values are identical.

DAT-Level SD Questions: These are almost always conceptual. "Which dataset has the larger standard deviation?" or "If every value is increased by 5, what happens to the SD?" (Answer: the SD stays the same — adding a constant shifts everything but doesn't change spread.)

Weighted Average

Used when different values have different importance (weights).

Weighted Mean = (w₁x₁ + w₂x₂ + ... + wₙxₙ) / (w₁ + w₂ + ... + wₙ)

Example: Course grade: Homework (20%) = 85, Midterm (30%) = 78, Final (50%) = 92. Weighted average = (0.20×85 + 0.30×78 + 0.50×92) / (0.20+0.30+0.50) = (17 + 23.4 + 46) / 1.0 = 86.4

Probability Fundamentals

Basic Probability: P(A) = (number of desired outcomes) / (total number of possible outcomes)

All probabilities are between 0 and 1 inclusive. P(A) = 0 means impossible; P(A) = 1 means certain.

Example: A bag has 3 red, 4 blue, and 5 green marbles. P(drawing a red) = 3/12 = 1/4 = 0.25.

Independent vs. Dependent Events

Independent events: The outcome of one does NOT affect the outcome of the other. P(A and B) = P(A) × P(B)

Example: Flip a coin and roll a die. P(heads and 6) = (1/2) × (1/6) = 1/12.

Dependent events: The outcome of one AFFECTS the outcome of the other (typically sampling without replacement). P(A and B) = P(A) × P(B|A) where P(B|A) is the probability of B given that A occurred.

Example: Draw two cards from a deck without replacement. P(both aces) = (4/52) × (3/51) = 12/2652 = 1/221.

The Complement Rule

Sometimes it's easier to calculate the probability of something NOT happening.

P(A) = 1 − P(not A)

Example: Roll a die twice. P(at least one 6) = 1 − P(no sixes) = 1 − (5/6)² = 1 − 25/36 = 11/36.

This is FAR easier than calculating P(exactly one 6) + P(two sixes) directly.

"At Least One" Problems

The complement rule makes these trivial: P(at least one) = 1 − P(none)

Example: A family has 3 children. P(at least one girl) = 1 − P(all boys) = 1 − (1/2)³ = 1 − 1/8 = 7/8.

Mutually Exclusive vs. Non-Mutually Exclusive

Mutually exclusive: Events cannot happen simultaneously. P(A or B) = P(A) + P(B) (no overlap to subtract)

Not mutually exclusive: Events CAN happen together. P(A or B) = P(A) + P(B) − P(A and B) (subtract the overlap to avoid double-counting)

Example: Draw one card. P(heart or king) = P(heart) + P(king) − P(king of hearts) = 13/52 + 4/52 − 1/52 = 16/52 = 4/13.

Expected Value

Expected value = sum of (each outcome × its probability).

Example: A game: win $100 with probability 0.1, win $20 with probability 0.3, lose $10 with probability 0.6. EV = (100 × 0.1) + (20 × 0.3) + (−10 × 0.6) = 10 + 6 − 6 = $10. Positive expected value — good game to play!

Common Traps

Trap 1 — Mean vs. Median Confusion. "What's the average?" The DAT may mean mean, or it may mean "typical." If the data has extreme outliers, the median is the better measure, and answer choices may include both. Read the context.

Trap 2 — Adding Probabilities for "AND" Events. P(A and B) is multiplied (for independent events), not added. Addition is for "OR" events (with overlap adjustment).

Trap 3 — Forgetting the Overlap Subtraction. When events aren't mutually exclusive, failing to subtract P(A and B) double-counts the overlap. P(heart or king) ≠ 13/52 + 4/52 (that counts king of hearts twice).

Trap 4 — Assuming Sampling WITH Replacement. "Two marbles drawn from a bag" — unless stated otherwise, assume WITHOUT replacement. The second probability changes because the first marble is gone.

Trap 5 — Gambler's Fallacy. Past independent events don't affect future ones. After 5 heads in a row, the probability of heads on the next flip is STILL 1/2. The coin has no memory. The DAT sometimes baits students who think the next flip is "due" to be tails.