Before You Begin

Arithmetic — averages and sums
Fractions & decimals — probability is expressed as fractions/decimals

📊 Statistics & Probability

📚 High School / AP Statistics 🎯 Key Concepts: Descriptive Stats, Probability Rules, Distributions

Why study this?

Statistics is how we make sense of data in an uncertain world, from medical research to election polling. Understanding statistics helps you critically evaluate claims.

Think of it intuitively

Statistics is the art of finding truth in data. But numbers mislead easily: one billionaire joining a group of average workers makes the mean salary enormous, but the median barely changes. The average is easily skewed by outliers.

Common Misconception

Common mistake: The mean (average) is always the best measure of center.

Reality: When data has outliers, the median is more representative. Always ask: which measure best represents this data?

1. Descriptive Statistics

Descriptive statistics summarize and describe a data set. The key measures are measures of center and measures of spread.

Measures of Center

Mean (average): \(\bar{x} = \dfrac{\sum x_i}{n}\) — sum of all values divided by the count
Median: the middle value when data is sorted. If n is even, average the two middle values.
Mode: the value that appears most often. A data set can have no mode, one mode, or multiple modes.

Data set: 4, 7, 7, 9, 11, 13, 15

Mean = \(\dfrac{4+7+7+9+11+13+15}{7} = \dfrac{66}{7} \approx 9.43\)
Median = 9 (4th value in sorted list)
Mode = 7 (appears twice)

Measures of Spread AP Exam

Range: maximum − minimum
Variance (\(\sigma^2\) or \(s^2\)): average of squared deviations from the mean
Standard deviation (\(\sigma\) or \(s\)): square root of variance — typical distance from the mean
IQR (Interquartile Range): Q3 − Q1 — spread of the middle 50% of data

Population Standard Deviation: \[ \sigma = \sqrt{\frac{\sum(x_i - \mu)^2}{N}} \] Sample Standard Deviation: \[ s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}} \]

Five-Number Summary and Box Plots

The five-number summary consists of: Minimum, Q1, Median (Q2), Q3, Maximum. A box plot displays these five values graphically.

Outlier Rule: A data point is an outlier if it falls below \(Q1 - 1.5 \times IQR\) or above \(Q3 + 1.5 \times IQR\).

2. Probability Fundamentals

Probability measures how likely an event is to occur. It ranges from 0 (impossible) to 1 (certain).

\[ P(A) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}} \]

Complement Rule

\[ P(A^c) = 1 - P(A) \]

Addition Rule AP Exam

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

If A and B are mutually exclusive (cannot both occur): \(P(A \cup B) = P(A) + P(B)\)

Multiplication Rule

\[ P(A \cap B) = P(A) \times P(B|A) \]

If A and B are independent: \(P(A \cap B) = P(A) \times P(B)\)

❌ Classic Mistake — Independent vs. Mutually Exclusive

If two events cannot happen at the same time, they must be independent Mutually exclusive events (P(A∩B)=0) are actually dependent — knowing A occurred tells you B didn't

Independent events: P(A|B) = P(A). Mutually exclusive events: P(A|B) = 0. These are opposite concepts, not the same.

Example: A bag has 4 red and 6 blue marbles. Probability of drawing 2 reds in a row (without replacement)?
\(P = \dfrac{4}{10} \times \dfrac{3}{9} = \dfrac{12}{90} = \dfrac{2}{15}\)

Conditional Probability

\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]

3. Counting Principles

Permutations — Order Matters

\[ P(n, r) = \frac{n!}{(n-r)!} \]

How many ways can 3 people finish 1st, 2nd, 3rd out of 8?
\(P(8,3) = \dfrac{8!}{5!} = 8 \times 7 \times 6 = 336\)

Combinations — Order Does Not Matter

\[ C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

How many ways to choose a 3-person committee from 8 people?
\(C(8,3) = \dfrac{8!}{3! \cdot 5!} = \dfrac{8 \times 7 \times 6}{6} = 56\)

4. Probability Distributions

Normal Distribution

The normal distribution is bell-shaped and symmetric about the mean. Described by mean (\(\mu\)) and standard deviation (\(\sigma\)).

Empirical Rule (68-95-99.7 Rule)

About 68% of data falls within 1σ of the mean
About 95% of data falls within 2σ of the mean
About 99.7% of data falls within 3σ of the mean

Z-Score (Standardized Score)

\[ z = \frac{x - \mu}{\sigma} \]

A test has mean 70 and standard deviation 10. A student scored 85. Find the z-score.
\(z = \dfrac{85 - 70}{10} = 1.5\) — the student is 1.5 standard deviations above the mean.

5. Practice Problems

Data: 3, 5, 5, 8, 10, 12. Find mean, median, and mode.
A fair die is rolled. What is the probability of rolling an even number?
A card is drawn from a standard 52-card deck. Find P(King or Heart).
How many 4-digit PIN codes are possible using digits 0–9 with no repetition?
A class has 10 students. How many ways can a team of 4 be selected?
IQ scores are normally distributed with μ = 100, σ = 15. What percent of scores fall between 85 and 115?

Answers

Mean = \(\dfrac{43}{6} \approx 7.17\); Median = \(\dfrac{5+8}{2} = 6.5\); Mode = 5
\(P = \dfrac{3}{6} = \dfrac{1}{2}\)
\(P(\text{K or H}) = \dfrac{4}{52} + \dfrac{13}{52} - \dfrac{1}{52} = \dfrac{16}{52} = \dfrac{4}{13}\)
\(P(10,4) = 10 \times 9 \times 8 \times 7 = 5{,}040\)
\(C(10,4) = 210\)
85 to 115 is within 1σ → approximately 68%

🔗 Bridge to Next Concept

How do you calculate probability for continuous data like height or weight?

Discrete probability uses counting, but continuous probability distributions require the area under a curve — which is exactly what integration (calculus) computes. That's the bridge from statistics to calculus.

Calculus

🔓 Master This to Unlock

Calculus — continuous probability, normal distribution integral Geometry — geometric probability, area models

Statistics is the foundation of data science and scientific reasoning. Mastering mean, variance, and probability now unlocks advanced topics like hypothesis testing, regression, and machine learning fundamentals.

Pre-Test Checklist

Variance = mean of squared deviations / Standard deviation = √variance
Addition rule: P(A∪B) = P(A)+P(B)−P(A∩B)
Independent events: P(A∩B) = P(A)×P(B)
Mean is sensitive to outliers; median is robust

🧠

Spaced Repetition — Ebbinghaus Curve

Review this material at increasing intervals to commit it to long-term memory.

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Calculus

The mathematics of change — calculus is the tool Newton invented to solve physics. The pinnacle of high school math.

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✓ Common Core State Standards aligned ✓ Reviewed Apr 2026 🔍 Accuracy verified Found an error? Let us know