Statistics and Probability
Master statistics and probability with this comprehensive flashcard deck covering every key concept from descriptive statistics to inferential analysis. Statistics is increasingly essential in every field — from medicine and economics to machine learning and social sciences — and it is a major component of AP Statistics, A-Level Mathematics, IB Mathematics, and undergraduate courses. This deck covers measures of central tendency and spread, probability rules, permutations and combinations, discrete and continuous probability distributions (including the normal distribution), hypothesis testing, confidence intervals, correlation and regression, and the central limit theorem. Based on publicly available resources including OpenStax Statistics and Khan Academy.
All Cards in This Deck
Question
What are mean, median, and mode?
Answer
Mean: Sum of values ÷ number of values (average). Median: Middle value when sorted (or average of two middle values if even count). Mode: Most frequently occurring value.
Question
What is range and interquartile range (IQR)?
Answer
Range: Maximum − Minimum. IQR: Q3 − Q1 (middle 50% of data). IQR is resistant to outliers, making it more reliable than range for skewed data.
Question
What is standard deviation (σ or s)?
Answer
A measure of how spread out data is from the mean. Population σ = √(Σ(x−μ)²/N). Sample s = √(Σ(x−x̄)²/(n−1)). Larger SD = more spread out.
Question
What is variance?
Answer
The square of the standard deviation. σ² (population) or s² (sample). Variance is in squared units; standard deviation brings it back to the original units.
Question
What is a box plot (box and whisker plot)?
Answer
A visual summary showing: minimum, Q1, median (Q2), Q3, maximum. The box spans Q1 to Q3 (IQR). Whiskers extend to min/max (or 1.5×IQR for outlier detection).
Question
What is the empirical rule (68-95-99.7 rule)?
Answer
In a normal distribution: 68% of data falls within 1 SD of the mean. 95% within 2 SDs. 99.7% within 3 SDs.
Question
What is a z-score and its formula?
Answer
z = (x − μ) / σ. It tells how many standard deviations x is from the mean. z=0 means at the mean. z=2 means 2 SDs above. Used to standardize and compare values.
Question
What is the normal distribution?
Answer
A symmetric, bell-shaped distribution defined by its mean (μ) and standard deviation (σ). The mean = median = mode. Described by: f(x) = (1/σ√2π) e^(−½((x−μ)/σ)²).
Question
What is probability and how is it measured?
Answer
The likelihood of an event occurring. P(event) = favorable outcomes / total possible outcomes. Always between 0 (impossible) and 1 (certain).
Question
What is the addition rule for probability?
Answer
P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events (cannot both occur): P(A or B) = P(A) + P(B).
Question
What are permutations and their formula?
Answer
Arrangements where order matters. P(n,r) = n! / (n−r)! — number of ways to arrange r items from n distinct items. Example: arranging 3 from 5 = 5!/2! = 60.
Question
What are combinations and their formula?
Answer
Selections where order does NOT matter. C(n,r) = n! / (r!(n−r)!) — also written as ⁿCᵣ or (n r). Example: choosing 3 from 5 = 10.
Question
What is a binomial distribution?
Answer
Models the number of successes in n independent trials, each with probability p of success. P(X=k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ. Mean = np. Variance = np(1−p).
Question
What is a Poisson distribution?
Answer
Models the number of events in a fixed interval when events occur at a constant mean rate λ. P(X=k) = (e⁻λ × λᵏ) / k!. Mean = λ. Variance = λ. Used for rare events.
Question
What is the Central Limit Theorem (CLT)?
Answer
If samples of size n are drawn from any population with mean μ and SD σ, the distribution of sample means approaches a normal distribution as n increases. Typically, n ≥ 30 is sufficient.
Question
What is a confidence interval?
Answer
A range of values that is likely to contain the true population parameter with a given confidence level. Example: 95% CI for a mean: x̄ ± z × (σ/√n).
Question
What is hypothesis testing?
Answer
A procedure to test a claim about a population parameter. Steps: 1. State H₀ (null) and H₁ (alternative). 2. Choose significance level α. 3. Calculate test statistic. 4. Find p-value. 5. Reject H₀ if p < α.
Question
What is a p-value?
Answer
The probability of observing results at least as extreme as the sample data, assuming H₀ is true. p < α (usually 0.05) → reject H₀. p ≥ α → fail to reject H₀.
Question
What is Type I and Type II error?
Answer
Type I (α): Rejecting H₀ when it is actually true (false positive). Type II (β): Failing to reject H₀ when it is actually false (false negative). Power = 1 − β.
Question
What is correlation (r) and what does it measure?
Answer
A measure of the linear relationship between two variables. r = +1: perfect positive correlation. r = −1: perfect negative correlation. r = 0: no linear correlation.
Question
What is the difference between correlation and causation?
Answer
Correlation means two variables tend to change together. Causation means one variable directly causes the other. Correlation does not imply causation — there may be confounding variables.
Question
What is linear regression and its equation?
Answer
A method to model the relationship between two variables with a straight line: ŷ = a + bx where b = slope and a = y-intercept. The least squares line minimizes the sum of squared residuals.