AP Statistics Overview
AP Statistics provides students with a comprehensive understanding of the major concepts and tools for collecting, analyzing, and drawing conclusions from data. This course prepares students for the AP Statistics exam and college-level statistics.
- Master probability concepts and distributions
- Understand statistical inference and hypothesis testing
- Develop skills in data analysis and interpretation
- Apply statistical methods to real-world problems
AP Statistics Standards
Probability Foundations
Standard 1: Finite Sample Spaces
Students solve probability problems with finite sample spaces by using the rules for addition, multiplication, and complementation for probability distributions. They understand the simplifications that arise with independent events.
P(at least one head in 3 tosses) = 1 - P(all tails) = 1 - (1/2)³ = 7/8
Standard 2: Conditional Probability
Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces. P(A|B) = P(A and B)/P(B)
If P(A) = 0.6, P(B) = 0.5, and P(A and B) = 0.3, then P(A|B) = 0.3/0.5 = 0.6
Standard 3: Discrete Random Variables
Students demonstrate an understanding of the notion of discrete random variables by using this concept to solve for the probabilities of outcomes, such as the probability of the occurrence of five or fewer heads in 14 coin tosses.
For discrete random variable X = number of heads in 3 tosses:
P(X = 0) = 1/8, P(X = 1) = 3/8, P(X = 2) = 3/8, P(X = 3) = 1/8
Distributions
Standard 4: Continuous Random Variables
Students understand the notion of a continuous random variable and can interpret the probability of an outcome as the area of a region under the graph of the probability density function associated with the random variable.
For a uniform distribution on [0,1], P(0.2 ≤ X ≤ 0.5) = area under f(x) = 1 from 0.2 to 0.5 = 0.3
Standard 5: Mean of Discrete Random Variable
Students know the definition of the mean of a discrete random variable and can determine the mean for a particular discrete random variable. μ = Σ[x · P(x)]
X = roll of fair die. Mean μ = (1+2+3+4+5+6)/6 = 3.5
Standard 6: Variance of Discrete Random Variable
Students know the definition of the variance of a discrete random variable and can determine the variance for a particular discrete random variable. σ² = Σ[(x - μ)² · P(x)]
X = roll of fair die. Variance σ² = [(1-3.5)² + (2-3.5)² + ... + (6-3.5)²]/6 ≈ 2.92
Standard 7: Standard Distributions
Students demonstrate an understanding of the standard distributions (normal, binomial, and exponential) and can use the distributions to solve for events in problems in which the distribution belongs to those families.
Binomial: X ~ B(10, 0.3) represents 10 trials with probability 0.3 of success
Normal: X ~ N(μ, σ) is symmetric bell-shaped curve
Exponential: Used to model time between events in Poisson process
Standard 8: Normal Distribution Parameters
Students determine the mean and the standard deviation of a normally distributed random variable.
If X ~ N(100, 15), then P(X < 115) ≈ 0.84 (using z-score z = (115-100)/15 = 1)
Standard 9: Central Limit Theorem
Students know the central limit theorem and can use it to obtain approximations for probabilities in problems of finite sample spaces in which the probabilities are distributed binomially.
Central Limit Theorem: Sample means of size n from any distribution approach normal as n increases
For n ≥ 30, the sampling distribution of x̄ is approximately N(μ, σ/√n)
Descriptive Statistics
Standard 10: Measures of Center and Spread
Students know the definitions of the mean, median, and mode of distribution of data and can compute each of them in particular situations.
Data: 2, 4, 4, 5, 7, 9
Mean = 31/6 ≈ 5.17, Median = (4+5)/2 = 4.5, Mode = 4
Standard 11: Variance and Standard Deviation
Students compute the variance and the standard deviation of a distribution of data.
Data: 2, 4, 4, 5, 7, 9
Standard deviation: σ = √[Σ(x-μ)²/n] ≈ 2.32
Standard 12: Least Squares Regression
Students find the line of best fit to a given distribution of data by using least squares regression. The regression line minimizes the sum of squared residuals.
For points (1,2), (2,3), (3,5), regression line ŷ = 0.5 + 1.5x minimizes Σ(y - ŷ)²
Standard 13: Correlation Coefficient
Students know what the correlation coefficient of two variables means and are familiar with the coefficient's properties. -1 ≤ r ≤ 1, where r measures strength and direction of linear relationship.
Correlation r = 0.85 indicates strong positive linear relationship
r = -0.92 indicates strong negative linear relationship
Standard 14: Organizing and Describing Data
Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line graphs and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.
Box-and-whisker plot displays: minimum, Q1, median, Q3, maximum
Histogram groups continuous data into bins and displays frequencies
Statistical Inference
Standard 15: Sampling Distributions
Students are familiar with the notions of a statistic of a distribution of values, of the sampling distribution of a statistic, and of the variability of a statistic.
If we take many samples of size n and compute x̄ for each, the distribution of these x̄ values is the sampling distribution
Standard 16: Sampling Distribution Properties
Students know basic facts concerning the relation between the mean and the standard deviation of a sampling distribution and the mean and the standard deviation of the population distribution. Standard error = σ/√n
If population has μ = 50, σ = 20, then sampling distribution of x̄ for n = 25:
Mean of x̄ = 50, Standard error = 20/√25 = 4
Standard 17: Confidence Intervals
Students determine confidence intervals for a simple random sample from a normal distribution of data and determine the sample size required for a desired margin of error.
95% Confidence Interval for mean with σ = 10, n = 100, x̄ = 50:
CI = 50 ± 1.96(10/√100) = 50 ± 1.96 = (48.04, 51.96)
Standard 18: P-values and Hypothesis Testing
Students determine the P-value for a statistic for a simple random sample from a normal distribution. The P-value is the probability of getting results at least as extreme as observed, assuming the null hypothesis is true.
Hypothesis test: H₀: μ = 45 vs Hₐ: μ ≠ 45
Test statistic: z = (50 - 45)/(10/√100) = 5
P-value < 0.001, so reject H₀ at α = 0.05
Standard 19: Chi-Square Test
Students are familiar with the chi-square distribution and chi-square test and understand their uses. Used to test independence in contingency tables and goodness of fit.
Chi-square test statistic: χ² = Σ[(Observed - Expected)²/Expected]
Used to test independence in a 2×2 contingency table with observed frequencies