Probability and Sampling Distributions Cheat Sheet

Understanding probability distributions and sampling distributions is crucial for making inferences about populations based on sample data. These concepts help quantify uncertainty and form the basis of statistical hypothesis testing.

Core Principles

• Probability is the long-run relative frequency of an event, scaled from 0 (impossible) to 1 (certain).
• Probability distributions describe the likelihood of different outcomes for a variable.
• Discrete variables have distinct values; continuous variables can take any value within a range.
• Probability for continuous distributions is represented by the area under the curve.
• The total area under any probability distribution curve equals 1.
• The Normal Distribution (bell curve) is a common continuous distribution where 68% of data falls within 1 standard deviation of the mean, 95% within 2, and >99% within 3.
• Sampling distributions describe the distribution of a statistic (like the mean) calculated from repeated samples.
• Sampling distributions are theoretical but can be understood through math and simulations.
• The Central Limit Theorem states that sampling distributions of the mean become approximately normal as sample size increases.
• The standard error of the mean measures the variability of sample means around the population mean.

Action Steps

• Identify the type of variable (discrete or continuous).
• Understand that probability is the area under the curve for continuous distributions.
• Recognize the properties of the Normal Distribution (68-95-99.7 rule).
• Calculate the z-score to standardize values.
• Use z-tables to find probabilities associated with z-scores.
• Apply the Central Limit Theorem to understand sampling distributions.
• Calculate the standard error of the mean to estimate variability.
• Use sampling distributions for hypothesis testing to assess uncertainty in estimates.

Formulas

• $z = \frac{y - \bar{y}}{s_y}$
• $ \sigma_{\bar{y}} \approx \frac{s_y}{\sqrt{n}} $
• $ s_y^2 = \frac{\sum_{i=1}^{n}(y_i - \bar{y})^2}{n - 1} $

Key Terms

• Probability: The measure of the likelihood that an event will occur, expressed as a number between 0 and 1.
• Probability Distribution: A function that describes the likelihood of obtaining the possible values that a random variable can assume.
• Discrete Variable: A variable that can only take on a finite number of values or a countably infinite number of values.
• Continuous Variable: A variable that can take on any value within a given range.
• Normal Distribution: A symmetric, bell-shaped probability distribution characterized by its mean and standard deviation.
• Z-score: A measure of how many standard deviations a particular data point is away from the mean.
• Sampling Distribution: A probability distribution of a statistic (e.g., sample mean) obtained from repeated sampling from a population.
• Central Limit Theorem: States that the sampling distribution of the sample mean approaches a normal distribution as the sample size gets larger.
• Standard Error of the Mean: The standard deviation of the sampling distribution of the sample mean.
• Hypothesis Testing: A statistical method used to determine if there is enough evidence in a sample of data to infer that the corresponding population is affected by the item in question.

Timeline

• February 3rd, 2026: Lecture on Probability Distributions and Sampling Distributions
• February 10th, 2026: Midterm Exam (Class time: 4:10-6pm)
• February 13th, 2026: Midterm Sleep Log Due
• February 24th, 2026: Assignment 5 Due; Reading Report 5 Due
• February 27th, 2026: Sleep Log 5 Due

People

• N/A: Instructor/TA (implied)