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1. Types of Data & Variables

  • Quantitative vs. Qualitative: Think of quantitative as “numbers you can crunch” (e.g., height in inches, number of cars). Qualitative is “descriptive” (e.g., favorite color, type of pet).

  • Discrete vs. Continuous: Discrete data is counted in whole numbers (e.g., 3 siblings, not 3.5). Continuous can take any value within a range (e.g., a person’s weight could be 150.7 lbs).

  • Tip: If you’re asked to classify data, check if it’s measurable (quantitative) or descriptive (qualitative), then decide if it’s countable (discrete) or measurable with decimals (continuous).

2. Sampling Methods & Bias

  • Sampling Methods: Random is the gold standard because it minimizes bias. Systematic (e.g., picking every 10th person) is easier but risks missing patterns. Stratified ensures representation from all groups (e.g., sampling from each grade level). Cluster is faster but less precise since you’re sampling whole groups (e.g., entire classrooms).

  • Bias: Response bias comes from how questions are asked (e.g., leading questions). Nonresponse bias happens when certain people don’t respond (e.g., busy people skipping surveys). Selection bias is when your sample doesn’t reflect the population (e.g., surveying only morning shoppers).

  • Tip: When evaluating a study, check if the sampling method matches the population and watch for bias clues in the setup.

3. Descriptive Statistics

  • Mean, Median, Mode: Mean is the average, sensitive to outliers (e.g., one huge value skews it). Median is the middle value, great for skewed data. Mode is the most common, useful for categorical data.

  • Range and Standard Deviation: Range is simple (biggest minus smallest). Standard deviation (σ) tells you how spread out data is—small σ means data huddles close to the mean, large σ means it’s all over the place.

  • Tip: For standard deviation, you don’t need to calculate it by hand for Regents, but know that it measures variability. Use a calculator if given raw data.

4. Normal Distribution & z-Scores

  • Normal Distribution: Picture a bell curve—most data clusters near the mean. The Empirical Rule (68-95-99.7) tells you how much data lies within 1, 2, or 3 standard deviations.

  • z-Scores: The formula ( z = \frac{x - \mu}{\sigma} ) shows how far a value ( x ) is from the mean ( \mu ) in terms of standard deviations ( \sigma ). Positive z means above average, negative means below.

  • Tip: z-scores let you compare data across different scales (e.g., test scores from different exams). Practice plugging numbers into the formula.

5. Correlation & Regression

  • Correlation Coefficient (r): Ranges from -1 to 1. Close to 1 or -1 means a strong linear relationship (positive or negative). Near 0 means no linear relationship.

  • Regression Line: The equation ( \hat{y} = a + bx ) predicts ( y ) (dependent variable) from ( x ) (independent variable). Slope ( b ) shows how much ( y ) changes per unit of ( x ).

  • Residuals: Difference between actual and predicted values. Positive means the point is above the line, negative means below.

  • Tip: For Regents, you’ll often interpret ( r ) or use a given regression line to predict values. Check if residuals are positive or negative to understand fit.

6. Probability Basics

  • Simple Probability: It’s just favorable outcomes over total outcomes. For example, rolling a 4 on a die: ( P(4) = \frac{1}{6} ).

  • Complement: The chance something doesn’t happen (e.g., not rolling a 4: ( 1 - \frac{1}{6} = \frac{5}{6} )).

  • Mutually Exclusive and Independent: Mutually exclusive events can’t happen together (e.g., rolling a 3 or 4). Independent events don’t affect each other (e.g., flipping a coin twice).

  • Conditional Probability: Probability of A given B happened. Use the formula when given joint and marginal probabilities.

  • Tip: Draw Venn diagrams or tables for “or” and “and” problems to keep track.

7. Probability Distributions

  • Binomial Probability: Used for scenarios with two outcomes (success/failure) over multiple trials (e.g., flipping a coin 5 times). The formula ( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} ) needs the number of trials ( n ), successes ( k ), and success probability ( p ).

  • Expected Value: Think of it as the “average” outcome over many trials. For binomial, it’s ( E(X) = n \cdot p ).

  • Tip: Use a calculator for binomial calculations (many have built-in functions). Focus on setting up the problem correctly.

8. Inference & Confidence Intervals

  • Margin of Error (MOE): How much your sample mean might differ from the true population mean. Bigger samples = smaller MOE.

  • Confidence Interval: Gives a range where the true mean likely lies. For example, a 95% CI of 50 ± 3 means you’re 95% confident the true mean is between 47 and 53.

  • Tip: Higher confidence (e.g., 99% vs. 95%) widens the interval because you’re being more cautious.