Random Sampling & Inferences
Use random sampling to draw inferences about a population. Compare two samples to support or refute conclusions.
D-S.1.1Larger random samples produce more reliable inferences about a population
Key Concepts — Sampling
Requirements for a Good Random Sample
- Random — every member of the population has an equal chance of being selected
- Representative — drawn from the full population, not a specific subgroup (avoid club members, volunteers, or convenience groups)
- Large enough — bigger samples reduce the effect of chance and produce more reliable results
How Sampling Affects Inferences
- A valid random sample lets you generalize to the whole population
- Biased samples lead to inaccurate predictions — the result reflects only the subgroup surveyed
- When comparing two surveys, use the one with the larger random sample
- To predict a population count: multiply the sample proportion by the total population size
Measures of Center & Variability
Calculate mean, median, mode, range, IQR, and MAD. Compare distributions using center and spread. Understand how skew affects which measure to use.
D-S.2Center: mean, median, mode · Variability: range, IQR (Q3−Q1), MAD · Skewed data → use median
Key Concepts — Center & Variability
Measures of Center
Describe the typical value of a data set.
- Mean — sum ÷ count (average); pulled by outliers
- Median — middle value when ordered; resistant to outliers
- Mode — most frequent value; can be none or multiple
Measures of Variability
Describe how spread out the data is.
- Range — max − min; affected by outliers
- IQR — Q3 − Q1; spread of the middle 50%; resistant to outliers
- MAD — average distance of each value from the mean; larger MAD = more spread
Skew, Center, and Choosing the Right Measure
- Right-skewed (tail stretches right): a few high values pull the mean up → mean > median → use median
- Left-skewed (tail stretches left): a few low values pull the mean down → mean < median → use median
- Symmetric: data is balanced, mean ≈ median → either measure works well
- Outliers pull the mean and inflate the range, but barely affect the median and IQR
Probability
Find theoretical and experimental probabilities of simple and compound events. Use complements. Understand probability boundaries.
D-S.3P(event) = favorable ÷ total · P(A and B) = P(A)×P(B) · P(not A) = 1−P(A) · 0 ≤ P ≤ 1
Key Concepts — Probability
Simple vs Compound Events
- Simple event — one outcome from one experiment (rolling a 4; drawing a red card)
- Compound event — two or more events combined (rolling AND flipping; drawing two cards in a row)
- For independent compound events: P(A and B) = P(A) × P(B)
- Events are independent if the result of one does not affect the other
Theoretical vs Experimental
- Theoretical — what should happen based on math: P = favorable outcomes ÷ total outcomes
- Experimental — what actually happened in a real trial: P = successes ÷ total trials
- The more trials you run, the closer experimental probability gets to theoretical probability
Complement (NOT Probability) & Probability Boundaries
- Complement: P(not A) = 1 − P(A) — the probability that an event does NOT happen
- Minimum probability = 0 — the event is impossible (e.g., rolling a 7 on a standard die)
- Maximum probability = 1 — the event is certain (e.g., rolling a number less than 10 on a standard die)
- All probabilities fall between 0 and 1, inclusive: 0 ≤ P(event) ≤ 1
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