Statistics and Probability: Sampling, Data Distributions, and Probability Models
What this domain tests
Statistics and Probability asks you to evaluate samples, compare data distributions using center and spread, and use theoretical and experimental probability to make predictions. Questions often require judging whether a conclusion is supported by the data.
Core rules
- A random sample gives every member of the population an equal chance of selection; it is more likely to represent the population than a convenience sample. Larger random samples generally produce more reliable estimates.
- Measures of center: mean, median, mode. The mean is affected by outliers; the median is resistant. Measures of spread include range, interquartile range (IQR), and mean absolute deviation (MAD).
- Compare distributions by both center and spread; two data sets can have the same mean but very different variability.
- Probability is a number from 0 to 1. Theoretical probability = number of favorable outcomes / total equally likely outcomes. Experimental probability = successes / trials.
- The complement rule: P(not A) = 1 − P(A).
- To predict a count, multiply the probability by the number of trials or the population size.
Worked example
A bag contains 4 red, 6 blue, and 2 green marbles. What is the probability of not drawing blue, and how many red marbles would you expect in 150 draws with replacement? Total marbles: 4 + 6 + 2 = 12. Blue: 6, so P(blue) = 6/12 = 1/2. P(not blue) = 1 − 1/2 = 1/2. P(red) = 4/12 = 1/3. Expected reds = 1/3 × 150 = 50. The experimental result may differ from 50 because probability predicts long-run frequency, not an exact count.
Traps
- Using counts instead of the total number of outcomes in a probability.
- Giving a probability greater than 1 or negative.
- Treating experimental probability as a guarantee for a small number of trials.
- Comparing two data sets only by their means while ignoring spread.
- Generalizing from a biased sample, such as surveying only students in one club.