Statistics and ProbabilityStatistics and Probability

SATMATH

Statistics and Probability

SAT Math Guide 6

Goal: Summarize data, interpret scatterplots and lines of best fit, compute probabilities, and judge claims about samples. Original study notes on standard SAT Math topics, with original examples. Not copied from any prep book. Big Picture

Data questions test interpretation more than calculation. Know what each statistic measures and what conclusions a study design supports.

Center and Spread

Mean = sum/number of values. Median = middle value when ordered. Mode = most frequent.

Range = max − min. IQR = Q3 − Q1. Standard deviation measures typical distance from the mean: larger means more spread. Outliers pull the mean toward them much more than the median. In a right-skewed set the mean is greater than the median. Adding the same number to every value shifts the mean and median but leaves the spread unchanged. Multiplying by k scales mean, median, and standard deviation by k.

Mean = total ÷ count. If the mean of 5 numbers is 12, their sum is 60.

Combined mean: weight by group size, not a simple average of the two means.

Scatterplots and Lines of Best Fit

The line of best fit is y = mx + b; interpret the slope as the predicted change in y for each one-unit increase in x, and the intercept as the predicted y when x = 0.

Correlation can be positive, negative, or none; it does not prove cause and effect.

Residual = actual value − predicted value.

Interpolation (inside the data range) is more reliable than extrapolation (outside it).

Probability

P(event) = favorable outcomes / total outcomes, between 0 and 1.

Complement: P(not A) = 1 − P(A).

Independent events: P(A and B) = P(A) × P(B).

Conditional probability: P(A | B) = P(A and B) / P(B). In a table, restrict to the given group first.

Cats Dogs Total

Freshmen 30 50 80

Seniors 45 25 70

Total 75 75 150

From the table: P(cat | senior) = 45/70 = 9/14 ≈ 0.64. P(senior and cat) = 45/150 = 0.30. They are different questions because the "given" changes the denominator.

Samples and Studies

A random sample supports conclusions about the whole population it came from, not about other populations. Random assignment to treatment groups supports cause-and-effect conclusions; observational studies do not. Margin of error: the estimate ± the margin gives a plausible range for the true value. A larger sample gives a smaller margin of error.

A biased sample (for example, surveying only people at one gym) cannot be generalized.

Check Yourself

Q: The data 2, 4, 4, 6, 14 have a mean of 6 and a median of 4. Why? A: The outlier 14 raises the mean. Q: A fair die is rolled once. What is P(a number greater than 4)? A: 2/6 = 1/3.

Q: Can a survey of students at one school describe all teenagers? A: Not reliably; the sample may not represent them. Key Terms

Term / Actor What it is and why it matters

Mean / median / mode Measures of center.

Standard deviation Typical distance from the mean.

Residual Actual minus predicted.

Conditional probability Probability given that another event occurred.

Margin of error Plus-or-minus range around a sample estimate.

Random assignment Assigning subjects to groups by chance; supports causal claims.

Making This Essay-Ready

Trap: Match the conclusion to the study design: random sample for population claims, random assignment for causation. Habit: Read the table headers before choosing the denominator.

SAT Math Guide 6 · Original SAT Math study notes