Statistics study guide
This guide is designed to help you understand the core ideas that commonly appear in introductory college Statistics courses. Read the explanation, follow the example, then try the practice questions without looking back. The goal is to understand the reasoning—not just memorize an answer.
Course requirements vary by school and instructor. Use this as a learning and review resource alongside your assigned textbook, lectures, and instructor guidance.
Descriptive statistics
Concept: Descriptive statistics organize and summarize data. The mean is the arithmetic average, the median is the middle ordered value, and the mode is the most frequent value. Range and standard deviation describe spread; quartiles and the interquartile range describe the middle half of the data.
Worked example: For 2, 4, 4, 6, 9: mean=25/5=5, median=4, mode=4, and range=9−2=7. An unusually large value would pull the mean upward more than the median.
Remember: Always order data before finding the median; use mean with standard deviation for roughly symmetric data; median and IQR are resistant to outliers.
Common mistakes: Using the number of gaps instead of number of observations for the mean; forgetting to sort for median; assuming mean and median must be equal.
Practice it
- Find mean and median of 3,5,8,8,11.
- Which measure is more resistant to an outlier: mean or median?
- Find the range of 12,18,7,21.
Probability basics
Concept: Probability measures how likely an event is, from 0 to 1. The complement rule is P(not A)=1−P(A). For independent events, multiply probabilities for “and”; for mutually exclusive events, add probabilities for “or.” Conditional probability updates the chance of an event given new information.
Worked example: A fair die has P(rolling 6)=1/6 and P(not rolling 6)=5/6. Two independent coin flips have P(two heads)=1/2×1/2=1/4.
Remember: Define the sample space; distinguish independent from mutually exclusive; use complements when “at least one” is easier to calculate indirectly.
Common mistakes: Adding when you should multiply; assuming events are independent without evidence; double-counting overlapping events.
Practice it
- What is P(not drawing a heart) from a standard deck?
- What is P(two tails) in two fair flips?
- If P(A)=0.35, find P(not A).
Normal distributions
Concept: A normal distribution is symmetric and bell-shaped. Its center is the mean, and the standard deviation controls spread. A z-score tells how many standard deviations a value is above or below the mean: positive z is above, negative z is below.
Worked example: If exam scores have mean 70 and standard deviation 10, a score of 85 has z=(85−70)/10=1.5, so it is 1.5 standard deviations above the mean.
Remember: Use z-scores to compare values on different scales; about 68% of normal data lies within 1 standard deviation of the mean; symmetry matters.
Common mistakes: Using variance instead of standard deviation in a z-score; reversing x−mean; assuming every dataset is normal.
Practice it
- Find z for x=60 when mean=50 and SD=5.
- About what percent lies within 2 SDs in a normal model?
- Is z=−1.2 above or below the mean?
Sampling & confidence intervals
Concept: A sample is used to learn about a population. Sampling variability means different random samples give different estimates. A confidence interval combines an estimate with a margin of error to give a plausible range for the population parameter. Larger samples usually reduce standard error and narrow the interval.
Worked example: A poll estimates support at 52% with a margin of error of 3 percentage points. A simple interval is 49% to 55%. The interval describes uncertainty in the estimation method, not a guarantee about one fixed parameter.
Remember: Random, representative sampling matters; higher confidence usually makes intervals wider; larger sample size usually makes them narrower.
Common mistakes: Saying 95% of individual observations lie in a 95% CI; treating convenience samples as representative; ignoring sample size.
Practice it
- If estimate=40 and margin=4, give the interval.
- What happens to interval width when sample size grows?
- Why can a biased sample give a misleading interval?
Hypothesis testing
Concept: Hypothesis testing compares a null hypothesis, usually “no effect/no difference,” with an alternative. A test statistic measures how far the sample result is from what the null predicts. The p-value is the probability, assuming the null model, of results at least as extreme as those observed.
Worked example: If α=0.05 and p=0.02, the result is statistically significant at the 5% level, so the null is rejected. This does not prove the alternative with certainty or measure practical importance.
Remember: State H0 and Ha before testing; compare p with α; statistical significance is not the same as practical importance.
Common mistakes: Saying p is the probability H0 is true; “accepting” H0 after a nonsignificant result; ignoring assumptions and effect size.
Practice it
- If p=.08 and α=.05, what decision is made?
- What does a small p-value indicate under H0?
- Why should effect size be considered too?
Correlation & regression
Concept: Correlation measures the direction and strength of a linear relationship between two quantitative variables. Regression models how an outcome changes with a predictor. The slope gives the predicted change in y for a one-unit increase in x, while residuals are observed minus predicted values.
Worked example: If ŷ=10+2x, increasing x by 1 raises the predicted y by 2. At x=5, predicted y=20. If observed y=23, the residual is 23−20=3.
Remember: Correlation ranges from −1 to 1; correlation does not establish causation; inspect scatterplots for nonlinear patterns and outliers.
Common mistakes: Claiming causation from correlation; extrapolating far outside the data; interpreting correlation as a slope.
Practice it
- For ŷ=4+3x, predict y when x=2.
- If observed=12 and predicted=10, find residual.
- Can r=0 still hide a nonlinear relationship?