First-year statistics study guide: the ideas that matter

What a first-year statistics course covers

Most introductory courses, whether in a math, business, psychology or life science department, follow the same arc. They start with describing data, move to probability and distributions, then spend the second half on inference: using a sample to say something about a population. The table shows the main units and what you should be able to do in each.

Formulas are usually given on a formula sheet. What is not given is knowing which procedure fits a question and what the result means. That is where to put your effort. Our statistics deck drills definitions and interpretations, and the statistics foundations course walks through each unit with short lessons.

Distributions and the normal model

A z-score tells you how many standard deviations a value sits from the mean: z = (x - μ)/σ. Once a quantity is roughly normal, z-scores turn every question into the same lookup.

The check line is worth copying into your own work. The 68-95-99.7 rule gives a rough answer in your head, which catches table-reading errors and sign mistakes before they cost you marks.

Sampling distributions and standard error

This is the unit that makes the rest of the course make sense, and the one students rush. Picture taking many random samples of 25 students and computing each sample's mean. Those means would not all match; they would cluster around the population mean with their own spread. That spread is the standard error, σ/√n for a mean and √(p(1 - p)/n) for a proportion.

• The standard error shrinks as the sample grows, but with √n: four times the sample size gives half the standard error.
• The central limit theorem says sample means are close to normal for large enough samples, even when the data are not. Small samples from skewed data are the exception to watch for.
• Standard deviation describes individual values. Standard error describes a statistic. Mixing the two up is one of the most common exam errors.

Confidence intervals: how to build and interpret one

The interpretation is where marks are won or lost. A correct version: "We are 95% confident that the mean monthly study time for all students at this school is between 69.1 and 75.7 hours." Here "95% confident" refers to the method. The NIST/SEMATECH engineering statistics handbook puts it plainly: an interval computed from a given sample either contains the true mean or it does not; if you repeated the sampling many times, about 95% of the intervals built this way would contain it.

Hypothesis tests and p-values, explained correctly

A test asks whether the data are surprising under a specific claim, the null hypothesis. The p-value is the probability of getting a result at least as extreme as the one observed, assuming the null hypothesis and the model are true.

In 2016 the American Statistical Association published a formal statement on p-values because they are misread so often, even in published research. Its principles are worth memorizing for any interpretation question:

• A p-value does not measure the probability that the null hypothesis is true, or the probability that the data arose by chance alone.
• A p-value does not measure the size of an effect or how important it is. A tiny, unimportant difference can give a small p-value in a huge sample.
• Decisions should not rest only on whether p crosses a threshold such as 0.05.
• A large p-value is not proof that the null hypothesis is true. It means the data are not strong evidence against it.

Greenland and colleagues (2016) list many more of these misreadings with examples, and it is a useful read before a final. A confidence interval often tells you more than the test: for the coin, a 95% interval for p runs from about 0.51 to 0.65, which shows both that 0.5 is outside it and how unfair the coin might plausibly be.

Common misconceptions and how to study for the exam

• Correlation means causation. A randomized experiment is the strongest basis for a causal claim. Observational data alone show association, not cause.
• A bigger sample fixes bias. It shrinks random error, not a biased sampling method. A large voluntary online poll is still a voluntary poll.
• Statistically significant means important. Check the effect size and the interval.
• Using z when you should use t. For a mean with an unknown population standard deviation, use t with n - 1 degrees of freedom.

For study, do problems and write the interpretation sentence in full every time, in context, naming the population and the parameter. Make flashcards for the interpretations, not just the formulas; our guide to making good flashcards shows how. If probability itself is the weak spot, the university probability course builds it up step by step.