Statistics answers questions about a population using data from a sample. This unit covers both halves of that: describing one variable with tables, graphs and summary statistics, and collecting data well, through random sampling and well-designed experiments, so the conclusions mean something. It is the largest unit on the exam.
A statistical study collects data from a sample to answer an investigative question about a population. Observational units carry variables, categorical or quantitative (discrete or continuous). A parameter describes a population; a statistic describes a sample.
Frequency tables give counts per category; relative frequency tables give proportions. Bar charts and pie charts display them, and both can be used to compare data sets on the same categorical variable.
Histograms, stem-and-leaf plots and dotplots show quantitative distributions in order. Describe a distribution by shape, center and variability, plus unusual features (outliers, gaps, clusters), always in context.
Center: mean and median. Position: quartiles and percentiles. Variability: range, IQR and standard deviation. Outliers lie more than 1.5 × IQR beyond a quartile (or more than 2 standard deviations from the mean). Median and IQR are resistant to outliers; mean, range and standard deviation are not.
A boxplot shows the five-number summary, with outliers marked separately. Skewed right usually means mean > median. Compare distributions by center, variability, shape and outliers. A z-score, (x − μ)/σ, gives a value's position in standard deviations from the mean.
An investigative question names the variable(s), the analysis (with the parameter and direction for a test or estimate), and the conclusions allowed. A census records everyone; an experiment imposes treatments; an observational study doesn't. Random selection allows generalizing to the population; random assignment allows cause-and-effect conclusions.
Random sampling methods are simple random, stratified, cluster and systematic, with or without replacement. Bias is a systematic error that pushes a statistic consistently too high or too low: voluntary response, undercoverage, nonresponse and response bias.
A well-designed experiment compares at least two treatments (one may be a control), randomly assigns treatments, uses replication and controls extraneous variables. Blinding and placebos limit bias. Designs include completely randomized, randomized block and matched pairs. Random assignment allows cause-and-effect conclusions.