Exploring One-Variable Data and Collecting Data
Describing a distribution
Always describe shape (roughly symmetric, skewed left/right, unimodal/bimodal, possible outliers), center (mean or median), and spread (range, IQR, standard deviation) - in that order. The mean is pulled toward skew and outliers; the median is resistant. For skewed data report median and IQR; for roughly symmetric data report mean and standard deviation. The five-number summary and boxplots display the IQR rule for outliers: a point is flagged if it lies more than 1.5×IQR below Q1 or above Q3. Standard deviation measures typical distance from the mean, so it is never negative and is affected by outliers.
Comparing groups and the normal model
When comparing two distributions, address differences in shape, center, and spread rather than just one number. The empirical (68-95-99.7) rule applies only to roughly normal distributions. Standardizing with z = (x − μ)/σ lets you compare values from different scales; a z-score states how many standard deviations a value sits from the mean.
Collecting data
Distinguish observational studies from experiments. Only a well-designed experiment supports causal conclusions; an observational study can reveal association but confounding variables may explain it. Random sampling (SRS, stratified, cluster, systematic) supports generalization to the population; random assignment to treatments supports causal inference. Bias sources - voluntary response, convenience, undercoverage, nonresponse, question wording - systematically miss the truth, unlike sampling variability, which shrinks as n grows.
Worked example. Data: 2, 4, 4, 5, 20. The median is 4 and Q1 = 3, Q3 = 5, so IQR = 2 and the upper fence is 5 + 3 = 8. The value 20 is an outlier; the median (4) resists it while the mean (7) is dragged upward.