Exploring One-Variable Data and Collecting Data

Notes

Starting with a question

Statistics starts with an investigative question that data can answer. A population is everyone you want to know about; a sample is the part you actually measure. A parameter describes a population; a statistic describes a sample.

• Categorical variables put individuals into groups (eye color). Quantitative variables are numbers you can average (height).

Describing a quantitative distribution

Always describe shape, center, variability and unusual features, in context.

• Shape: symmetric, skewed left (long tail left), skewed right, unimodal or bimodal.
• Center: mean $\bar x$ or median. Variability: standard deviation $s$, range, or IQR $= Q_3 - Q_1$.
• Outliers: below $Q_1 - 1.5\,\text{IQR}$ or above $Q_3 + 1.5\,\text{IQR}$.
• The median and IQR are resistant; the mean and standard deviation are pulled by skew and outliers. In right-skewed data, the mean is usually greater than the median.
• A z-score $\displaystyle z = \frac{x - \bar x}{s}$ says how many standard deviations a value is from the mean.

Graphs

• Categorical: bar charts, pie charts. Quantitative: dotplots, stemplots, histograms, boxplots (five-number summary: min, $Q_1$, median, $Q_3$, max).
• When comparing groups, use comparison words: "The median for group A is about 5 points higher than group B."

Collecting data

• Simple random sample (SRS): every group of size $n$ is equally likely. Stratified: SRS within each similar group (stratum). Cluster: randomly pick whole groups. Systematic: every $k$th individual from a random start.
• Bias: voluntary response, convenience, undercoverage, nonresponse, and response bias (wording or lying).
• Observational study: observe without assigning treatments; shows association only. Experiment: randomly assign treatments; can show cause and effect.
• Good experiments use comparison, random assignment, replication and control. Blocking groups similar units first to reduce variability. Double-blind means neither subjects nor evaluators know who got which treatment.