This unit is the base of the course: what a statistical study is, how to display and summarize one variable, and how data should be collected. It is the heaviest unit on the exam (20–30% of multiple-choice questions). Expect to describe distributions in context, compute summary statistics and outlier fences, and judge sampling methods and experimental designs.
- statistical study
- Collecting data from a sample to answer an investigative question about a larger population. It is needed when the population is too large or too hard to measure completely.
- Example: A state can't time every adult's grocery trip, so a researcher times the trips of 400 randomly chosen adults.
- frequency and relative frequency tables
- A frequency table gives the count in each category of a categorical variable. A relative frequency table gives each category's proportion; percentages and ratios carry the same information.
- Example: Of 80 diners, 36 ordered pasta: frequency 36, relative frequency 36/80 = 0.45, or 45%.
- histogram
- Groups a quantitative variable's values into ordered intervals (bins). Each bar's height is the count or proportion in its bin; changing the bin width can change the picture.
- Example: Ages of 60 museum visitors in 10-year bins look unimodal; 5-year bins reveal a second peak of young children.
- mean
- The sum of the values divided by the number of values: x̄ = Σxᵢ/n for a sample. With the median, one of the two common measures of center.
- Example: Quiz scores 6, 8, 8, 9, 10: x̄ = 41/5 = 8.2.
- comparing distributions
- Comparing two or more distributions of the same variable means comparing shape, center, variability and unusual features, using comparative words (greater than, less than) and context.
- Example: Tomato plants given more light have a higher median height (24 cm vs 19 cm) and a larger IQR (8 cm vs 5 cm).
- census
- Recording information from every item or individual in the population, not just a sample.
- Example: A club with 45 members asks all 45 which night they prefer for meetings.
- sampling with and without replacement
- Without replacement, a selected unit is set aside and can't be chosen again. With replacement, it returns to the population and could be chosen again.
- Example: Drawing 3 raffle names and keeping them out is without replacement; putting each name back first is with replacement.
- well-designed experiment
- An experiment with comparison of at least two treatment groups (one may be a control), random assignment of treatments, replication, and direct control of extraneous sources of variation.
- Example: Twenty bean plants randomly assigned to two lamp types, 10 per lamp, all in identical pots, soil and water.
- population and sample
- The population is all items or individuals of interest (size N). The sample is the subset data come from (size n). One piece of information is a datum.
- Example: Population: all 2,300 employees of a hospital, N = 2,300. Sample: 150 randomly chosen employees who are surveyed, n = 150.
- bar chart and pie chart
- Displays for one categorical variable. Bar heights show each category's count or proportion; pie slice areas show each category's share of the whole (100%).
- Example: Favorite seasons of 120 students: bars of heights 42, 30, 24 and 24, or pie slices of 35%, 25%, 20% and 20%.
- stem-and-leaf plot
- A display that splits each value into a stem (leading digit or digits) and a leaf (usually the last digit), both in order, so every data value stays visible.
- Example: Scores 62, 67, 71, 75, 75 → stem 6 | leaves 2 7; stem 7 | leaves 1 5 5.
- median
- The middle value of the data ordered from smallest (minimum) to largest (maximum). With an odd n take the middle value; with an even n average the two middle values.
- Example: Four puppies weigh 3, 5, 8 and 12 pounds: the median weight is (5 + 8)/2 = 6.5 pounds.