Explore — Unit Review

Unit 1 is about looking before you leap: understanding what your data are, how they were collected, and what an honest summary looks like — long before any inference. This review ties the five chapters together and drills the central skill: matching a question to the right display and summary.

The big picture

Exploring data is a short, repeatable workflow:

  1. Know your data — cases and variables, and each variable’s type (categorical or numerical). The type decides everything that follows.
  2. Know where it came from — study design. How the data were collected sets the ceiling on what you can conclude: an experiment with random assignment can support a cause-and-effect claim; an observational study can show association but is vulnerable to confounding. Random sampling governs how far results generalize.
  3. Summarize honestly — one variable at a time (visualizing distributions, describing distributions) and then relationships between two, choosing displays and numbers that fit the variable types and don’t mislead.

Which display and which summary?

This single table is the heart of Unit 1. The variables you have — how many, and what type — pick the row:

What you have Typical display Numerical summary
One categorical bar chart, pie/waffle counts, proportions
One numerical histogram, dotplot, boxplot center (mean/median), spread (SD/IQR), shape
Two categorical two-way table, stacked/side-by-side bars conditional proportions
Categorical + numerical side-by-side boxplots group means/medians
Two numerical scatterplot correlation \(r\), regression line

Mean vs. median, SD vs. IQR. For roughly symmetric data, the mean and standard deviation describe center and spread well. When the distribution is skewed or has outliers, prefer the median and IQR — they are resistant to extreme values.

A quick decision guide

  1. How many variables are in the question — one or two?
  2. What type is each — categorical or numerical?
  3. Read the matching row above for the display, then pick the summary; if a single numerical variable is skewed, switch from mean/SD to median/IQR.
  4. Separately ask the design question: experiment or observational? (causation vs. association) and how was the sample drawn? (generalizability).

Key ideas to carry forward

Case · variable · categorical vs. numerical · explanatory vs. response · observational study vs. experiment · random assignment (→ causation) · random sampling (→ generalizability) · confounding · distribution · center / spread / shape · skew · outlier · conditional proportion · correlation.

Common pitfalls

  • Association is not causation — only a randomized experiment licenses a cause-and-effect claim.
  • A representative sample is about how it was drawn, not how big it is — a huge convenience sample can still be biased.
  • Pie charts hide comparisons — bar charts are almost always easier to read.
  • The mean is pulled toward the tail — in skewed data it can misrepresent a “typical” value; report the median.
  • Correlation describes only linear association — a strong curved pattern can have \(r\) near 0.

Exercises

The Unit 1 skill is matching a question to the right display and summary. Answers are provided so you can check your work.

  1. Which display and which summary? For each scenario, name (i) an appropriate display and (ii) an appropriate numerical summary.

    1. The distribution of commute times (minutes) for 500 employees.
    2. The breakdown of students by major (six categories).
    3. Whether passing a course is related to attending a review session (each yes/no).
    4. How exam score relates to hours studied.
    5. Comparing salaries across three departments.

    Show answer
    Display Summary
    a histogram or boxplot median & IQR (commute times are usually right-skewed)
    b bar chart counts / proportions per major
    c two-way table, side-by-side or stacked bars conditional proportions (pass rate by attendance)
    d scatterplot correlation \(r\) (and a regression line)
    e side-by-side boxplots group medians (or means) and spreads

  2. Mean or median? SD or IQR? A dataset of household incomes is strongly right-skewed with a few very high values. Which measures of center and spread should you report, and why?

    Show answer

    Report the median and IQR. Both are resistant to the extreme high incomes; the mean and SD would be inflated by the long right tail and would overstate a “typical” household’s income.

  3. What can you conclude? A study finds that students who use a tutoring center have higher GPAs than those who don’t. Two designs are possible: (A) students chose whether to attend (observational); (B) students were randomly assigned to attend or not (experiment). For each, state whether a causal claim (“tutoring raises GPA”) is justified.

    Show answer

    (A) No — observational; motivated students may both attend tutoring and earn higher GPAs (confounding), so only association is supported. (B) Yes — random assignment balances other factors across groups, so a difference can be attributed to the tutoring itself.

  4. Spot the misleading choice. A report uses a pie chart to compare the market share of eight competing brands, several of which are close in size. Why is this a poor choice, and what’s better?

    Show answer

    Human eyes judge lengths far better than angles/areas, so near-equal slices are almost impossible to rank in a pie chart — especially with eight categories. A bar chart (ordered by size) makes the comparison immediate.

  5. Scope of inference. A campus survey is collected by polling students as they leave the gym one afternoon. To what population, if any, do the results generalize?

    Show answer

    This is a convenience sample of gym-goers at one time — not a random sample of all students. Results may describe these respondents but do not reliably generalize to the whole student body (gym-goers likely differ systematically). Sample size doesn’t fix this; the sampling method does.