Textbook: Interactive Statistics, 3rd ed. (Sullivan & Woodbury), Sections 2.2–2.4
Chapter Summary
Raw data is just a pile of numbers — this chapter turns it into pictures and tables that reveal the story. You’ll organize quantitative data into frequency distributions (tables counting how many values fall in each class) and then display them as graphs: histograms, frequency polygons, ogives, stem-and-leaf plots, and dotplots. You’ll also learn how graphs can lie: Section 2.4 shows the tricks (truncated axes, 3D effects, misleading pictographs) that make small differences look huge.
The skill that pays off all semester is learning to read a distribution’s shape. Is it symmetric and bell-shaped, skewed left, skewed right, or uniform? That shape tells you which average to trust in Chapter 3 (the mean gets dragged toward a skew’s tail), whether the normal model of Chapter 7 applies, and even which test to pick later on. Every time you meet new data this semester, your first move should be the same: make a graph and look at the shape. A concrete example: the ages of 40 students in a night class might form a histogram skewed right, with most students in their 20s and a long tail of students in their 40s and 50s — that tail immediately warns you the mean age will overstate what a “typical” student looks like.
Study Guide
Key terms and definitions
- Frequency distribution — a table listing classes (categories or intervals) and the number of observations in each.
- Relative frequency — class frequency ÷ total observations; cumulative frequency — running total of frequencies up to and including the current class.
- Class width — the difference between consecutive lower class limits.
- Histogram — bars for quantitative data where bars touch (no gaps), height = frequency.
- Frequency polygon — line graph connecting class midpoints at their frequencies.
- Ogive — cumulative frequency polygon; useful for finding percentiles.
- Stem-and-leaf plot — displays every data value while showing shape (stem = leading digits, leaf = trailing digit).
- Dotplot — one dot per observation stacked above the number line.
- Distribution shape: symmetric, skewed left (tail to the left), skewed right (tail to the right), uniform (flat).
- Bar chart vs. histogram: bar charts are for qualitative data with gaps between bars; histograms are for quantitative data with touching bars.
Essential formulas
- Class width = (largest value − smallest value) ÷ (number of classes), then round up to a convenient number.
- Relative frequency = f ÷ n, where f is the class frequency and n is the total number of observations.
Step-by-step procedures
- Build a frequency distribution: (1) choose the number of classes (usually 5–15); (2) compute class width and round up; (3) set the lower limit of the first class (at or below the smallest value); (4) list classes and tally frequencies; (5) add relative and cumulative frequency columns if needed.
- Describe a distribution’s shape: look at the histogram — symmetric (mirror image), skewed left (long tail left, like exam scores with a few low outliers), skewed right (long tail right, like incomes), or uniform (roughly equal bars).
- Spot a misleading graph: check whether the vertical axis starts at zero, whether 3D or pictures distort area, and whether the title/axis labels match the data.
Common mistakes to avoid
- Leaving gaps between histogram bars (that’s a bar chart — for qualitative data).
- Using the unrounded class width, which can leave the largest value with no class.
- Calling a distribution “skewed left” because the hump is on the left — skew is named for the tail, not the hump.
- Forgetting that a picture is not proof: graphs suggest patterns; tests in later chapters confirm them.
Self-check: I can…
- [ ] Build a frequency distribution from raw data with an appropriate class width.
- [ ] Draw and interpret a histogram, frequency polygon, and stem-and-leaf plot.
- [ ] Describe a distribution’s shape (symmetric, skewed left/right, uniform).
- [ ] Explain the difference between a histogram and a bar chart.
- [ ] Identify at least two ways a graph can mislead.
StatCrunch How-To
- Frequency table: Stat > Tables > Frequency — select the column; StatCrunch builds counts and relative frequencies.
- Histogram: Graph > Histogram — select the column, choose bins; compare with different bin widths to see how shape changes.
- Stem-and-leaf: Graph > Stem and Leaf — a quick way to see shape while keeping every value visible.
- Dotplot: Graph > Dotplot — useful for small data sets.
- Tip: to compare two groups (e.g., test scores for two sections), use Graph > Histogram and put the data “grouped by” the section column.
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