Textbook: Interactive Statistics, 3rd ed. (Sullivan & Woodbury), Sections 3.1–3.5
Chapter Summary
Chapter 2 gave us pictures; this chapter gives us numbers that summarize a whole data set in a line or two. Measures of center — the mean, median, and mode — tell you where the “middle” is, and measures of spread — the range, variance, and standard deviation — tell you how scattered the values are around that middle. The standard deviation is the star of the chapter: it becomes the ruler for probability in Chapters 6 and 7, the building block of confidence intervals and tests, and the yardstick for deciding what’s “unusual.”
The chapter’s deeper lesson is about resistance. The mean and standard deviation use every value, so one extreme outlier can drag them far from the bulk of the data — they are not resistant. The median and interquartile range ignore extremes, so they stay honest when the data are skewed. That’s why you always pair your numbers with a graph from Chapter 2: if the histogram is skewed right (like household incomes, where a few very high earners pull the tail), report the median; if it’s roughly symmetric (like heights), the mean is fine. You’ll also meet the z-score, which converts any value into “how many standard deviations from the mean” — the universal language for comparing values from completely different data sets, like comparing a 78 on a hard exam to a 91 on an easy one.
Study Guide
Key terms and definitions
- Mean (x̄ for a sample, μ for a population) — the arithmetic average; uses every value.
- Median — the middle value of ordered data; resistant to outliers.
- Mode — the most frequent value.
- Range — maximum − minimum; quick but uses only two values.
- Variance (s² sample, σ² population) — the average squared deviation from the mean.
- Standard deviation (s, σ) — the square root of the variance, in the original units.
- Resistant — a statistic whose value isn’t much affected by extreme values (median, IQR). The mean and standard deviation are not resistant.
- Empirical Rule (68–95–99.7) — for bell-shaped data: ~68% within 1 SD, ~95% within 2 SDs, ~99.7% within 3 SDs of the mean.
- Chebyshev’s inequality — for any shape: at least (1 − 1/k²) of the data lies within k standard deviations of the mean.
- z-score — number of standard deviations a value is from the mean; z = (x − μ)/σ.
- Quartiles (Q1, Q2, Q3) — divide ordered data into fourths; Q2 is the median. Percentile — the value below which a given percent of data falls.
- Interquartile range (IQR) = Q3 − Q1; the spread of the middle 50%.
- Outlier fences: lower fence = Q1 − 1.5(IQR), upper fence = Q3 + 1.5(IQR). Values outside are potential outliers.
- Five-number summary — min, Q1, median, Q3, max; displayed in a boxplot.
Essential formulas
- Sample mean: x̄ = Σx ÷ n. Population mean: μ = Σx ÷ N.
- Sample variance: s² = Σ(x − x̄)² ÷ (n − 1). Population variance: σ² = Σ(x − μ)² ÷ N.
- Standard deviation: s = √s² (sample), σ = √σ² (population).
- z-score: z = (x − μ) ÷ σ (population) or z = (x − x̄) ÷ s (sample).
- Empirical Rule: P(μ − σ < X < μ + σ) ≈ 0.68; within 2σ ≈ 0.95; within 3σ ≈ 0.997.
- Chebyshev: proportion within k SDs ≥ 1 − 1/k² (e.g., k = 2 → at least 75%).
- IQR = Q3 − Q1. Fences: Q1 − 1.5(IQR) and Q3 + 1.5(IQR).
Step-by-step procedures
- Find quartiles: (1) order the data; (2) find the median (Q2); (3) Q1 is the median of the lower half, Q3 the median of the upper half (not including the median itself when n is odd, per the textbook’s method).
- Build a boxplot: (1) compute the five-number summary; (2) draw the box from Q1 to Q3 with a line at the median; (3) extend whiskers to the most extreme values inside the fences; (4) mark values outside the fences with an asterisk as outliers.
- Use the Empirical Rule: (1) confirm the data are roughly bell-shaped (check a histogram!); (2) compute μ ± 1σ, μ ± 2σ, μ ± 3σ for the 68/95/99.7% intervals.
Common mistakes to avoid
- Dividing by n instead of n − 1 when computing the sample variance/standard deviation.
- Applying the Empirical Rule to skewed data — it’s only for bell-shaped distributions (use Chebyshev otherwise).
- Thinking a large z-score is automatically “wrong” — it just means unusual; |z| > 2 is uncommon, |z| > 3 is very unusual.
- Reporting the mean for skewed data without the median. Always check shape first.
Self-check: I can…
- [ ] Compute and interpret mean, median, and mode, and explain which to use when.
- [ ] Compute sample and population variance and standard deviation.
- [ ] Explain resistant vs. non-resistant statistics with examples.
- [ ] Apply the Empirical Rule and Chebyshev’s inequality appropriately.
- [ ] Compute z-scores and use them to compare values or flag unusual ones.
- [ ] Find quartiles, the IQR, and build a boxplot with outlier fences.
StatCrunch How-To
- Summary statistics: Stat > Summary Stats > Columns — select your column, then click “Statistics” to choose mean, median, std. dev., variance, min, max, Q1, Q3, IQR, and more. Click Compute.
- Boxplot: Graph > Boxplot — select the column; check “Draw boxes horizontally” if you like. Outliers appear as dots.
- Histogram check: Graph > Histogram before using the Empirical Rule — confirm the bell shape first.
- z-scores for a whole column: Data > Compute > Expression — enter
(ColumnName - mean(ColumnName)) / stdev(ColumnName)(use the sample values from Summary Stats, or the built-in functions) and name the new column “z-scores”.
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