Mathfolis

Quantitative Variable Displays

Unit 1 · Exploring One-Variable Data

What AP Stats asks here

Quantitative data is summarized visually by dotplot, stem-and-leaf, histogram, or boxplot. The right display depends on sample size and goal — small datasets get dotplots that preserve every value; medium-to-large datasets get histograms or boxplots; multiple groups get side-by-side boxplots. AP problems often hinge on a bad bin-width choice obscuring distribution shape.

Display by sample size and goal

Small n (≤ 50)
dotplot, stem-and-leaf\text{dotplot, stem-and-leaf}
Medium / large n
histogram\text{histogram}
Compare groups
side-by-side boxplots\text{side-by-side boxplots}

Histogram bin width

Sensible target
5≤bins≤155 \le \text{bins} \le 15
Boundary convention
value at 160→[160,165)\text{value at } 160 \to [160, 165)
AP Tip: Stem-and-leaf plots preserve raw values while showing shape — but only stay readable for n ≤ 50. Always include a key like 7 | 2 = 72 so readers know the place value.
Caution: Histogram bars touch because the underlying variable is continuous. If you draw a histogram with gaps, you have communicated 'categorical', not 'quantitative'.
Type 1

Display choice by sample size

Match the display to the sample size and the question. Small n preserves individual values; large n shows shape; group comparisons need boxplots.

Example 1
For each scenario, pick the most informative display. (i) 25 students' test scores, teacher wants to spot individual outliers. (ii) Income distributions across 5 cities, n = 1000 each. (iii) Body weights for 5000 hospital patients. Which choice matches the three? (A) (i) histogram; (ii) histogram; (iii) histogram (B) (i) dotplot; (ii) side-by-side boxplots; (iii) histogram (C) (i) pie chart; (ii) side-by-side bar charts; (iii) histogram (D) (i) dotplot; (ii) histogram; (iii) dotplot

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Type 2

Read or construct a display

Pull a specific value or describe a constructed display. For boxplots from a five-number summary, draw the box between Q1 and Q3 with the median inside; whiskers extend to min and max in the absence of outliers.

Example 2
A dataset has min 4, Q1=10Q_1 = 10, median 15, Q3=22Q_3 = 22, max 35. Which best describes the boxplot? (A) Box from 4 to 35 with the median in the middle; symmetric distribution. (B) Box from 10 to 22 with the median at 15; right-skewed because the upper whisker (22→35) is longer than the lower whisker (4→10). (C) Box from 10 to 22 with the median at 22; left-skewed. (D) The five-number summary cannot be drawn as a boxplot without all 30 raw values.

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Type 3

Bin-width critique

Identify the failure mode of an extreme bin choice. Too few bins flatten the shape; too many bins look like noise. The 5–15 bin guideline keeps shape visible.

Example 3
Two histograms are drawn of the same 200-observation dataset. Histogram A uses bin width 1 and produces 80 bins; histogram B uses bin width 50 and produces 2 bins. Which critique is correct? (A) A is fine because more bins always show more detail. (B) B is fine because fewer bins are easier to read. (C) Both fail — A is too noisy (most bins 0–1 observations); B destroys shape information by aggregating into two height values. (D) Both are fine — bin width does not affect the conclusion.

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