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AP Statistics 1.9 Comparing Distributions of a Quantitative Variable Study Notes

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LEARNING OBJECTIVE

  • Graphical representations and statistics allow us to identify and represent key features of data

Key Concepts:

  • Comparing Distributions of a Quantitative Variable

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Comparing Distributions of a Quantitative Variable

Purpose

In AP Statistics, when comparing two or more distributions, you must describe similarities and differences using a structured approach. This is often remembered by the acronym SOCS: Shape, Outliers, Center, Spread.

1. Shape

Look for symmetry, skewness, modality (number of peaks), clusters, and gaps.

  • Distributions can be:
    • Symmetric (bell-shaped, uniform).
    • Skewed right (longer right tail).
    • Skewed left (longer left tail).
    • Unimodal, bimodal, or roughly uniform.
  • When comparing → note whether one group is more skewed, more symmetric, or shows multiple modes.

2. Outliers

Report unusual observations (extremely high or low values) or clusters separated by gaps.

  • Identify using the IQR rule:

    Outlier if \( x < Q_1 – 1.5 \times IQR \) or \( x > Q_3 + 1.5 \times IQR \).

  • In comparisons → note if one group has more outliers than another, since outliers can distort measures of center and spread.

3. Center

Describe the typical value using median or mean.

  • Which group tends to have higher (or lower) values?
  • Choice of summary statistic depends on shape:
    • If symmetric with no outliers → use mean.
    • If skewed or has outliers → use median.

4. Spread (Variability)

Describe how spread out the data are.

  • Numerical summaries:
    • Range = max – min.
    • IQR = \( Q_3 – Q_1 \) → resistant measure of spread.
    • Standard deviation → sensitive to skew and outliers.
  • When comparing → state which distribution is more variable and by how much.

5. How to Write a Comparison

  • Always mention both groups explicitly (don’t describe them separately in isolation).
  • Example phrasing: “Group A has a higher median score than Group B, but Group B is more spread out with an outlier.”
  • A complete comparison uses all of SOCS: Shape, Outliers, Center, Spread.

Example 

Two classes took the same test. Their scores were summarized in boxplots:

  • Class A: Min = 55, Q1 = 65, Median = 75, Q3 = 85, Max = 95
  • Class B: Min = 50, Q1 = 60, Median = 70, Q3 = 80, Max = 100

Compare the two classes’ test score distributions using SOCS (Shape, Outliers, Center, Spread).

▶️ Answer / Explanation
  • Shape: Both appear roughly symmetric (median near the center of each box, whiskers of similar length). No clear skewness.
  • Outliers: None are indicated by the five-number summaries.
  • Center: Class A median = 75, Class B median = 70. Class A’s typical score is higher.
  • Spread:
    • Class A IQR = 85 – 65 = 20; Range = 95 – 55 = 40.
    • Class B IQR = 80 – 60 = 20; Range = 100 – 50 = 50.
    • Both have equal IQR, but Class B is more variable overall because its range is wider.

Final Comparison: Class A scored higher on average (median 75 vs 70), while Class B’s scores are more spread out with a wider range.

Example 

Students in two grade levels reported their daily screen time (in hours). The data were summarized:

  • Grade 9: Min = 1, Q1 = 2, Median = 3, Q3 = 4, Max = 8
  • Grade 12: Min = 2, Q1 = 3, Median = 4, Q3 = 5, Max = 12 (with an outlier at 12)

 Compare the distributions of screen time for Grade 9 and Grade 12 students using SOCS.

▶️ Answer / Explanation
  • Shape: Grade 9 appears slightly right-skewed (longer whisker to the right). Grade 12 is also right-skewed due to the high outlier at 12.
  • Outliers: Grade 9 has none. Grade 12 has one outlier (12 hours).
  • Center: Grade 9 median = 3 hours, Grade 12 median = 4 hours. On average, Grade 12 students spend more time on screens.
  • Spread:
    • Grade 9 IQR = 4 – 2 = 2; Range = 8 – 1 = 7.
    • Grade 12 IQR = 5 – 3 = 2; Range = 12 – 2 = 10.
    • Both groups have equal IQRs, but Grade 12 has a larger overall range due to the outlier.

Final Comparison: Grade 12 students generally spend more time on screens (median 4 vs 3 hours), but their data are more variable and include an outlier.

Example 

Two track teams recorded their 5k race times (in minutes):

  • Team X: Min = 16, Q1 = 17, Median = 18, Q3 = 20, Max = 24
  • Team Y: Min = 15, Q1 = 18, Median = 21, Q3 = 25, Max = 35

Use SOCS to compare the 5k times of Team X and Team Y.

▶️ Answer / Explanation
  • Shape: Team X appears slightly right-skewed (longer whisker on the right). Team Y is strongly right-skewed (long upper whisker, possible high extreme value at 35).
  • Outliers: Not explicitly indicated, but 35 is unusually large and could be an outlier by IQR rule.
  • Center: Team X median = 18 minutes, Team Y median = 21 minutes. Team X is generally faster.
  • Spread:
    • Team X IQR = 20 – 17 = 3; Range = 24 – 16 = 8.
    • Team Y IQR = 25 – 18 = 7; Range = 35 – 15 = 20.
    • Team Y is more variable, with a wider spread of times.

Final Comparison: Team X is faster on average (median 18 vs 21) and more consistent (smaller spread). Team Y has more variation and slower typical times.

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