AP Statistics (AP Stats): Five?Number Summary and Outliers (1.5×IQR Rule)

AP Statistics – Five?Number Summary and Outliers (1.5×IQR Rule)

AP Statistics: Five-Number Summary and Outliers (1.5×IQR Rule) – Exam-Ready Study Guide

What This Is

The five-number summary (minimum, Q1, median, Q3, maximum) and the 1.5×IQR rule for outliers are fundamental tools for describing and analyzing the distribution of a quantitative dataset. On the AP exam, you’ll use these to summarize data, identify potential outliers, and justify conclusions about variability. For example, a school administrator might analyze student test scores to determine if a few unusually low scores (outliers) are skewing the class average, or a quality control manager might use the IQR to assess consistency in product dimensions.

Key Terms & Formulas

• Five-number summary: A set of five values that divide a dataset into four equal parts: Minimum (Min), Q1 (First Quartile), Median (Q2), Q3 (Third Quartile), Maximum (Max).
• IQR (Interquartile Range): IQR = Q3 – Q1. Measures the spread of the middle 50% of the data.
• Outlier (1.5×IQR Rule):
• Lower bound = Q1 – 1.5×IQR
• Upper bound = Q3 + 1.5×IQR
• Any data point below the lower bound or above the upper bound is considered an outlier.
• Boxplot (Box-and-Whisker Plot): A graphical display of the five-number summary. Whiskers extend to the smallest and largest values within 1.5×IQR of Q1 and Q3; outliers are plotted as individual points.
• TI-84: 1-Var Stats: Use STAT-CALC-1-Var Stats to compute the five-number summary (scroll down to see Min, Q1, Med, Q3, Max).
• TI-84: Boxplot: Enter data in L1, then 2ND-Y= (STAT PLOT)-Plot1-On-Boxplot (5th icon)-Xlist: L1-Zoom-ZoomStat (9).
• Resistant measure: A statistic (e.g., median, IQR) that is not strongly affected by outliers or skewness.
• Skewness and the five-number summary:
• Right-skewed: Median < Mean; longer right whisker.
• Left-skewed: Median > Mean; longer left whisker.
• Symmetric: Median-Mean; whiskers roughly equal.

Step-by-Step / Process Flow

How to solve a typical AP FRQ involving the five-number summary and outliers:

1. Compute the five-number summary:

2. Use 1-Var Stats on your TI-84 or calculate manually:

   • Median (Q2): Middle value (or average of two middle values for even n).
   • Q1: Median of the lower half (exclude Q2 if n is odd).
   • Q3: Median of the upper half (exclude Q2 if n is odd).
   • Min/Max: Smallest and largest data points.

3. Calculate the IQR and outlier bounds:

4. IQR = Q3 – Q1
5. Lower bound = Q1 – 1.5×IQR

6. Upper bound = Q3 + 1.5×IQR

7. Identify outliers:

8. List any data points below the lower bound or above the upper bound.

9. Interpret in context:

10. Example: "The outlier at 120 suggests one student scored significantly higher than the rest, which may inflate the class average."

11. Draw a boxplot (if required):

12. Label the five-number summary and outliers.

13. Whiskers extend to the smallest/largest non-outlier values.

14. Compare distributions (if multiple groups):

15. Compare medians (center), IQRs (spread), and outliers (unusual values).

Common Mistakes

• Mistake: Forgetting to exclude the median when calculating Q1/Q3 for an odd number of data points. Correction: For n = 9, the lower half is the first 4 values (not 5), and Q1 is the median of those 4.

• Mistake: Using the mean instead of the median to describe the center when outliers are present. Correction: The median is resistant to outliers; use it for skewed data or when outliers exist.

• Mistake: Mislabeling whiskers as extending to Min/Max instead of the outlier bounds. Correction: Whiskers extend to the smallest/largest values within 1.5×IQR of Q1/Q3; outliers are plotted separately.

• Mistake: Calculating IQR as (Max – Min)/2 or confusing it with range. Correction: IQR = Q3 – Q1, not the full range.

• Mistake: Ignoring units or context when interpreting outliers. Correction: Always state what the outlier means (e.g., "One tree is 5 meters taller than the rest, possibly due to better soil conditions").

AP Exam Insights

• FRQs often ask you to:
• Compute the five-number summary and IQR from a dataset (or use 1-Var Stats).
• Identify outliers using the 1.5×IQR rule and justify their impact (e.g., "The outlier may skew the mean upward").
• Compare two distributions using boxplots (e.g., "Group A has a higher median and less variability than Group B").

• Explain why the median/IQR are better than the mean/standard deviation for skewed data.

• Tricky distinctions:

• Outliers vs. influential points: Outliers are extreme values; influential points change a regression line significantly (not the same thing!).

• IQR vs. standard deviation: IQR measures spread of the middle 50%; SD measures spread around the mean. Use IQR for skewed data.

• Calculator pitfalls:

• 1-Var Stats gives the five-number summary, but you must scroll down to see Q1 and Q3.

• When drawing boxplots, turn off other plots (Y=-clear functions) to avoid errors.

• Common FRQ setup:

• "A biologist measures the heights of 20 plants. The five-number summary is Min=12, Q1=15, Med=18, Q3=22, Max=30. Identify any outliers and explain how they might affect the mean."
• "Two classes took the same test. Draw side-by-side boxplots and compare their distributions."

Quick Check Questions

1. Multiple Choice: A dataset has Q1 = 40, Q3 = 70, and Max = 120. Using the 1.5×IQR rule, which of the following is an outlier? (A) 10 (B) 20 (C) 110 (D) 120 Answer: (D) 120. Upper bound = 70 + 1.5×(70–40) = 115; 120 > 115.

2. FRQ Part: The five-number summary for a dataset is Min=5, Q1=12, Med=18, Q3=25, Max=40. (a) Calculate the IQR. (b) Identify any outliers. (c) Explain how the presence of outliers might affect the mean. Answers: (a) IQR = 25 – 12 = 13. (b) Lower bound = 12 – 1.5×13 = –7.5 (no outliers); Upper bound = 25 + 1.5×13 = 44.5; 40 is not an outlier. (c) If outliers existed, they would pull the mean away from the median (e.g., a high outlier would increase the mean).

3. Multiple Choice: Which of the following is not part of the five-number summary? (A) Mean (B) Median (C) Q1 (D) Maximum Answer: (A) Mean. The five-number summary includes Min, Q1, Med, Q3, Max.

Last-Minute Cram Sheet

1. Five-number summary: Min, Q1, Med, Q3, Max.
2. IQR = Q3 – Q1 (spread of middle 50%).
3. Outlier bounds: Q1 – 1.5×IQR (lower), Q3 + 1.5×IQR (upper).
4. Boxplot whiskers: Extend to non-outlier min/max.
5. TI-84: 1-Var Stats-Scroll for Q1/Q3.
6. TI-84: Boxplot-STAT PLOT-Boxplot icon-ZoomStat.
7. Median/IQR are resistant to outliers; mean/SD are not.
8. Right-skewed: Longer right whisker; Left-skewed: Longer left whisker.
9. Always label outliers on boxplots (plot as dots).
10. Compare distributions using medians (center) and IQRs (spread).