AP Statistics (AP Stats): Measures of Center (Mean, Median, Mode) and Their Resistances

AP Statistics – Measures of Center (Mean, Median, Mode) and Their Resistances

AP Statistics: Measures of Center (Mean, Median, Mode) and Their Resistances

Exam-Ready Study Guide

What This Is

Measures of center (mean, median, mode) summarize the "typical" value in a dataset, but they behave differently when outliers or skewness are present. The mean is the arithmetic average, the median is the middle value, and the mode is the most frequent value. On the AP exam, you’ll need to: - Choose the best measure of center for skewed vs. symmetric data. - Explain why one measure is more resistant (less affected by outliers) than another. - Use these concepts in real-world contexts (e.g., comparing salaries, analyzing test scores, or evaluating housing prices).

Example: A company claims its average salary is $80,000, but most employees earn $50,000. The median ($50,000) better represents typical earnings because the mean is inflated by a few high earners (outliers).

Key Terms & Formulas

• Mean (x? or ?):
• Formula: x? = (?x?) / n (sample mean) or ? = (?x?) / N (population mean).
• x? = individual data points, n = sample size, N = population size.

• Calculator: STAT-CALC-1-Var Stats (enter data in L1).

• Median (M):

• The middle value when data is ordered. If n is even, average the two middle numbers.

• Resistant to outliers (unlike the mean).

• Mode:

• The most frequent value(s) in a dataset. Can be unimodal, bimodal, or multimodal.

• Resistant measure:

• A statistic (e.g., median, IQR) that is not strongly affected by outliers or skewness.

• Skewed left (negatively skewed):

• Tail on the left; mean < median.

• Skewed right (positively skewed):

• Tail on the right; mean > median.

• Symmetric distribution:

• Mean-median-mode.

• Outlier:

• A data point far from the rest (e.g., 1.5×IQR above Q3 or below Q1).

• 1-Var Stats (TI-84):

• STAT-CALC-1-Var Stats-Enter list (e.g., L1)-ENTER.

• Returns x? (mean), Med (median), n, min, max, Q1, Q3.

• Effect of adding/subtracting a constant (a):

• Mean and median shift by a; spread (range, IQR, SD) does not change.

• Effect of multiplying/dividing by a constant (b):

• Mean, median, range, IQR, and SD all scale by b.

Step-by-Step / Process Flow

How to answer an FRQ about measures of center:
1. Describe the distribution’s shape (symmetric, skewed left/right, uniform). - Example: "The histogram is skewed right with a few high outliers."
2. Calculate and compare the mean and median. - Use 1-Var Stats on your TI-84 for quick calculations.
3. Explain which measure is more appropriate based on skewness/outliers. - Example: "The median is a better measure of center because the data is skewed right."
4. Interpret the chosen measure in context. - Example: "The median salary of $50,000 better represents typical earnings because the mean is inflated by a few high salaries."
5. Discuss resistance (if asked). - Example: "The median is resistant to outliers, while the mean is not."

Common Mistakes

• Mistake: Using the mean for skewed data without justification.

• Correction: Always check the shape first! For skewed data, the median is usually better. The AP exam loves testing this.

• Mistake: Forgetting to order data before finding the median.

• Correction: The median requires sorted data. For even n, average the two middle numbers.

• Mistake: Assuming the mode is always a single number.

• Correction: Datasets can be bimodal (two modes) or have no mode (all values unique).

• Mistake: Misinterpreting "resistant" as "always better."

• Correction: Resistance is a trade-off. The mean uses all data points (good for symmetric data), while the median ignores extreme values (good for skewed data).

• Mistake: Not using the calculator efficiently.

• Correction: Always use 1-Var Stats to avoid arithmetic errors. Double-check your list (L1) for typos.

AP Exam Insights

• Tricky Distinction: The AP exam often asks you to compare the mean and median in context (e.g., "Why is the median a better measure for this dataset?").
• Common FRQ Setup:
• A dataset is given (e.g., salaries, test scores), and you must:
  1. Calculate mean/median.
  2. Describe the shape.
  3. Justify which measure is more appropriate.
• Calculator Pitfall: If you enter data incorrectly in L1, 1-Var Stats will give wrong answers. Always verify your list!
• Real-World Context: Expect questions about income, housing prices, or test scores—all classic examples of skewed data where the median is preferred.

Quick Check Questions

1. Multiple Choice: A dataset has a mean of 50 and a median of 45. Which of the following is most likely true about the distribution? (A) It is symmetric. (B) It is skewed left. (C) It is skewed right. (D) It has no outliers.

Answer: (C) It is skewed right. Explanation: When the mean > median, the data is typically skewed right.

1. FRQ Part: A real estate agent claims the "average" home price in a neighborhood is $350,000. The prices (in $1,000s) are: 250, 260, 270, 280, 300, 320, 350, 400, 500, 1200
2. Calculate the mean and median.
3. Which measure better represents a "typical" home price? Justify your answer.

Answer: - Mean = $413,000 (use 1-Var Stats). - Median = $310,000 (average of 300 and 320). - The median better represents a typical price because the data is skewed right by the $1.2M outlier, which inflates the mean.

Last-Minute Cram Sheet

1. Mean = (?x?)/n (not resistant).
2. Median = middle value (resistant).
3. Mode = most frequent value (can be multiple).
4. Skewed right? Mean > median.
5. Skewed left? Mean < median.
6. Symmetric? Mean-median.
7. Outliers? Median is better.
8. TI-84: STAT-CALC-1-Var Stats for mean/median.
9. Always check shape before choosing mean/median!
10. Resistant-always better—context matters!