AP Statistics (AP Stats): Empirical Rule (68?95?99.7%)

AP Statistics – Empirical Rule (68?95?99.7%)

AP Statistics: Empirical Rule (68-95-99.7%) – Exam-Ready Study Guide

What This Is

The Empirical Rule (or 68-95-99.7% Rule) describes how data is distributed in a normal distribution (bell-shaped, symmetric). It states that: - ~68% of data falls within 1 standard deviation (?) of the mean (?). - ~95% falls within 2? of ?. - ~99.7% falls within 3? of ?.

This is essential on the AP exam because it’s used to: - Estimate probabilities (e.g., "What percent of SAT scores are between 1100 and 1300 if-= 1200 and-= 100?"). - Check for outliers (e.g., "Is a data point 3? from the mean unusual?"). - Solve normal distribution problems (a major FRQ topic).

Real-world example: A factory produces bags of chips with a mean weight of 16 oz and-= 0.2 oz. Using the Empirical Rule, we can estimate what percent of bags weigh between 15.6 oz and 16.4 oz (within 2?).

Key Terms & Formulas

• Normal Distribution (Bell Curve): A symmetric, unimodal distribution defined by its mean (?) and standard deviation (?).
• Empirical Rule (68-95-99.7% Rule):
• 68% of data falls within ? ± ?.
• 95% falls within ? ± 2?.
• 99.7% falls within ? ± 3?.
• Z-Score: z = (x – ?) / ?-Measures how many standard deviations a value is from the mean.
• Standard Normal Distribution: A normal distribution with ? = 0 and ? = 1 (used for z-scores).
• normalcdf(lower, upper, ?, ?) (TI-84): Finds the probability that a value falls between lower and upper in a normal distribution.
• invNorm(area, ?, ?) (TI-84): Finds the value corresponding to a given percentile in a normal distribution.
• Outlier (Empirical Rule Definition): A data point more than 3? from ? (falls outside the 99.7% range).
• Percentile: The percent of data below a given value (e.g., the 90th percentile is the value where 90% of data falls below it).

Step-by-Step / Process Flow

How to Solve Empirical Rule Problems (FRQ Style)

1. Identify-and ? from the problem (or calculate them if given data).
2. Determine the range (e.g., "within 1? of ?"-? ± ?).
3. Apply the Empirical Rule to estimate the percent of data in that range.
4. If the problem asks for probability, use normalcdf(lower, upper, ?, ?).
5. If the problem asks for a specific value (e.g., "What score is at the 90th percentile?"), use invNorm(area, ?, ?).
6. Interpret in context (e.g., "About 95% of bags weigh between 15.6 oz and 16.4 oz").
7. Check for outliers (if asked): Any value >-+ 3? or <-– 3? is unusual.

Example FRQ Setup: "Heights of adult men are normally distributed with-= 70 inches and-= 3 inches. What percent of men are between 64 and 76 inches tall?" Solution:
1.-= 70,-= 3.
2. 64 = 70 – 2?, 76 = 70 + 2?-within 2?.
3. Empirical Rule: 95% of data falls within 2?.
4. Answer: About 95% of men are between 64 and 76 inches tall.

Common Mistakes

AP Exam Insights

What’s Frequently Tested? - FRQs often ask: - "What percent of [data] falls between [two values]?" (Use normalcdf or Empirical Rule.) - "Is [value] an outlier?" (Check if it’s > 3? from ?.) - "Find the [percentile] of [data]." (Use invNorm.) - Multiple-choice traps: - Giving a skewed distribution and asking about the Empirical Rule (trick question—it doesn’t apply!). - Asking for probability when the answer is a percentile (or vice versa).

Calculator Pitfalls: - normalcdf vs invNorm: - normalcdf-Probability (e.g., "What’s the chance a value is between 100 and 120?"). - invNorm-Value (e.g., "What score is at the 90th percentile?"). - Forgetting to set ? and ? in invNorm (default is ?=0, ?=1, which is wrong if your data isn’t standardized!).

Tricky Distinctions: - Empirical Rule vs. Chebyshev’s Theorem: - Empirical Rule-Only for normal distributions (gives exact %). - Chebyshev’s-Any distribution (gives a minimum % within k?, e.g., at least 75% within 2?). - Z-Score Interpretation: - A z-score of 2 means the value is 2? above ? (not "2% above").

Quick Check Questions

1. Multiple Choice

Heights of adult women are normally distributed with-= 65 inches and-= 2.5 inches. What percent of women are shorter than 62.5 inches? (A) 16% (B) 32% (C) 50% (D) 68% (E) 84%

Answer: (A) 16% Explanation: 62.5 is 1? below ? (z = -1). Since ~68% is within 1?, ~34% is above-+ ?, and ~34% is below-– ?. Thus, ~16% is below-– ?.

2. FRQ Part

A machine fills soda bottles with a mean of 12 oz and-= 0.1 oz. The amounts are normally distributed. (a) What percent of bottles contain between 11.8 oz and 12.2 oz? (b) A quality control inspector rejects bottles with less than 11.7 oz. What percent of bottles are rejected?

Answer (a): ~95% Explanation: 11.8 =-– 2?, 12.2 =-+ 2?-within 2?-~95% (Empirical Rule).

Answer (b): ~0.15% Explanation: 11.7 =-– 3?-normalcdf(-1E99, 11.7, 12, 0.1) = 0.00135-~0.15% (use normalcdf for exact value).

Last-Minute Cram Sheet

1. Empirical Rule: 68% (1?), 95% (2?), 99.7% (3?) only for normal distributions.
2. Z-Score Formula: z = (x – ?) / ?-How many ?’s from
3. normalcdf(lower, upper, ?, ?)-Probability between two values.
4. invNorm(area, ?, ?)-Value at a given percentile.
5. Outliers: > 3? from? (Empirical Rule definition).
6. Always check if the distribution is normal before using the Empirical Rule.
7. invNorm defaults to ?=0, ?=1-Set-and-if your data isn’t standardized!
8. Within 1? = 68%, but below-–-= 16% (half of 32%).
9. Chebyshev’s Theorem is for any distribution (at least 75% within 2?).
10. FRQ Tip: If asked for a percent, use normalcdf or Empirical Rule. If asked for a value, use invNorm.