AP Statistics (AP Stats): Z?Scores (z = (x – ?)/?) and Standard Normal Distribution

AP Statistics – Z?Scores (z = (x – ?)/?) and Standard Normal Distribution

AP Statistics Study Guide: Z-Scores & Standard Normal Distribution

What This Is

A z-score measures how many standard deviations a data point (x) is from the mean (?). The standard normal distribution (mean = 0, SD = 1) allows us to find probabilities and percentiles using z-scores. This is essential for hypothesis testing, confidence intervals, and comparing data across different scales (e.g., comparing SAT scores to ACT scores, or determining if a factory’s product defect rate is unusually high).

Key Terms & Formulas

• z-score formula: z = (x – ?) / ?
• x = individual data point
• ? = population mean

• ? = population standard deviation

• Standard Normal Distribution (Z-distribution):

• Mean (?) = 0, Standard Deviation (?) = 1

• Used to find probabilities for normal distributions via normalcdf() or invNorm().

• Empirical Rule (68-95-99.7 Rule):

• ~68% of data falls within 1? of ?, ~95% within 2?, ~99.7% within 3?.

• normalcdf(lower, upper, ?, ?) (TI-84):

• Finds the probability that X falls between lower and upper in a normal distribution.

• For standard normal, use normalcdf(lower, upper, 0, 1).

• invNorm(area, ?, ?) (TI-84):

• Finds the x-value corresponding to a given percentile (area to the left).

• For standard normal, use invNorm(area, 0, 1).

• Standardizing Data:

• Converting a normal distribution to Z (standard normal) by subtracting-and dividing by ?.

• Percentile:

• The percentage of data below a given value (e.g., 90th percentile = 90% of data is below this value).

• Outlier Rule (Using z-scores):

• A data point is an outlier if |z| > 3 (or sometimes 2, depending on context).

Step-by-Step / Process Flow

Solving a Typical AP FRQ (Finding Probabilities or Percentiles)

1. Identify the distribution:
2. Is it normal? (Check with a graph or given info.)

3. What are ? and ??

4. Standardize (if needed):

5. Convert x to z using z = (x – ?) / ?.

6. Use normalcdf() or invNorm():

7. For probabilities: normalcdf(lower, upper, ?, ?)

8. For percentiles: invNorm(area, ?, ?)

9. Interpret in context:

10. Example: "There is a 15% chance that a randomly selected battery lasts less than 8 hours."

11. Check for reasonableness:

12. Does the answer make sense? (e.g., a z-score of 5 is almost impossible in real data.)

Common Mistakes

• Mistake: Forgetting to standardize before using normalcdf() or invNorm().

• Correction: Always convert to z first (or input ? and ? into the calculator).

• Mistake: Mixing up normalcdf() and invNorm().

• Correction:

  • normalcdf()-probability/area
  • invNorm()-x-value/percentile

• Mistake: Using sample statistics (x?, s) instead of population parameters (?, ?) in z-score formula.

• Correction: Only use z = (x – ?) / ? when ? and ? are known. For samples, use t-scores.

• Mistake: Misinterpreting percentiles (e.g., saying the 90th percentile means 90% of data is above it).

• Correction: The 90th percentile means 90% of data is below it.

• Mistake: Assuming all distributions are normal.

• Correction: Check for normality (graph, given info, or large n for CLT).

AP Exam Insights

• Frequent FRQ Setups:
• Given a normal distribution, find probabilities (e.g., "What % of lightbulbs last > 1000 hours?").
• Find percentiles (e.g., "What score separates the top 10% of test-takers?").

• Compare z-scores across different distributions (e.g., "Is a 1200 SAT or 28 ACT more impressive?").

• Tricky Distinctions:

• z vs. t: Use z when ? is known; use t when ? is unknown (and estimated by s).

• Probability vs. Percentile: normalcdf() gives probability; invNorm() gives a value.

• Calculator Pitfalls:

• Forgetting to input ? and ? in normalcdf()/invNorm() (defaults to Z-distribution).
• Using invNorm() for two-tailed probabilities (must divide-by 2 first).

Quick Check Questions

1. MCQ: A normal distribution has ? = 50 and ? = 5. What is the probability that a randomly selected value is between 45 and 55?
2. (A) 0.341
3. (B) 0.682
4. (C) 0.954
5. (D) 0.997

6. Answer: (B) 0.682 (Empirical Rule: ~68% within 1?).

7. FRQ: The weights of newborn babies are normally distributed with ? = 7.5 lbs and ? = 1.2 lbs.

8. (a) What is the probability a baby weighs less than 6 lbs?
9. (b) What weight is at the 90th percentile?
10. Answer:
    • (a) normalcdf(-1E99, 6, 7.5, 1.2)-0.1056 (10.56%).
    • (b) invNorm(0.90, 7.5, 1.2)-9.03 lbs.

Last-Minute Cram Sheet

1. z-score formula: z = (x – ?) / ? (standardizes data).
2. Standard Normal: ? = 0, ? = 1.
3. normalcdf(lower, upper, ?, ?)-probability.
4. invNorm(area, ?, ?)-x-value for percentile.
5. Empirical Rule: 68% (1?), 95% (2?), 99.7% (3?).
6. Always check if the distribution is normal (or n-30 for CLT).
7. Use z for known ?; use t for unknown ?.
8. Outliers: |z| > 3 (or 2, depending on context).
9. Percentile = % below (not above).
10. Default normalcdf()/invNorm() is Z-distribution (?=0, ?=1).