AP Statistics (AP Stats): Assessing Normality (Normal Probability Plot)

AP Statistics – Assessing Normality (Normal Probability Plot)

AP Statistics: Assessing Normality (Normal Probability Plot) – Exam-Ready Study Guide

What This Is

A normal probability plot (or quantile-quantile plot) is a graphical tool used to assess whether a dataset follows a normal distribution—a key assumption for many statistical tests (e.g., t-tests, confidence intervals for means). On the AP exam, you’ll often need to justify normality before running a test or constructing an interval. For example, a pharmaceutical company might test whether a new drug’s effect on blood pressure follows a normal distribution to ensure valid t-test results.

Key Terms & Formulas

• Normal Distribution: A symmetric, bell-shaped distribution defined by its mean (?) and standard deviation (?). Notation: N(?, ?).
• Normal Probability Plot (NPP): A scatterplot of observed data vs. expected z-scores under normality. If the points lie close to a straight line, the data is approximately normal.
• Skewness: Asymmetry in a distribution. Right-skewed (long right tail) or left-skewed (long left tail) data will deviate from the line in an NPP.
• Outliers: Points far from the line in an NPP may indicate outliers or non-normality.
• 68-95-99.7 Rule: In a normal distribution, ~68% of data falls within 1? of ?, ~95% within 2?, and ~99.7% within 3?.
• TI-84 Command for NPP: STAT PLOT-Type: Last icon (normal probability plot)-Xlist: L1-Data Axis: X-ZoomStat.
• Shapiro-Wilk Test (for reference): A formal test for normality (not on AP exam, but useful to know). H?: Data is normal, H?: Data is not normal.
• Central Limit Theorem (CLT): For large n (typically n-30), the sampling distribution of the sample mean is approximately normal, even if the population isn’t.

Step-by-Step / Process Flow

How to Assess Normality on the AP Exam

1. Plot the Data:
2. Enter data into L1 (TI-84: STAT-Edit).
3. Create a normal probability plot (2nd-Y=-Plot1-Type: Last icon-Xlist: L1-ZoomStat).
4. Interpret the Plot:
5. Normal data: Points lie close to a straight line (with minor deviations).
6. Non-normal data: Systematic curvature (e.g., S-shape = skewed) or outliers.
7. Check Sample Size:
8. If n-30, the CLT may justify normality for means (but not always for other tests).
9. If n < 30, normality must be assessed via the NPP or context.
10. Write a Justification (FRQs):
11. Example: "The normal probability plot shows points roughly in a straight line with no systematic curvature, so the data is approximately normal."
12. State Assumptions for Tests:
13. For a t-test/interval, write: "The data is approximately normal (verified by NPP), so the t-procedures are appropriate."

Common Mistakes

• Mistake: Assuming all large samples (n-30) are normal. Correction: The CLT applies to sampling distributions of the mean, not individual datasets. Always check the NPP for small samples.

• Mistake: Ignoring outliers in the NPP. Correction: Outliers can distort normality. Note them in your response (e.g., "The NPP shows one outlier, but the rest of the points follow a linear pattern.").

• Mistake: Confusing "normal" with "symmetric." Correction: Symmetric data isn’t always normal (e.g., uniform distributions). Use the NPP to confirm.

• Mistake: Forgetting to mention normality in FRQs. Correction: Always explicitly state whether the data is normal (or if the CLT applies) before running a test.

AP Exam Insights

• FRQs often ask you to:
• Create and interpret an NPP (e.g., "Construct a normal probability plot and use it to assess normality.").
• Justify normality for a t-test/interval (e.g., "Explain why a t-test is appropriate for these data.").
• Tricky Distinction: The CLT applies to means, not individual data points. For small samples (n < 30), always check the NPP.
• Calculator Pitfall: Don’t confuse the NPP with a scatterplot. The NPP’s y-axis is expected z-scores, not raw data.
• Common Setup: A problem gives a small dataset (n = 10) and asks if a t-test is valid. You must assess normality via the NPP.

Quick Check Questions

1. Multiple Choice

A normal probability plot for a dataset of 20 observations shows points that curve upward at the ends. Which of the following is most likely true? (A) The data is normally distributed. (B) The data is right-skewed. (C) The data is left-skewed. (D) The data has outliers. Answer: (C) The upward curve at the ends suggests a left-skewed distribution (long left tail).

2. FRQ Part

A researcher collects the following sample of 15 test scores: 82, 85, 88, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 105 (a) Construct a normal probability plot for these data. (b) Based on your plot, is it reasonable to assume the data is approximately normal? Justify your answer. Answer: (a) [Sketch or describe the NPP: Points should lie close to a straight line with minor deviations.] (b) Yes, the normal probability plot shows points roughly in a straight line with no systematic curvature, so the data is approximately normal.

Last-Minute Cram Sheet

1. Normal Probability Plot (NPP): Check if points lie on a straight line-normal.
2. TI-84 NPP: STAT PLOT-Last icon-Xlist: L1-ZoomStat.
3. CLT: For n-30, sampling distribution of the mean is ~normal (but check NPP for small n).
4. Skewness: Curved NPP = non-normal (S-shape = skewed).
5. Outliers: Points far from the line may indicate outliers.
6. FRQ Justification: "The NPP shows a linear pattern, so the data is approximately normal."
7. Don’t assume normality for small samples—always check the NPP!
8. t-tests require normality (or large n via CLT).
9. Right-skewed: NPP curves downward at the right.
10. Left-skewed: NPP curves upward at the left.