5.2 The Normal Distribution, Revisited

Turn the interval into areas under the bell by standardizing and using the standard normal (z) table or your calculator. Steps: 1. Write the mean μ and standard deviation σ for the normal model. 2. Convert each endpoint x to a z-score: z = (x − μ)/σ. Call them z_a and z_b. 3. Look up the cumulative probabilities Φ(z_a) and Φ(z_b) (table A or calculator’s normalcdf / normalcdf(lower, upper, μ, σ) on exam-allowed calculators). 4. The probability that X is between the two numbers is P(a < X < b) = Φ(z_b) − Φ(z_a). Quick example: μ = 50, σ = 8, want P(46 < X < 62). z46 = (46−50)/8 = −0.5 → Φ(−0.5) ≈ 0.3085 z62 = (62−50)/8 = 1.5 → Φ(1.5) ≈ 0.9332 P = 0.9332 − 0.3085 = 0.6247 (about 62.5%). This matches the AP CED idea that area under the normal curve = probability (see Topic 5.2 study guide: https://library.fiveable.me/ap-statistics/unit-5/normal-distribution-revisited/study-guide/dx4vMcx3WjSw68f1Ov66). For more practice, try problems at (https://library.fiveable.me/practice/ap-statistics).

Z-score formula (population value): z = (x − μ) / σ. More generally for a statistic: z = (statistic − parameter) / (standard error). Example: for a sample mean, z = (x̄ − μ) / (σ/√n); for a sample proportion, z = (p̂ − p) / √[p(1−p)/n]. When to use them: - Use z-scores to standardize values from a normal (or approximately normal) distribution so you can find probabilities as areas under the standard normal curve (VAR-6.A). - Use them to convert an X value to how many standard deviations it is from the mean (percentiles, tails, confidence bounds—VAR-6.B). - On the AP exam you can use z-tables or technology (tables are provided; you may also use a graphing calculator) to get areas or inverse areas. For a quick topic review, see the Topic 5.2 study guide (https://library.fiveable.me/ap-statistics/unit-5/normal-distribution-revisited/study-guide/dx4vMcx3WjSw68f1Ov66). For broader Unit 5 review and practice questions, check (https://library.fiveable.me/ap-statistics/unit-5) and (https://library.fiveable.me/practice/ap-statistics).

Use the normal when the distribution you’re modeling is roughly bell-shaped and symmetric or when the sampling distribution becomes approximately normal by the Central Limit Theorem (CLT). Practical rules for AP Stats: - For sample means: use a normal (z or t as appropriate) when the population is normal OR n is “large” so the CLT applies. If σ is unknown and n is small, use t; if n is large, the sampling distribution of x̄ is approximately normal with σx̄ = σ/√n (or s/√n). - For proportions: approximate the sampling distribution of p̂ with a normal when np ≥ 10 and n(1 − p) ≥ 10 (use standard error √[p̂(1−p̂)/n] on the exam). - For discrete binomial-to-normal approximations: use continuity correction (add/subtract 0.5) when n is large and the np rules above hold. - Don’t force a normal fit for skewed, multimodal, or small-sample raw data unless CLT or known normal population justifies it. On the AP exam you’ll standardize (z = (value − μ)/σ) or use technology/z-tables (CED VAR-6.B). For a focused review check the Topic 5.2 study guide (https://library.fiveable.me/ap-statistics/unit-5/normal-distribution-revisited/study-guide/dx4vMcx3WjSw68f1Ov66), the Unit 5 overview (https://library.fiveable.me/ap-statistics/unit-5), and practice questions (https://library.fiveable.me/practice/ap-statistics).

P(X < a) is the area under the normal curve to the left of a—the probability a value is at or below a. P(X > a) is the area to the right of a—the probability a value is above a. Because the whole area is 1, P(X > a) = 1 − P(X < a). To compute either: standardize a to z = (a − μ)/σ, then use the standard normal table or calculator. (The AP table gives area below z, so it directly gives P(X < a).) Quick example: μ = 100, σ = 15, a = 115 → z = 1.0. Table value P(Z < 1.0) ≈ 0.8413, so P(X < 115) ≈ 0.8413 and P(X > 115) ≈ 0.1587. If a = μ, symmetry implies P(X < μ) = P(X > μ) = 0.5. For more practice and exact steps aligned to the CED (VAR-6.A, VAR-6.B), see the Topic 5.2 study guide (https://library.fiveable.me/ap-statistics/unit-5/normal-distribution-revisited/study-guide/dx4vMcx3WjSw68f1Ov66) and try problems on Fiveable’s practice page (https://library.fiveable.me/practice/ap-statistics).

Think of a continuous random variable (like height) as being spread smoothly along the x-axis. The normal curve is just a picture that assigns relative likelihoods across values. The key rule from the CED: the area under the curve over an interval equals the probability the value falls in that interval (VAR-6.A.3). So P(14 < X < 16) = area under the bell from 14 to 16. For any single exact value (e.g., P(X = 15)) the area is essentially 0—we only use intervals for probabilities (VAR-6.A.1). To compute those areas you standardize: z = (x − μ)/σ and use a z-table or technology to get cumulative area (VAR-6.B.1). For example, P(X < x) = area left of z; P(a < X < b) = area left of b minus area left of a. On the AP exam you can use your calculator or the provided z-table (check calculator policy in the CED). For a clear walkthrough and practice, see the Topic 5.2 study guide (https://library.fiveable.me/ap-statistics/unit-5/normal-distribution-revisited/study-guide/dx4vMcx3WjSw68f1Ov66) and try practice problems (https://library.fiveable.me/practice/ap-statistics).

Start by turning the percent/area into a probability and decide which inequality fits (left, right, between, or most-extreme). Then use the standard normal (z) table or your calculator’s inverse normal (invNorm) to get the z-boundary, and finally unstandardize: x = μ + zσ. Quick recipe: - Left tail: P(X < xa) = p → find z with area p to the left, then xa = μ + zσ. - Right tail: P(X > xb) = p → find z with area 1 − p to the left, xb = μ + zσ. - Between: P(xa < X < xb) = p → find lower z and upper z that give that central area (or split tails). - Most extreme p%: split p/2 in each tail; for 90% central area you’d use z = ±1.645 (so xa = μ −1.645σ, xb = μ +1.645σ). For 95% central use ±1.96. AP tip: you can use z-tables or calculator invNorm; on the exam tables and a graphing calculator are allowed. For more practice and examples, see the Topic 5.2 study guide (https://library.fiveable.me/ap-statistics/unit-5/normal-distribution-revisited/study-guide/dx4vMcx3WjSw68f1Ov66) and extra practice (https://library.fiveable.me/practice/ap-statistics).

"Lowest 25% of values" means the cutoff point x so that 25% of the distribution lies to the left of x (a left-tail area). In symbols: P(X < x) = 0.25. For a normal model you standardize: P(Z < (x − μ)/σ) = 0.25. Use a z-table or technology (inverse normal). The z that gives area 0.25 is about −0.674. So solve x = μ + zσ = μ + (−0.674)σ. Example: if μ = 100 and σ = 12, x ≈ 100 − 0.674(12) ≈ 91.9—the lowest 25% weigh less than ~91.9. This is exactly what VAR-6.B in the CED expects: use z-scores or technology to find interval boundaries. For practice, try problems in the Unit 5 study guide (https://library.fiveable.me/ap-statistics/unit-5/normal-distribution-revisited/study-guide/dx4vMcx3WjSw68f1Ov66) and more practice questions (https://library.fiveable.me/practice/ap-statistics).

Steps—quick and practical: 1. Standardize (if you prefer): compute z1 = (a − μ)/σ and z2 = (b − μ)/σ. (This uses the CED idea of standardization and z-scores.) 2. Use your graphing calculator (AP exam requires one): on a TI-84/TI-83 use normalcdf(lower, upper, mean, sd). - If you standardized: normalcdf(z1, z2, 0, 1). - If you didn’t standardize: normalcdf(a, b, μ, σ). Example: P( a < X < b ) = normalcdf(a, b, 100, 15) returns the area between a and b. 3. Interpret the result as the probability (area under the normal curve between a and b), per VAR-6.A.3. If you don’t have a calculator, standardize and use the standard normal table (find area below each z, subtract: Φ(z2) − Φ(z1)). For more review and practice problems, check the Topic 5.2 study guide (https://library.fiveable.me/ap-statistics/unit-5/normal-distribution-revisited/study-guide/dx4vMcx3WjSw68f1Ov66) and the Unit 5 overview (https://library.fiveable.me/ap-statistics/unit-5).

Use these quick checks to decide if a normal curve is appropriate: - Shape: look at a histogram or boxplot. The normal model fits distributions that are roughly symmetric and bell-shaped with no strong skew or heavy outliers (CED VAR-6.C.1). - Empirical rule: if about 68%, 95%, 99.7% of data fall within 1, 2, 3 SDs of the mean, that’s a good sign. - Sample size & CLT: for sampling distributions of a mean, the Central Limit Theorem says sample means are approximately normal for large n. A common rule of thumb is n ≥ 30, but if the population is already roughly normal you can use smaller n (see Topic 5.3). - For proportions use the success–failure rule: np ≥ 10 and n(1 − p) ≥ 10 to justify normal approximation. - Use a normal probability plot (QQ-plot) or test skewness if unsure. For AP exam guidance and practice, check the Topic 5.2 study guide (https://library.fiveable.me/ap-statistics/unit-5/normal-distribution-revisited/study-guide/dx4vMcx3WjSw68f1Ov66), Unit 5 overview (https://library.fiveable.me/ap-statistics/unit-5), and lots of practice problems (https://library.fiveable.me/practice/ap-statistics).

Z-scores are the paper-and-pencil way to standardize: z = (x − μ)/σ converts any normal X to the standard normal Z so you can read area from a z-table. That’s great for showing work, understanding symmetry/percentiles, and for problems that ask you to explain steps (CED VAR-6.A/VAR-6.B). Technology (calculator/computer) does the same math faster: use normalcdf or normalcdf(lower, upper, μ, σ) to get probabilities, and invNorm (inverse normal) to find cutoffs for a given area. It avoids extra rounding and works directly with μ and σ. On the AP exam you’re expected to bring a graphing calculator and you may use these commands (see exam calculator policy in the CED). When to use which: use z-scores to show understanding and to check work; use technology to save time and get more precise answers on long problems. For more practice and examples, see the Topic 5.2 study guide (https://library.fiveable.me/ap-statistics/unit-5/normal-distribution-revisited/study-guide/dx4vMcx3WjSw68f1Ov66) and extra problems (https://library.fiveable.me/practice/ap-statistics).

You divide the percentage by 2 when the question asks for the most extreme p% of values (i.e., the extreme values on both tails) rather than just one tail. Because the normal distribution is symmetric, “most extreme 5%” means 2.5% in the left tail and 2.5% in the right tail. So for p% most extreme, use P(X < x_left) = (1/2)(p/100) and P(X > x_right) = (1/2)(p/100) to find the two cutoff z-scores (VAR-6.B.2.d in the CED). If the problem asks for the lowest p% or the highest p% only, don’t divide by 2—use the full p% for that single tail. In practice: for “most extreme 1%,” find z for 0.5% (0.005) and −z for 99.5% (0.995). Use inverse-normal (calculator or table) to get the z’s, then convert with x = μ + zσ. For more examples and AP-aligned practice, see the Topic 5.2 study guide (https://library.fiveable.me/ap-statistics/unit-5/normal-distribution-revisited/study-guide/dx4vMcx3WjSw68f1Ov66) and the Unit 5 overview (https://library.fiveable.me/ap-statistics/unit-5).

When you get computer/calculator output for a normal problem, read it like this: the software gives you either a CDF (area to left) or an inverse (quantile). Match that to the probability statement in the problem. - If it reports μ and σ, standardize: z = (x−μ)/σ. Then either use the CDF value (P(X < x) = CDF) or convert to tail area: P(X > x) = 1 − CDF. - If the output gives a z and a probability, remember table/tech output gives P(Z < z). Example: z = 1.25 → P(Z < 1.25) ≈ 0.8944, so P(Z > 1.25) ≈ 0.1056. - For “what value has top 5%?” use inverse normal (quantile): top 5% cutoff has z ≈ 1.645, then x = μ + zσ. - Watch wording: left vs. right tail, “between” means subtract two CDFs. Use continuity correction when approximating discrete with normal. AP rules: you may use a graphing calculator on the exam; know both CDF and inverse-normal commands. For more examples and step-by-step practice, see the Topic 5.2 study guide (https://library.fiveable.me/ap-statistics/unit-5/normal-distribution-revisited/study-guide/dx4vMcx3WjSw68f1Ov66) and thousands of practice items (https://library.fiveable.me/practice/ap-statistics).

Good question—for a normal (continuous) distribution P(X < xa) and P(X ≤ xa) are the same because any single exact value has probability 0. The CED’s VAR-6.A.1 and VAR-6.A.3 explain that probabilities come from area under the curve (intervals), not individual points, so ≤ vs < makes no difference when X is continuous. That’s why z-tables and calculators report "probability lying below z" (see the formula sheet/table in the CED). It does matter, though, when you deal with discrete variables (counts) or when you approximate a discrete distribution with a normal: use a continuity correction (add or subtract 0.5) to go between ≤ and < for better accuracy. For AP exam work, treat normals as continuous and don’t worry about the ≤ vs <; only use continuity corrections when approximating binomial/Poisson with a normal (Topic 5.3/5.5). For extra practice, check the Topic 5.2 study guide (https://library.fiveable.me/ap-statistics/unit-5/normal-distribution-revisited/study-guide/dx4vMcx3WjSw68f1Ov66) and more problems (https://library.fiveable.me/practice/ap-statistics).

You can use a normal approximation when the distribution you’re modeling is basically symmetric, unimodal, and not too heavy-tailed or full of outliers—in short, “bell-shaped.” Practically, check a histogram/boxplot: the shape should be roughly symmetric about the center, one peak, and few extreme values. Use the empirical-rule idea (68–95–99.7) as a rough sanity check: about 68% of values within 1 SD, 95% within 2 SDs, etc. For sample means, the Central Limit Theorem lets you use normal models when n is large (even if the population isn’t perfect), and for proportions require np and n(1−p) both at least about 10. For discrete-to-continuous approximations (like binomial → normal) use a continuity correction. If you’re unsure, rely on plots or compute skewness; strong skew or clear outliers means don’t approximate. For AP-aligned review see the Topic 5.2 study guide (https://library.fiveable.me/ap-statistics/unit-5/normal-distribution-revisited/study-guide/dx4vMcx3WjSw68f1Ov66) and try practice problems (https://library.fiveable.me/practice/ap-statistics).

Write the probability you want as an inequality first, then translate area to z-scores and solve for x. Use these templates from the CED (Topic 5.2): - Left tail: P(X < xa) = p → xa = μ + z_p·σ (z_p is the z with area p to its left). - Between: P(xa < X < xb) = p → find z-values that give that middle area, then xa = μ + z_a·σ, xb = μ + z_b·σ. - Right tail: P(X > xb) = p → xb = μ + z_{1−p}·σ (since area left of xb is 1−p). - “Most extreme” p%: split into two tails: P(X < xa) = p/2 and P(X > xb) = p/2. Quick example: “middle 90%” → P(xahttps://library.fiveable.me/ap-statistics/unit-5/normal-distribution-revisited/study-guide/dx4vMcx3WjSw68f1Ov66) and over 1,000 practice problems (https://library.fiveable.me/practice/ap-statistics).