5.2 The Normal Distribution, Revisited

The normal distribution is a symmetric, bell-shaped curve where the area under the curve over an interval gives you the probability that a value falls in that interval.

Why This Matters for the AP Statistics Exam

The normal distribution is the engine behind almost everything in the second half of AP Statistics. Once you can calculate the area under a normal curve, you can find probabilities for sampling distributions, build confidence intervals, and run hypothesis tests later in the course. On exam questions you will be asked to recognize when a distribution can be treated as approximately normal, compute a probability from an interval, or work backward from a percentile to find a cutoff value. Showing clear setup, correct notation, and an answer interpreted in context is important for clear exam work on free-response questions.

Key Takeaways

• A normal distribution is symmetric and bell-shaped, with the mean, median, and mode all equal at the center.
• The total area under any density curve is 1, and the area over an interval equals the probability a value lands in that interval.
• A z-score, z = (x - μ)/σ, tells how many standard deviations a value is above or below the mean. Always state the direction (positive or negative).
• The empirical rule (68-95-99.7) gives quick estimates for areas within 1, 2, and 3 standard deviations of the mean.
• You can find an interval from a given area using z-scores, a standard normal table, or technology like normalcdf and invNorm.
• A distribution can be approximated by a normal model when it is roughly symmetric and bell-shaped, or when sample-size conditions are met.

The Normal Model

A normal distribution is a continuous probability distribution shown as a bell-shaped curve that is symmetric around the mean. The mean, median, and mode are all equal, and the curve is unimodal (one peak).

To describe a normal model, talk about center, shape, and spread:

• Center: the mean, the arithmetic average.
• Shape: bell-shaped and symmetric around the mean.
• Spread: the standard deviation, which measures how far values typically fall from the mean.

A continuous random variable can take any value within its domain, and every interval within that domain has a probability attached to it. For a normal random variable, the area under the curve over an interval represents the probability that a value lies in that interval.

A model such as a sampling distribution is approximately normal when these conditions are met:

Because normal distributions are symmetric and bell-shaped, they can also be used to approximate other distributions that have similar shape.

Empirical Rule Review

The area under any density curve, including the normal curve, equals 1, or 100% of the data. There is 50% of the data on each side of the mean.

• 68% of the data is within 1 standard deviation of the mean (34% on each side).
• 95% of the data is within 2 standard deviations of the mean (an additional 13.5% on each side).
• 99.7% of the data is within 3 standard deviations of the mean (an additional 2.35% on each side).

Z-Scores

A z-score indicates how many standard deviations above or below the mean a value is. A z-score of -2 means the value is 2 standard deviations below the mean. A z-score of 1 means it is 1 standard deviation above the mean. When you interpret a z-score, always indicate the direction (positive or negative). The size of the z-score also matters, especially when making inferences.

Z-scores let you compare values from two different sets of data, such as SAT and ACT scores. They strip away units, so the units do not matter and the z-score stays the same even if you convert units. Think of z-scores as a way to compare across different scales.

The formula is:

z = (x - μ) / σ

Finding Intervals from a Given Area

In a normal distribution, you can find the interval associated with a given area by assigning the right inequalities to the boundaries.

Using P(X) notation:

• P(X < xa) = p/100 means the lowest p% of values lie to the left of xa.
• P(xa < X < xb) = p/100 means p% of values lie between xa and xb.
• P(X > xb) = p/100 means the highest p% of values lie to the right of xb.

For example, to find the interval that captures the middle 0.95 (95%) of values, use the z-scores that match that area. You can get those z-scores from a standard normal table or from a calculator or software. Then convert each z-score back into a value with x = μ + zσ.

For instance, if the mean is 100 and the standard deviation is 15, the interval for the middle 95% is found by plugging the matching z-scores into x = μ + zσ.

To find the most extreme p% of values, split the area into two equal pieces on the two ends of the distribution:

P(X < xa) = (1/2)(p/100) and P(X > xb) = (1/2)(p/100)

This means half of the p% most extreme values lie to the left of xa, and half lie to the right of xb.

How to Use This on the AP Statistics Exam

Problem Solving

• Standardize first when you have a raw value: compute z = (x - μ)/σ, then find the area.
• Use normalcdf(lower, upper, μ, σ) to find a probability between two values. For a left tail, use a very small lower bound; for a right tail, use a very large upper bound.
• Use invNorm(area, μ, σ) when you are given an area and need to find the cutoff value.
• Sketch the curve and shade the region you want. A quick picture keeps you from finding the wrong tail.

Common Trap

• "At least" and "more than" both point to the right tail, while "at most" and "less than" point to the left tail. Read the wording carefully.
• For the most extreme p% of values, remember to split the area in half before looking up z-scores.

Showing Clear Work

• Write your probability statement with correct notation, such as P(X > 72).
• State your final answer with units and in context, not just a bare decimal.

Practice Problems

(1) A study is conducted to compare the heights of men and women in a certain population. The mean height of men is 70 inches, with a standard deviation of 3 inches. The mean height of women is 65 inches, with a standard deviation of 2.5 inches. Calculate the z-score for a man who is 72 inches tall.

(2) A study is conducted to compare the GPA of students at two different colleges. The mean GPA of students at College A is 3.0, with a standard deviation of 0.5. The mean GPA of students at College B is 2.5, with a standard deviation of 0.4. Calculate the z-score for a student at College A who has a GPA of 3.5.

(3) A study is conducted to compare the number of hours spent studying per week by students at two different universities. The mean number of hours spent studying per week at University A is 15 hours, with a standard deviation of 3 hours. The mean number of hours spent studying per week at University B is 12 hours, with a standard deviation of 2 hours. Calculate the z-score for a student at University A who studies 21 hours per week.

(4) A study is conducted to compare the heights of men and women in a certain population. The mean height of men is 70 inches, with a standard deviation of 3 inches. The mean height of women is 65 inches, with a standard deviation of 2.5 inches. Calculate the z-score for a woman who is 62 inches tall.

(5) Going back to problem (2), calculate the z-score for a student at College B who has a GPA of 3.0.

Answers

(1) Use the formula:

z = (x - μ) / σ

Plugging in the values:

z = (72 - 70) / 3 = 2 / 3 = 0.67

The z-score for a man who is 72 inches tall is 0.67.

(2) Plugging in the values:

z = (3.5 - 3.0) / 0.5 = 0.5 / 0.5 = 1

The z-score for a student at College A with a GPA of 3.5 is 1.

(3) Plugging in the values:

z = (21 - 15) / 3 = 6 / 3 = 2

The z-score for a student at University A who studies 21 hours per week is 2.

(4) Plugging in the values:

z = (62 - 65) / 2.5 = -3 / 2.5 = -1.2

The z-score for a woman who is 62 inches tall is -1.2.

(5) Plugging in the values:

z = (3.0 - 2.5) / 0.4 = 0.5 / 0.4 = 1.25

The z-score for a student at College B with a GPA of 3.0 is 1.25.

Common Misconceptions

• The height of the normal curve is not a probability. Probability comes from area under the curve over an interval, not from a single point.
• The probability of an exact single value for a continuous variable is 0, since a single point has no width. That is why P(X < a) and P(X ≤ a) give the same answer here.
• A larger z-score does not always mean "more." A z-score of -3 is far below the mean. Direction matters, so always note the sign.
• The empirical rule is only an estimate and only applies to normal (or roughly normal) distributions. For exact areas, use a table or technology.
• Being symmetric and bell-shaped is what lets you use a normal model, not just having a single peak. A skewed unimodal distribution is not normal.
• Forgetting to convert z-scores back into the original values is a common slip. After finding z from an area, use x = μ + zσ to get the actual cutoff.

Related AP Statistics Guides

• Unit 5 Overview: Sampling Distributions
• 5.1 Introducing Statistics: Why Is My Sample Not Like Yours?
• 5.5 Sampling Distributions for Sample Proportions
• 5.3 The Central Limit Theorem
• 5.4 Biased and Unbiased Point Estimates
• 5.7 Sampling Distributions for Sample Means