1.10 The Normal Distribution

The normal distribution is a mound-shaped, symmetric model defined by the population mean $\mu$ and population standard deviation $\sigma$, written as $N(\mu, \sigma)$. You use z-scores to standardize values, the empirical rule (68-95-99.7) to estimate proportions, and technology to find exact proportions and percentiles for normally distributed data.

AP Stats Normal Distribution

In AP Stats, a normal distribution is a symmetric, mound-shaped model that can describe some population distributions. It is written as N(mu, sigma), where mu is the population mean and sigma is the population standard deviation. The standard normal distribution is N(0, 1), and z-scores convert values from any normal model onto that common scale.

For Topic 1.10, you should be able to compare a data distribution to a normal model, use the empirical rule for quick estimates, and use technology or a standard normal table to find proportions and percentiles. Do not use a normal model unless the data are approximately symmetric and unimodal.

Why This Matters for the AP Statistics Exam

This topic gives you the tools to model real data with a theoretical curve and to compare positions across different data sets. On the AP Statistics exam, you will need to recognize when a normal model fits, convert values to z-scores, apply the empirical rule, and find proportions or percentiles using technology or a standard normal table.

Z-scores and percentiles also let you compare relative position, which shows up whenever two values come from different distributions with different means and standard deviations. The normal model returns later in the course (sampling distributions and inference), so building fluency here pays off across multiple units.

Key Takeaways

• A normal distribution is mound-shaped and symmetric, with parameters mu (population mean) and sigma (population standard deviation), written N(mu, sigma).
• A z-score tells you how many standard deviations a value sits above or below the mean. Positive means above, negative means below.
• The empirical rule applies only to normal distributions: about 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD.
• Use a calculator, a standard normal table, or computer output to find proportions and percentiles for normal data.
• Percentiles and z-scores let you compare relative position within or between data sets.
• Always check that data is approximately symmetric and unimodal before using a normal model.

Z-Scores: Standardizing Values

A z-score (also called a standardized score) measures how many standard deviations a data value falls above or below the mean. You calculate it by subtracting the mean from the value and dividing by the standard deviation.

For a population, the z-score formula is:

$z = \frac{x_i - \mu}{\sigma}$

where $z$ is the z-score, $x_i$ is a data value, $\mu$ is the population mean, and $\sigma$ is the population standard deviation.

When you work with sample data, you use the sample mean and sample standard deviation instead:

$z = \frac{x - \bar{x}}{s}$

Example: a data set has a mean of 50 and a standard deviation of 10. A value of 70 gives:

$z = \frac{70 - 50}{10} = 2$

That value sits 2 standard deviations above the mean.

Z-scores are useful because they let you compare values from different data sets on the same scale. A positive z-score means the value is above the mean, and a negative z-score means it is below. The farther a z-score is from 0 (in either direction), the more unusual that value is relative to the rest of the data.

How Standardizing Affects the Distribution

When you standardize, you shift values by the mean and rescale by the standard deviation. These two actions affect a distribution differently.

• Shifting (adding or subtracting a constant) moves the center and other position measures like percentiles, minimum, and maximum by that same amount. The shape and spread stay the same.
• Rescaling (multiplying or dividing by a constant) changes the mean, minimum, maximum, range, IQR, and standard deviation. The shape stays the same; it just looks stretched or squeezed.

Multiple-choice questions often test whether you know how shifting and rescaling change shape, center, and spread, so be ready for that trap.

The Normal Model

Some data sets can be described as approximately normally distributed. A normal curve is mound-shaped and symmetric, and it works best for distributions that are symmetric and unimodal.

The normal model has two parameters: the population mean $\mu$ and the population standard deviation $\sigma$. It is often written as $N(\mu, \sigma)$. A parameter is a numerical summary of a population, so these values describe the model, not a specific data set.

Many real-world variables can be modeled by a normal distribution. Common examples include body temperature and the weight of a loaf of bread. Other variables often modeled this way include height, blood pressure, and birth weight, though whether a normal model fits always depends on the actual data.

Do not apply a normal model without checking that the data is roughly symmetric and unimodal. To check, look at a histogram (it should be roughly symmetric with a single peak in the middle) or make a normal probability plot. If the data is clearly skewed or has multiple peaks, a normal model is not a good fit.

The Standard Normal Model

The standard normal model is a normal distribution with a mean of 0 and a standard deviation of 1, written as $N(0, 1)$. Z-scores are based on this model.

When data is normally distributed, converting values to z-scores standardizes them onto the same scale, which makes comparison and analysis easier. To standardize, you subtract the mean and divide by the standard deviation, exactly as the z-score formula shows.

The Empirical Rule (68-95-99.7)

For a normal distribution, the empirical rule describes how data clusters around the mean:

• About 68% of observations fall within 1 standard deviation of the mean.
• About 95% of observations fall within 2 standard deviations of the mean.
• About 99.7% of observations fall within 3 standard deviations of the mean.

This rule works only for normal models. Do not apply it to skewed distributions, where it will fail.

When sketching a normal model, start at the center and extend the tails outward. You do not need to draw past three standard deviations, since very little area is left beyond that point. The curve never touches the horizontal axis because it extends forever in both directions. The point where the bell shape changes from curving down to curving out (the inflection point) sits exactly one standard deviation from the mean.

Finding Proportions and Percentiles

The area under a normal curve over an interval gives the proportion of values in that interval. You can find these areas using a calculator, a standard normal table (z-table), or computer-generated output.

The general process:

1. Convert your value to a z-score using $z = \frac{x_i - \mu}{\sigma}$.

2. Use technology or the z-table to find the area (proportion) to the left, right, or between values.

You can also work backward. If you know the area (a proportion or percentile), you can use an inverse normal function or a z-table to find the matching value or to estimate a parameter for the population.

The $p$th percentile is the value that has $p$% of the data less than or equal to it. Percentiles and z-scores both describe relative position, so you can use them to compare points within one data set or across two different data sets, even when those sets have different means and standard deviations.

How to Use This on the AP Statistics Exam

Problem Solving

• Write the z-score formula, plug in the value, mean, and standard deviation, and keep units consistent throughout.
• State what your z-score means in context, for example "this value is 1.5 standard deviations above the mean."
• For proportion or percentile questions, show the z-score step and clearly state what area you are finding (left, right, or between). Clear setup makes your work easy to follow.

MCQ

• Expect questions on how shifting and rescaling change shape, center, and spread. Remember: shifting changes center and position but not spread; rescaling changes spread and center but not shape.
• Know the empirical rule percentages cold so you can estimate proportions quickly without a calculator.

Common Trap

• Using the empirical rule on a distribution that is not normal. Check for symmetry and a single peak first.
• Forgetting that a negative z-score is valid and simply means the value is below the mean.

Common Misconceptions

• A z-score is not resistant to outliers. Because z-scores use the mean and standard deviation, extreme values affect them. A z-score is "unitless," but that does not make it resistant.
• The empirical rule is not a general rule. It applies only to approximately normal distributions, not to skewed or multimodal data.
• Mu and sigma are parameters, not statistics. They describe a population model, not a specific sample. Sample values use $\bar{x}$ and $s$.
• A normal model needs a normality check. Mound-shaped and symmetric is required; do not assume any data set is normal without looking at a histogram or normal probability plot.
• A high z-score does not mean an error. A z-score of 9, for example, is valid and simply signals a very unusual value relative to the rest of the data.

Practice Problems

(1) A sample of 50 students at a school took a math test, and the mean score was 75 out of 100. The standard deviation of the scores was 10.

A. Calculate the z-score for a student who scored a 90 on the test.

B. Interpret the z-score. What does this mean in the context of statistics and test scores?

(2) A baseball player has a batting average of 0.300, which is the mean number of hits per at-bat over the course of a season. The standard deviation of the player's batting average is 0.050. In a recent game, the player had 4 at-bats and scored 3 hits. Calculate the z-score for the player's performance in this game.

(3) Another important idea in this topic is that percentiles and z-scores may be used to compare relative positions of points within a data set or between data sets.

A study was conducted to determine the average number of hours per week that college students spend studying. The study found that the average number of hours per week spent studying is 15 hours, with a standard deviation of 4 hours. A random sample of 25 college students was selected and the number of hours they spent studying per week was recorded. The sample mean was found to be 13 hours per week, with a z-score of -1.5.

Based on the information provided, what is the average number of hours per week that college students spend studying?

Answers

(1) A. First, subtract the mean score from the student's score to find the difference:

90 - 75 = 15

Next, divide the difference by the standard deviation:

15 / 10 = 1.5

B. The z-score is 1.5, which means the student scored 1.5 standard deviations above the mean test score.

Note: When calculating z-scores, use the same units for the mean, standard deviation, and data point. In this example, all values are in points on the test. If the mean and standard deviation were in different units (for example, percent instead of points), you would convert to the same units before calculating the z-score.

(2) First, calculate the player's batting average for the game by dividing hits by at-bats:

batting average = 3 / 4 = 0.750

Next, subtract the mean batting average from the game batting average:

0.750 - 0.300 = 0.450

Finally, divide the difference by the standard deviation:

0.450 / 0.050 = 9

The z-score for the player's performance in this game is 9, which means the performance was 9 standard deviations above the mean. That signals a very strong (and very unusual) game.

(3) Use the z-score formula:

z = (x - x̄) / standard deviation

Here the mean is 15 hours per week and the standard deviation is 4 hours per week. The value you are solving for is x.

Substituting in:

z = (x - 15) / 4

You are given that the z-score is -1.5, so:

-1.5 = (x - 15) / 4

Multiply both sides by 4:

-6 = x - 15

Add 15 to both sides:

x = 9

So the value with a z-score of -1.5 is 9 hours per week.

Related AP Statistics Guides

• Unit 1 Overview: Exploring One-Variable Data
• 1.1 Introducing Statistics: What Can We Learn from Data?
• 1.3 Representing a Categorical Variable with Tables
• 1.8 Graphical Representations of Summary Statistics
• 1.9 Comparing Distributions of a Quantitative Variable
• 1.4 Representing a Categorical Variable with Graphs