Density curves, the normal model, the empirical rule, z-scores, areas under the curve, percentiles, technology, and assessing normality — the geometry and algebra of probability for continuous data
Every measurement in the natural world varies. Heights, test scores, reaction times, rainfall totals, manufacturing tolerances, and blood pressure readings all fluctuate from one individual to the next. But that variation is rarely chaotic or structureless. For an enormous range of real phenomena, the variation follows a recognizable pattern: most values cluster near the center, fewer values appear at the extremes, and the distribution is roughly symmetric. This pattern is called the normal distribution, and it is one of the most important ideas in all of statistics.
Chapter 11 introduces normal distributions as mathematical models of variation. You will study what a density curve is, why area under a curve represents probability, how the normal distribution's shape is controlled by its mean and standard deviation, and how to use the empirical rule and standardization to calculate probabilities and percentiles. You will also learn to assess whether a normal model is reasonable for real data and to use technology to find probabilities and values. By the end of this chapter, you will be able to connect geometry, probability, and algebra into a unified framework for reasoning about continuous data.
A standardized exam is taken by thousands of students each year. The scores are approximately normally distributed with a mean of 500 and a standard deviation of 100. A student scores 650. How unusual is that score? What percentage of students scored higher? What score would place a student in the top 10 percent? These questions require a model — a smooth mathematical description of how scores vary — and a method for using that model to find probabilities and percentiles. By the end of this chapter, you will be able to answer all of them.
When you collect quantitative data, you get a list of values. You can display those values in a dot plot, a histogram, or a box plot. These displays show you the actual data you collected — the exact numbers from the actual individuals in your sample. But real data are imperfect. Sample sizes are limited. Histograms depend on how wide you draw the bars. Two different data sets might show similar patterns but look slightly different just because of chance variation in who happened to be in each sample.
A distribution model is a smooth mathematical description of how the values of a variable tend to behave in a population. Instead of describing the exact counts or frequencies in a particular sample, a distribution model describes the overall pattern: where values tend to cluster, how far they spread, and how likely various ranges of values are.
Think of a distribution model the way you might think of a map. A map is not the exact ground you walk on. It is a simplified, idealized representation of the ground that is designed to help you navigate. A distribution model is not the exact data. It is a smooth, idealized representation designed to help you reason about probability and variability. Distribution models are particularly useful because they allow us to ask and answer probability questions. Instead of asking "How many of the 38 students in our class scored above 80?" — a question that depends on that particular group — a distribution model lets us ask "For students in this population, what proportion are expected to score above 80?" That is a more general and more powerful question.
The two most important features of any distribution model are its shape and the specific parameters that control where it is centered and how spread out it is. For the normal distribution, those parameters are the mean (μ) and the standard deviation (σ).
A density curve is a smooth curve drawn over a histogram that models the distribution of a continuous variable. The curve represents an idealized version of what the data look like in the full population, not just in a particular sample.
1. The curve is always at or above the horizontal axis. No part of the curve goes below zero.
2. The total area under the entire curve equals exactly 1.
Area under the curve between two values = probability that a randomly chosen individual falls in that range.
These two properties make sense when you think about probability. If the total area under the curve equals 1, and probability is always between 0 and 1, then areas under sections of the curve can represent probabilities. Probability and area become the same thing — and this is a profound connection between geometry and statistics.
You can describe a density curve the same way you describe any distribution: shape (symmetric? skewed? uniform?), center (where the median splits area equally; the mean is the balance point), and spread (how wide the curve is and how quickly it falls toward the tails). For a symmetric, bell-shaped density curve, the mean and the median are equal and both sit at the central peak.
Draw a smooth, hill-shaped curve symmetric about a central value. The curve rises from near zero on the left, reaches its highest point at the center, and falls back toward zero on the right. Label the center as the mean μ. Label the horizontal axis with the variable name. Shade a region between two values on either side of center. The shaded area represents the probability that a randomly selected individual falls in that range.
A density curve is a mathematical model, not a histogram of actual data. Actual data may show gaps, bumps, and irregularities. The density curve smooths these out to reveal the underlying pattern of the population.
Area = width × height = 4 × 0.10 = 0.40The normal distribution is the most important density curve in statistics. It is sometimes called a bell curve because of its distinctive shape: symmetric, single-peaked, and smoothly tapering toward both tails. A normal distribution is completely described by two numbers: its mean μ and its standard deviation σ. We write N(μ, σ) to name a specific normal distribution.
| Property | Description |
|---|---|
| Shape | Symmetric, bell-shaped, single-peaked (unimodal), tapering smoothly to both tails |
| Center | Mean μ = median = mode; all three are at the center peak |
| Spread | Controlled by standard deviation σ; larger σ → wider, flatter curve |
| Tails | Extend to ±∞ in theory; virtually all probability lies within ±3σ of the mean |
| Total area | Exactly 1 (by definition of a density curve) |
Draw three overlapping bell-shaped curves on the same horizontal axis. One is narrow and tall (small σ). One is medium width and height (medium σ). One is wide and flat (large σ). Label the center of each with its mean. Note that all three have the same total area (= 1). This shows how σ controls spread without changing the total probability.
The normal distribution appears naturally when a variable is the sum of many small, independent influences — dozens of genes, environmental factors, and measurement errors all adding together. Each small source of variation contributes a little. When you add many independent sources of variation, the result tends to become normally distributed. This is formalized by the Central Limit Theorem, which you will study in Chapter 12.
Not every variable is normally distributed. Income distributions are typically right-skewed. Reaction times are right-skewed. Some biological measurements are skewed or bimodal. Always check whether a normal model is reasonable before applying it, rather than assuming all data are normal.
Every normal distribution is completely described by its mean (μ) and standard deviation (σ). The mean locates the center and the point of symmetry. The standard deviation controls how spread out the distribution is — a small σ produces a tall, narrow curve while a large σ produces a short, wide curve.
A useful geometric fact: the inflection points of the bell curve — the points where the curve changes from bending downward to bending upward — occur exactly one standard deviation on either side of the mean, at μ − σ and μ + σ. Knowing this lets you sketch a reasonable normal curve by hand.
1. Mark the mean μ at center.
2. Mark μ ± σ one standard deviation out (inflection points).
3. Mark μ ± 2σ two standard deviations out.
4. Mark μ ± 3σ three standard deviations out.
5. Draw a smooth bell curve peaking at μ, passing through the inflection points at μ ± σ.
6. The curve approaches but never touches the horizontal axis beyond ±3σ.
μ = 64 inches
μ ± 1σ = 64 ± 3 → 61 and 67 inches (inflection points)
μ ± 2σ = 64 ± 6 → 58 and 70 inches
μ ± 3σ = 64 ± 9 → 55 and 73 inches
Once you know the mean and standard deviation of a normal distribution, you can estimate probabilities without additional calculation using the empirical rule — also called the 68-95-99.7 rule.
For any normal distribution N(μ, σ):
≈ 68% of values lie within 1 standard deviation of the mean: between μ − σ and μ + σ
≈ 95% of values lie within 2 standard deviations of the mean: between μ − 2σ and μ + 2σ
≈ 99.7% of values lie within 3 standard deviations of the mean: between μ − 3σ and μ + 3σ
Draw a normal curve. Draw vertical lines at μ, and at μ ± σ, μ ± 2σ, μ ± 3σ. Label the inner band (μ ± σ) as 68%. Label the band μ ± 2σ as 95%. Label the band μ ± 3σ as 99.7%. Note that 0.3% of the area is in the two tails beyond three standard deviations. By symmetry, each outer tail holds 0.15%, and each inner region (between 1 and 2 SDs from center) holds 13.5% on each side.
(a) Between 400 and 600: within 1 SD → ≈ 68%
(b) Above 700: beyond 2 SDs above mean.
95% are between 300 and 700.
5% are in both tails. By symmetry: 5%/2 = 2.5% above 700.
(c) Below 300: same symmetry → ≈ 2.5%
P(500 < X < 600) ≈ 68% / 2 = 34%The empirical rule gives exact percentages only at exactly μ ± σ, μ ± 2σ, and μ ± 3σ. Do not apply these percentages to values that are not at exactly one, two, or three standard deviations from the mean. For any other value, you must compute a z-score and use the standard normal distribution.
The empirical rule covers only three special cases. For any other value, you need a more general method. The key insight is that all normal distributions are related to each other by a simple transformation. If you know how to find probabilities for one specific normal distribution, you can find probabilities for any normal distribution. The specific normal distribution that serves as the universal reference is called the standard normal distribution.
The standard normal distribution is the normal distribution with mean μ = 0 and standard deviation σ = 1. It is written N(0, 1). The variable in the standard normal distribution is traditionally called z, and values of z are called z-scores.
The standard normal table (z-table) gives the cumulative area to the left of a z-score. That is, for any z-value, the table gives P(Z ≤ z) — the area to the left of z under the N(0,1) curve. This is called the cumulative probability or left-tail probability.
Draw a normal curve centered at 0. Label the horizontal axis as z. Mark z = 0 at center and mark z = 1, 2, 3 to the right and z = −1, −2, −3 to the left. Shade the area to the left of a given z-value. Label that shaded area as the cumulative probability P(Z ≤ z). This left-tail area is what the standard normal table provides.
A z-score measures how many standard deviations a value is above or below the mean of its distribution. A z-score of 0 means the value equals the mean. A positive z-score means the value is above the mean. A negative z-score means the value is below the mean. A z-score of 1.5 means the value is 1.5 standard deviations above the mean.
Standardization is the process of converting original values into z-scores. Standardization removes the original units and scale, allowing values from different normal distributions to be directly compared on the same scale.
Rahel (Exam A): z = (91 − 75) / 8 = 16/8 = 2.00
Diego (Exam B): z = (650 − 500) / 100 = 150/100 = 1.50
x = 500 + (−1.3)(100) = 500 − 130 = 370The formula x = μ + zσ converts z-scores back to original values. Use it whenever you are given a percentile or probability and need to find the original value — as in Section 11.9.
Once you have a z-score, you can find the corresponding probability using a standard normal table or technology. The standard normal table gives left-tail areas: the proportion of the distribution at or below a given z-value. Three types of probability problems arise with different shading directions.
| Problem Type | Goal | Method |
|---|---|---|
| Type 1 | P(X < a) — less than a value | Compute z; look up left-tail area directly |
| Type 2 | P(X > a) — greater than a value | Compute z; take 1 − P(Z ≤ z) (complement rule) |
| Type 3 | P(a < X < b) — between two values | Compute z₁ and z₂; subtract areas: P(Z ≤ z₂) − P(Z ≤ z₁) |
Always draw a sketch before calculating a probability. The sketch shows which direction the shading goes and helps you catch errors where you should subtract from 1 but forget to. Label the mean, the value of interest, and the shaded region before doing any arithmetic.
z = (65 − 70) / 3 = −5/3 ≈ −1.67
P(Z ≤ −1.67) ≈ 0.0475
z = (76 − 70) / 3 = 2.00
P(Z ≤ 2.00) ≈ 0.9772
P(X > 76) = 1 − 0.9772 = 0.0228
z₁ = (67 − 70) / 3 = −1.00 → P(Z ≤ −1.00) ≈ 0.1587
z₂ = (73 − 70) / 3 = +1.00 → P(Z ≤ +1.00) ≈ 0.8413
P(67 < X < 73) = 0.8413 − 0.1587 = 0.6826
Sometimes the question is reversed: instead of being given a value and asked for a probability, you are given a probability or percentile and asked for the value. These are inverse normal problems, and they use the reverse standardization formula x = μ + zσ.
Step 1: Identify the cumulative probability (area to the left) corresponding to the percentile.
Step 2: Find the z-score that gives that left-tail area — use the z-table in reverse, or use technology.
Step 3: Convert the z-score back to the original scale using: x = μ + zσ
Step 4: Interpret in context with units.
From z-table: z ≈ 1.28 since P(Z ≤ 1.28) ≈ 0.8997 ≈ 0.90
x = μ + zσ = 500 + (1.28)(100) = 500 + 128 = 628
From z-table: z ≈ 1.645 since P(Z ≤ 1.645) ≈ 0.95
x = 64 + (1.645)(3) = 64 + 4.935 ≈ 68.9 inches
P(Z ≤ z) = 0.025 → z ≈ −1.96
P(Z ≤ z) = 0.975 → z ≈ +1.96
Lower: 500 + (−1.96)(100) = 304
Upper: 500 + (+1.96)(100) = 696
In practice, statisticians use technology to calculate normal probabilities and inverse normal values. Technology gives more precise answers and handles any value of μ, σ, and x quickly. Understanding the z-score conversion remains important for interpretation and for later topics in inference.
| Purpose | Typical Command | Input Format |
|---|---|---|
| Find P(a < X < b) | normalcdf / norm.cdf | normalcdf(a, b, μ, σ) |
| Find P(X < b) | normalcdf | normalcdf(−∞, b, μ, σ) |
| Find P(X > a) | normalcdf | normalcdf(a, +∞, μ, σ) |
| Find x given P(X ≤ x) = p | invNorm / norm.ppf | invNorm(p, μ, σ) |
normalcdf(100, 130, 120, 14) → ≈ 0.7361invNorm(0.10, 120, 14) → ≈ 102.1Technology gives a number, but you still need to interpret it in context. Always ask: Does this number make sense given what I know about the distribution? What does this probability or value mean for the situation I am studying?
Before applying normal distribution methods, you should check whether the normal model is appropriate for the data. Using a normal model for strongly skewed, bimodal, or bounded data can produce misleading probability estimates.
| Method | What to Look For | Suggests Normal If… |
|---|---|---|
| Histogram | Shape of the distribution | Roughly symmetric, single-peaked, bell-shaped |
| Box plot | Median position and whisker lengths | Median near box center, whiskers roughly equal, few outliers |
| 68-95-99.7 check | % of data within 1, 2, 3 SDs | Close to 68%, 95%, 99.7% |
| Normal probability plot | Points vs. a diagonal reference line | Points fall close to a straight line |
Draw two normal probability plots side by side. In the first, points fall very close to a straight diagonal line — this represents approximately normal data. In the second, the points curve away in an S-shape — this represents skewed data. Label both clearly. When points are close to the line, the normal model is reasonable. When they curve systematically, consider a different model.
Within ±1 SD (26°F to 38°F): 22 days = 73% (expected: 68%)
Within ±2 SD (20°F to 44°F): 29 days = 97% (expected: 95%)
Within ±3 SD (14°F to 50°F): 30 days = 100% (expected: 99.7%)
No real data set will follow the normal model perfectly. The question is whether the data are close enough to normal that the model is useful as an approximation. Small samples may appear non-normal even when they come from a roughly normal population. Always use professional judgment alongside formal checks.
| Field | Application | Why Normal Model? |
|---|---|---|
| Manufacturing | Part dimensions, fill weights, tolerances | Small random variation in production processes; many independent sources |
| Standardized Testing | SAT, ACT, IQ scores | Tests are scaled to produce approximately normal distributions; facilitates percentile ranking |
| Scientific Measurement | Repeated measurements of physical quantities | Random measurement errors are often normally distributed |
| Biology | Heights, birth weights, blood markers | Additive effects of many genetic and environmental factors |
| Finance | Short-term asset returns | Approximation for short time horizons; but heavy tails are common — model has known limitations |
Applying a normal model without checking whether it is appropriate can lead to serious errors. In medicine, a diagnostic test that assumes normal variation might classify a healthy person as sick if their measurement follows a skewed distribution. In finance, underestimating extreme events because of an assumed normal model contributed to major financial crises. Always check whether the model fits the context before drawing conclusions.
Correction: Always compute z = (x − μ)/σ before using the standard normal table. The table only works for N(0,1).
Correction: The table gives the left-tail area. For the right tail, take 1 − P(Z ≤ z). Always draw a sketch first.
Correction: Check with a histogram, box plot, and 68-95-99.7 count before applying the normal model. Not all data are normal.
Correction: The empirical rule applies exactly only at μ ± σ, μ ± 2σ, and μ ± 3σ. For any other value, compute a z-score and use the standard normal distribution.
Correction: A z-score of 3 corresponds to a probability of about 0.0013 in the upper tail — small, but not zero. Events in the tails of normal distributions are unlikely but do occur.
Correction: A percentile is a rank in the distribution, not a percentage correct. The 80th percentile means you scored above 80% of all test-takers — your actual score may be any number.
Correction: invNorm gives left-tail values. For the top 5%, enter invNorm(0.95, μ, σ) — the value that cuts off the bottom 95% and leaves the top 5% in the right tail.
Chapter 11 has focused on normal distributions as models for the variation of individual observations in a population. But statistics often works with summaries of samples rather than individual measurements. When you take a random sample and compute the sample mean x̄ or sample proportion p̂, those statistics vary from sample to sample. The distribution of those statistics across all possible samples is called a sampling distribution.
Here is the crucial connection between Chapter 11 and what follows in Chapter 12:
Sampling distributions are often approximately normal — even when the individual data are not.
This result, known as the Central Limit Theorem, explains why normal distributions are so central to statistical inference. Even if the population data are skewed, the distribution of sample means from large enough samples will be approximately normal. Everything you learned in Chapter 11 — z-scores, areas, percentiles — transfers directly to sampling distributions in Chapter 12.
Use the 68-95-99.7 rule. Show which reference value each boundary corresponds to.
Calculate the z-score for each value. State whether it is above or below the mean and by how many standard deviations.
For each problem, draw a sketch and shade the region of interest. Compute a z-score and find the probability.
Each problem shows a student error. Identify the mistake and write the correct solution.
What error did the student make? What is the correct probability?
What did the student do wrong? What is the correct original-scale answer?
Identify the error in this reasoning and rewrite the statement correctly.
What is wrong with this approach? What should the student check first?
Identify the error. What should the student have entered? What is the correct cutoff?
Answers may be placed in the answer key.
Chapter 11 developed the normal distribution as a model for how individual observations in a population vary. You studied how to find probabilities for a single observation drawn from a normal distribution.
Chapter 12 shifts the question: instead of asking about individual values, you will ask about sample statistics — especially sample means x̄ and sample proportions p̂. When you draw a random sample from a population and compute the sample mean, you get one value. But if you drew another random sample of the same size, you would get a slightly different sample mean. The distribution of all possible sample means across all possible samples is called the sampling distribution of the sample mean.
The Central Limit Theorem says that this sampling distribution is approximately normal — even when the population distribution is not — provided the sample size is large enough. Everything you learned in Chapter 11 transfers directly: z-scores, areas, percentiles, the empirical rule. The only change is that the standard deviation of the sampling distribution (called the standard error) depends on both the population standard deviation and the sample size. This is why the normal distribution is so central to statistical inference, and why Chapter 11 is the essential preparation for Chapters 12 and 13.
Preview: If a population has μ = 100 and σ = 20, and you take a random sample of size n = 100, what is the approximate distribution of the sample mean x̄? How does taking a larger sample make the sample mean more reliable?