Normal Distributions

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Quick Reference

Try This First

A standardized reading test is built so that the scores follow a bell-shaped distribution with a center at 500 and a typical spread of 100 points. The picture below shows the bell, with tick marks placed every 100 points away from the center.

                       ____
                     /      \
                   /          \
                 /              \
               /                  \
            __/                      \__
        ___/                            \___
   ____/                                    \____
  |------|------|------|------|------|------|------|
  200    300    400    500    600    700    800
                       (center)

Before reading any definition, work these on paper.

1. The bell is symmetric. What single score sits exactly at the center, with half of all test-takers scoring below it and half scoring above it?
2. The tick marks are spaced one typical spread (100 points) apart. A rough rule says that almost everyone (about 95 out of 100 people) scores within two of those spacings of the center. Between which two scores does that put almost everyone?
3. Predict, before checking: a score of 700 is two spacings above the center. Do you expect that score to be common or rare? Then keep reading.

You have just read the two numbers that define a Normal distribution off a picture: its center (the mean) and its typical spread (the standard deviation). You also used a rough version of the 68-95-99.7 rule.

An Intuitive Introduction

A density curve is a smooth idealized picture of a distribution, drawn so that the total area beneath it equals 1 and so that the area above any interval equals the proportion of observations in that interval. The Normal curves are one particular family of density curves, and they are the most important family in the course.

A Normal curve has a single recognizable shape: a symmetric bell, highest at the center and tapering off smoothly toward both ends. What makes the Normal curves a family rather than a single curve is that two numbers are free to change. The first number, the mean, says where the center of the bell sits along the horizontal axis. Slide the mean up and the whole bell shifts to the right without changing shape. The second number, the standard deviation, says how wide the bell is. A small standard deviation gives a tall narrow bell with the data packed near the center; a large standard deviation gives a short wide bell with the data spread out.

Think of the curve as a process that takes a value on the horizontal axis and gives back a height. Two settings, the mean and the standard deviation, fully determine that process. Once those two numbers are fixed, the entire curve is fixed: its peak, its width, and the proportion of area in any region. No other information is needed. This is why a Normal distribution is often written in shorthand as $N(\mu, \sigma)$, naming only those two numbers.

The standard deviation has a feature worth noticing on the picture. Starting from the peak, the curve falls away steeply at first; then, as it nears the bottom, it changes from curving downward to curving outward. The point on each side where that change happens, called an inflection point, sits exactly one standard deviation away from the mean. So the standard deviation is not an abstract number for a Normal curve; it is a distance you can mark on the graph.

Prerequisite Hub

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Cross-Course Prerequisites

None. The density-curve, center, and spread ideas are all in the earlier MATH380 nodes above.

The Official Definition

Normal distribution. The Normal distributions are described by a special family of bell-shaped, symmetric density curves called Normal curves. A particular Normal distribution is completely specified by two numbers, its mean $\mu$ and its standard deviation $\sigma$. The mean is located at the center of the symmetric curve and is the same as the median; the standard deviation $\sigma$ controls the spread of the curve, and is the distance from the center to the change-of-curvature (inflection) points on either side.

(Moore, McCabe, Craig, IPS 9th edition, Section 1.4, Density Curves and Normal Distributions; corroborated by OpenStax Introductory Statistics 2e, 6.1 The Standard Normal Distribution, https://openstax.org/books/introductory-statistics-2e/pages/6-1-the-standard-normal-distribution.)

Three features inside this definition are easy to skim past and worth holding onto.

• A Normal distribution is fixed by exactly two numbers. Give the mean and the standard deviation and the curve is completely determined. Nothing else is needed or allowed.
• The mean equals the median. This is true because a Normal curve is symmetric, so its balance point and its equal-area midpoint are the same value. For a skewed distribution the mean and median would differ, but never for a Normal one.
• The standard deviation is a distance on the graph, not just a calculated number. It is the horizontal distance from the center out to the inflection point, the place where the curve switches from bending down to bending out.

Reading the Two Numbers Off a Curve

Changing $\mu$ slides the whole bell left or right without reshaping it. Changing $\sigma$ stretches or squeezes the bell without moving its center. Translating between the two numbers and the picture of the curve is the core skill for reading a Normal distribution.

The 68-95-99.7 Rule

The single most useful fact about every Normal distribution is that the proportion of area in a band around the mean depends only on how many standard deviations wide that band is, never on the particular values of the mean or the standard deviation.

The 68-95-99.7 rule (empirical rule for Normal distributions). In any Normal distribution with mean $\mu$ and standard deviation $\sigma$:

• approximately 68 percent of the observations fall within $\sigma$ of the mean $\mu$ (that is, between $\mu - \sigma$ and $\mu + \sigma$);
• approximately 95 percent of the observations fall within $2\sigma$ of $\mu$ (between $\mu - 2\sigma$ and $\mu + 2\sigma$);
• approximately 99.7 percent of the observations fall within $3\sigma$ of $\mu$ (between $\mu - 3\sigma$ and $\mu + 3\sigma$).

(Moore, McCabe, Craig, IPS 9th edition, Section 1.4, Density Curves and Normal Distributions: the 68-95-99.7 rule.)

The rule is the same for every Normal distribution because the only thing that changes from one Normal curve to another is the scale on the horizontal axis. Measuring distance in units of $\sigma$ removes that scale, so the proportions are identical.

A picture keeps the three bands straight.

                    ___
                  /     \
                /         \
              /             \
            /                 \
          /                     \
       __/                        \__
   ___/   |       |       |        \___
  |-------|-------|-------|-------|-------|
 mu-3s  mu-2s   mu-s     mu     mu+s   mu+2s  mu+3s

  |<-------------- 68% -------------->|   within 1 sigma
  |<------------------- 95% ------------------->|  within 2 sigma
  |<------------------------ 99.7% ----------------------->| within 3 sigma

Two consequences follow directly from the symmetry of the curve and are used constantly.

• Because the curve is symmetric, each central band splits evenly across the mean. The 68 percent within one standard deviation means 34 percent sits just below the mean and 34 percent just above it.
• The area outside a band is shared equally by the two tails. Since 95 percent lies within two standard deviations, 5 percent lies outside; by symmetry that splits into 2.5 percent in the low tail and 2.5 percent in the high tail.

These splits turn the three headline percentages into the proportion above or below any point that lands on a whole number of standard deviations from the mean.

Worked Examples

Example 1: Reading the Two Numbers and Marking the Bands

A Normal distribution models the heights of adult women in a population, with mean $\mu = 64$ inches and standard deviation $\sigma = 2.5$ inches, written $N(64, 2.5)$.

Predict first. The center of the bell is 64 inches. One standard deviation is 2.5 inches, so the inflection points sit at $64 - 2.5 = 61.5$ and $64 + 2.5 = 66.5$ inches. Most heights should cluster within a few inches of 64.

Step 1: Mark one, two, and three standard deviations on each side.

Step 2: Attach the percentages. By the 68-95-99.7 rule, about 68 percent of women have heights between 61.5 and 66.5 inches, about 95 percent between 59 and 69 inches, and about 99.7 percent between 56.5 and 71.5 inches.

Read it back. Almost every adult woman in this model (99.7 percent) falls between 56.5 and 71.5 inches. That matches the prediction that heights cluster tightly around 64. The numbers 64 and 2.5 alone produced this entire description.

Example 2: Proportion Below a Point Using Symmetry

Scores on a placement exam follow $N(500, 100)$: mean 500, standard deviation 100. What proportion of test-takers score below 600?

Predict first. The point 600 is one standard deviation above the mean. More than half the area lies below 600, because 600 is above the center, so the answer should be larger than 50 percent but not close to 100 percent.

Step 1: Locate 600 in standard-deviation units. Since $600 = 500 + 100 = \mu + \sigma$, the point 600 is exactly one standard deviation above the mean.

Step 2: Use the 68 percent band and symmetry. The 68-95-99.7 rule says 68 percent of scores fall between $\mu - \sigma = 400$ and $\mu + \sigma = 600$. By symmetry, half of that 68 percent, which is 34 percent, lies between the mean 500 and the point 600.

Step 3: Add the lower half. Half of all area, 50 percent, lies below the mean 500. Add the 34 percent between 500 and 600:

\[ 50\% + 34\% = 84\%. \]

Read it back. About 84 percent of test-takers score below 600. This is larger than 50 percent and well short of 100 percent, exactly as predicted, because 600 sits one standard deviation above the center.

Example 3: Proportion in a Tail

A machine fills bottles so that the fill volume follows $N(500, 5)$: mean 500 milliliters, standard deviation 5 milliliters. What proportion of bottles contain more than 510 milliliters?

Predict first. The point 510 is two standard deviations above the mean. Two standard deviations out is far, so the proportion beyond it should be small, a couple of percent at most.

Step 1: Locate 510 in standard-deviation units. Since $510 = 500 + 10 = 500 + 2(5) = \mu + 2\sigma$, the point 510 is two standard deviations above the mean.

Step 2: Use the 95 percent band. The rule says 95 percent of bottles fall within two standard deviations of the mean, between 490 and 510 milliliters. So 5 percent fall outside that band.

Step 3: Split the outside area between the two tails. By symmetry the 5 percent outside the band splits evenly: 2.5 percent below 490 and 2.5 percent above 510.

\[ \frac{5\%}{2} = 2.5\%. \]

Read it back. About 2.5 percent of bottles contain more than 510 milliliters. The answer is small, as predicted, because 510 is two standard deviations above the center, near the upper edge of where almost all bottles fall.

Example 4: Same Rule, Different Curves

Two Normal distributions describe two manufacturing lines. Line A has $N(200, 4)$ and Line B has $N(50, 4)$, both measured in grams. For each line, between which two values does the middle 95 percent of output fall?

Predict first. The two lines have the same standard deviation but different means. The 95 percent bands should have the same width (they share $\sigma = 4$) but be centered at different places.

Step 1: Apply the two-standard-deviation band to each line. The middle 95 percent runs from $\mu - 2\sigma$ to $\mu + 2\sigma$.

\[ \text{Line A: } 200 - 2(4) = 192 \quad \text{to} \quad 200 + 2(4) = 208. \]

\[ \text{Line B: } 50 - 2(4) = 42 \quad \text{to} \quad 50 + 2(4) = 58. \]

Read it back. Line A holds its middle 95 percent between 192 and 208 grams; Line B between 42 and 58 grams. Both bands are 16 grams wide, because both lines share $\sigma = 4$. Only the center moved, which confirms that the standard deviation sets the width of the band and the mean sets its location. The 68-95-99.7 rule produced the same proportions for two different curves.

Common Misconceptions

Practice Problems

Mastery Checklist

Novice (Level 1-2):

• I can read the mean and the standard deviation off the notation $N(\mu, \sigma)$ and name what each one does to the curve.
• I can mark $\mu \pm \sigma$, $\mu \pm 2\sigma$, and $\mu \pm 3\sigma$ on a number line and locate the inflection points.
• I can state the 68-95-99.7 rule and apply the one-standard-deviation band.

Competent (Level 3-4):

• I can convert a given value into a whole number of standard deviations from the mean.
• I can use symmetry to split a central band into the area above and below the mean.
• I can find the proportion in a middle band whose two edges are different distances from the mean.

Proficient (Level 5):

• I can reason from a known tail proportion back to the mean and the standard deviation.
• I can explain why the 68-95-99.7 rule is the same for every Normal distribution.
• I can construct a question on a given Normal distribution to produce a target proportion and justify the answer.

Connections

Looking back:

• Density Curves establish that area under a smooth curve represents proportion, the idea a Normal curve specializes.
• Measures of Center: Mean and Median supply the mean that sits at the center of the bell and equals the median.
• Measures of Spread: Variance and Standard Deviation supply the standard deviation that sets the width of the bell.

Looking ahead:

• Standardizing and z-Scores convert any Normal distribution into the standard Normal $N(0, 1)$ so that one table handles every Normal curve. The 68-95-99.7 rule is the first set of standard cutoffs.

Real-world connections:

• Standardized-test scores (such as SAT sections built around a mean of 500 and a standard deviation of 100) are reported on a Normal scale, so the 68-95-99.7 rule turns a raw score into a sense of how common it is.
• Manufacturing quality control sets acceptable limits at two or three standard deviations from a target so that the proportion of parts outside the limits is known in advance.
• Biological measurements such as heights, birth weights, and blood pressure are often modeled as Normal, which lets a clinician judge whether a measurement is typical or unusual from its distance in standard deviations.

Resources

Last updated: 2026-06-16