The Normal Curve and the Empirical Rule

Lesson 1 of 2: Fit Data to a Normal Model

In this lesson:

• Recognize the bell curve fixed by mean and SD
• Estimate percentages with the 68-95-99.7 rule

Learning Objectives for This Unit

By the end of this unit, you should be able to:

1. Recognize the bell curve, fixed by mean and SD
2. Apply the empirical rule to estimate percentages
3. Compute a z-score and find areas with technology
4. Judge when a normal model is appropriate

Recall: Mean and Standard Deviation

• Mean: the center, the balance point of the data
• Standard deviation (SD): typical distance a value sits from the mean
• Small SD → values huddle close; large SD → spread out

Two numbers — center and spread — define every curve today.

Estimate a Percentage Without Listing

A test has scores following a bell curve: mean 500, SD 100.

• What percentage of students scored above 600?
• Listing thousands of scores is impossible

Two numbers — mean and SD — are enough. Let's see how.

A Bell Curve Over Real Data

• Heights: mean 64 in, SD 2.5 in
• The smooth curve idealizes this familiar shape

Many real distributions closely approximate this normal curve.

Defining Features of the Curve

The normal curve is:

• Symmetric about the mean — mirror-image halves
• Mean = median — both at the central peak
• Single-peaked — one mode at the center
• Tails approach the axis but never touch

These features are what make a curve "normal."

Inflection Points Mark One SD

At the inflection points, the curve's bend reverses.

• They sit exactly one SD from the mean, each side
• For heights (mean 64, SD 2.5): at 61.5 and 66.5 in

The SD is the distance from peak to where the bend reverses.

The Mean Slides, the SD Stretches

• Change the mean → curve slides left or right
• Change the SD → curve stretches wider or narrower

Two numbers fully specify the curve: where, and how wide.

Area Under the Curve Is Proportion

The area under the curve over a range = the proportion of the population there.

• 30% of the area between two values → 30% of the population there
• The total area under the whole curve is 100%

To find a percentage, find an area. Everything depends on this.

Quick Check: Read the Curves

Three bell curves:

• A: tall, narrow, centered at 50
• B: short, wide, centered at 50
• C: tall, narrow, centered at 80

Which has the largest SD? Which two share a mean?

Answers: B has the largest SD (widest); A and B share the mean (both at 50).

The Empirical Rule Is Three Areas

"Area = proportion" gives us three areas worth memorizing.

• They are the proportions within 1, 2, and 3 SDs of the mean
• The rule isn't a new fact — it's three specific areas of the curve

Let's see exactly what they are.

About 68% Within One SD

• About 68% of data lies within 1 SD of the mean
• For scores (mean 500, SD 100): about 68% between 400 and 600

About 95% and 99.7% Wider Out

Widening the region around the mean:

• About 95% within 2 SD — scores 300 to 700
• About 99.7% within 3 SD — scores 200 to 800

68, 95, 99.7 — the whole rule. Memorize these three.

The Symmetry Split for Tails

68% lies between 400 and 600, so 32% is outside — split by symmetry.

• The two tails are equal: 16% above 600, 16% below 400
• So about 16% scored above 600 — our hook answered!

Symmetry splits the leftover evenly into the two tails.

Remember: The Rule Is Approximate

The figures are approximate and are percentages of area, not counts.

• For a count, multiply the percentage by the sample size
• The rule holds only for roughly bell-shaped data

"Normal" is a specific shape — not a synonym for "ordinary."

Your Turn: Estimate a Range

Battery life: roughly normal, mean 40 hours, SD 5 hours.

What percent of batteries last between 35 and 45 hours?

How far are 35 and 45 from the mean? Estimate, then advance.

Answer: 35 and 45 are each one SD from 40, so about 68% of batteries last between them.

Your Turn: A One-Sided Tail

Same batteries: mean 40 hours, SD 5 hours.

What percent last more than 45 hours?

45 is one SD above the mean. Use the symmetry split, then advance.

Answer: 68% are within one SD, leaving 32% in the tails; half is above 45, so about 16%.

Full Task: Estimate a Tail

Heights: roughly normal, mean 170 cm, SD 8 cm.

What percent of people are taller than 186 cm?

Find how many SDs out, then split.

Answer: 186 is two SDs above the mean; 95% lie within two SDs, leaving 5% in the tails, so about 2.5% above.

Key Takeaways From This Lesson

✓ A normal curve is fixed by mean and SD

✓ Area = proportion — read percentages off the shape

✓ The rule: about 68, 95, 99.7% within 1, 2, 3 SD

The figures are approximate area percentages

The rule needs bell-shaped data

Next: any value with z-scores.