Concept Lesson

Empirical Rule: 68–95–99.7 Rule for Normal Distributions

A lesson in the 68-95-99.7 rule that moves from intuition and definitions to worked reasoning, error correction, and independent practice.

Course status: Revised 2026-27 course
Updated: July 18, 2026
Practice: Easy, Tough and Toughest

Lesson Goals: Empirical Rule

The 68-95-99.7 rule is an approximation for distributions that are reasonably normal and cannot be applied automatically to skewed, multimodal, or heavy-tailed data.

Reader taskone-, two-, and three-standard-deviation regions and appropriate use

Planned modules6

Mathematics2 expressions

Worked checks60

Boundary: The empirical rule is an approximation for roughly normal distributions, not all data.

68–95–99.7 rule

68–95–99.7 rule in empirical rule: The standardized score is z=-1.50; the modeled proportion at or below x is approximately 0.0668. The value is 1.50 standard deviations below the model mean.

Worked reasoning

For 68–95–99.7 rule in empirical rule, In a constructed normal model for an online-course completion sample, $\mu=73$ and $\sigma=10$ . Analyze $x=58.0$ using 68–95–99.7 rule.

z=\frac{58.0-73}{10}=-1.50,\qquad P(X\le 58.0)=\Phi(-1.50)\approx0.0668.

When the idea is valid

For 68–95–99.7 rule in empirical rule, A normal probability statement requires a reasonable normal model; standardization alone does not establish shape.

Misconception to remove

For 68–95–99.7 rule in empirical rule, reject this error: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

One, two, and three standard deviations

One, two, and three standard deviations in empirical rule: The standardized score is z=-1.00; the modeled proportion at or below x is approximately 0.1587. The value is 1.00 standard deviations below the model mean.

Worked reasoning

For One, two, and three standard deviations in empirical rule, In a constructed normal model for a greenhouse germination experiment, $\mu=63$ and $\sigma=8$ . Analyze $x=55.0$ using one, two, and three standard deviations.

z=\frac{55.0-63}{8}=-1.00,\qquad P(X\le 55.0)=\Phi(-1.00)\approx0.1587.

When the idea is valid

For One, two, and three standard deviations in empirical rule, A normal probability statement requires a reasonable normal model; standardization alone does not establish shape.

Misconception to remove

For One, two, and three standard deviations in empirical rule, reject this error: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Estimating counts and percentages

Estimating counts and percentages in empirical rule: The standardized score is z=0.50; the modeled proportion at or below x is approximately 0.6915. The value is 0.50 standard deviations above the model mean.

Worked reasoning

For Estimating counts and percentages in empirical rule, In a constructed normal model for a quality-control inspection, $\mu=70$ and $\sigma=13$ . Analyze $x=76.5$ using estimating counts and percentages.

z=\frac{76.5-70}{13}=0.50,\qquad P(X\le 76.5)=\Phi(0.50)\approx0.6915.

When the idea is valid

For Estimating counts and percentages in empirical rule, A normal probability statement requires a reasonable normal model; standardization alone does not establish shape.

Misconception to remove

For Estimating counts and percentages in empirical rule, reject this error: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Reverse problems

Reverse problems in empirical rule: The standardized score is z=1.25; the modeled proportion at or below x is approximately 0.8944. The value is 1.25 standard deviations above the model mean.

Worked reasoning

For Reverse problems in empirical rule, In a constructed normal model for a greenhouse germination experiment, $\mu=73$ and $\sigma=8$ . Analyze $x=83.0$ using reverse problems.

z=\frac{83.0-73}{8}=1.25,\qquad P(X\le 83.0)=\Phi(1.25)\approx0.8944.

When the idea is valid

For Reverse problems in empirical rule, A normal probability statement requires a reasonable normal model; standardization alone does not establish shape.

Misconception to remove

For Reverse problems in empirical rule, reject this error: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

When not to use it

When not to use it in empirical rule: The standardized score is z=2.00; the modeled proportion at or below x is approximately 0.9772. The value is 2.00 standard deviations above the model mean.

Worked reasoning

For When not to use it in empirical rule, In a constructed normal model for a greenhouse germination experiment, $\mu=60$ and $\sigma=13$ . Analyze $x=86.0$ using when not to use it.

z=\frac{86.0-60}{13}=2.00,\qquad P(X\le 86.0)=\Phi(2.00)\approx0.9772.

When the idea is valid

For When not to use it in empirical rule, A normal probability statement requires a reasonable normal model; standardization alone does not establish shape.

Misconception to remove

For When not to use it in empirical rule, reject this error: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Practice

Practice in empirical rule: The standardized score is z=-1.50; the modeled proportion at or below x is approximately 0.0668. The value is 1.50 standard deviations below the model mean.

Worked reasoning

For Practice in empirical rule, In a constructed normal model for a manufacturing fill-volume check, $\mu=63$ and $\sigma=12$ . Analyze $x=45.0$ using practice.

z=\frac{45.0-63}{12}=-1.50,\qquad P(X\le 45.0)=\Phi(-1.50)\approx0.0668.

When the idea is valid

For Practice in empirical rule, A normal probability statement requires a reasonable normal model; standardization alone does not establish shape.

Misconception to remove

For Practice in empirical rule, reject this error: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Formula and Notation Reference

Empirical rule within one standard deviation

P(\mu-\sigma<X<\mu+\sigma)\approx0.68

Empirical rule within one standard deviation in Empirical Rule: Define the event or random variable first; complements, conditioning, and trial assumptions determine which probability expression applies.

Empirical rule within two standard deviations

P(\mu-2\sigma<X<\mu+2\sigma)\approx0.95

Empirical rule within two standard deviations in Empirical Rule: Define the event or random variable first; complements, conditioning, and trial assumptions determine which probability expression applies.

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Guided, Independent and Challenge Practice

Every question in Empirical Rule: 68–95–99.7 Rule for Normal Distributions is newly written from the revised framework and the logic visible in public College Board materials. Constructed numerical settings are identified as instructional scenarios and are never represented as measurements from a real population. No released or secure question wording is reproduced.

Easy Practice

Easy 1: When not to use it

Question P32-Easy-1. In a constructed normal model for a greenhouse germination experiment, $\mu=67$ and $\sigma=11$ . Analyze $x=50.5$ using when not to use it.