68-95-99.7 Rule Explained
The 68-95-99.7 rule works because the normal curve has a fixed relative shape: when distance from the mean is measured in standard deviations, fixed areas under the curve—68%, 95%, and 99.7%—always contain the data.
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If every normal distribution can have a different mean or spread, why are 68%, 95%, and 99.7% always the same?
- normal-curve
- The bell-shaped density curve of a normal distribution; the total area underneath it equals 1, or 100%.
- sd-units
- Distance from the mean expressed in standard deviations (z-scores); each tick mark is one SD.
- area-probability
- The area between two values on a density curve equals the probability of falling in that interval.
- fixed-shape
- After rescaling by the mean and SD, every normal curve has the same relative shape, so the same SD intervals capture the same area.
- empirical-rule
- For a normal distribution, about 68%, 95%, and 99.7% of data lie within 1, 2, and 3 SDs of the mean.
The 68-95-99.7 rule applies to every data set.
Show that the rule is only valid when data are approximately normal; non-normal shapes break the percentages.
The percentages in the 68-95-99.7 rule are arbitrary.
Show that the percentages are exact areas under the normal density curve, not chosen at random.
Each standard-deviation step adds the same amount of data.
Show that the normal curve is highest near the mean, so the first SD interval covers much more area than the next.
- Mean and standard deviation
- Basic reading of a bell-shaped histogram
- Percentages as parts of a whole
- calculus integration behind the normal formula
- confidence intervals and hypothesis testing
- non-normal distribution families in detail
- Explain where the 68%, 95%, and 99.7% values come from
- Estimate the percentage of data between the mean and 1 SD
- Decide whether the rule can be used for a given histogram
- When looking at an approximately normal histogram or distribution, use 68-95-99.7 to make quick mental probability estimates and to flag unusual values beyond 3 SDs.
An introductory statistics learner who can interpret a bell curve and knows what standard deviation means, but has not studied calculus.
- 01The 68-95-99.7 PatternslideOrientationObserve
Open with the surprising pattern that every normal distribution shows the same percentages, then set up the driving question of the course.
- Bell-shaped data always show the same 68-95-99.7 pattern
- Mean and spread can differ, but the percentages do not
- Goal: find out where the numbers come from
- 02Commit to a GuessquizPredictionPredict
Before any explanation, learners commit to their guess for the percentage within one standard deviation, creating a need to know the answer.
- About what share of normal data lies within 1 SD of the mean?
- Choose an answer before seeing the derivation
- 03Area Under the CurveslideModel buildingObserve
Build the key idea that a density curve's total area is 100%, and a slice between two values represents the percentage of data in that range.
- Total area under the normal curve = 100%
- Area between two points = share of data in that interval
- This is how probabilities become visible on the curve
- 04Discover the Areas YourselfinteractiveModel buildingConstruct
Use a slider to move shaded boundaries along the normal curve and read the area under it, discovering that the percentages are fixed by the curve itself.
- Adjust the boundary from 0 to 1 SD and see about 34%
- Try 1 to 2 SDs and compare the much smaller area
- The percentages are built into the curve, not guessed
- 05Same Shape, Any BellslideModel buildingObserve
Show why changing the mean or standard deviation does not change the relative areas: it only shifts or stretches the same fixed shape.
- Changing the mean slides the whole curve left or right
- Changing the SD stretches or squeezes the x-axis
- In SD units, every normal curve has the same shape and areas
- 06The Rule in One PictureslideModel buildingObserve
Draw the boundaries at 1, 2, and 3 standard deviations and label the shaded areas to reveal the 68-95-99.7 rule.
- [Chart] Normal curve with ±1, ±2, and ±3 SD regions shaded
- ±1 SD shades about 68%
- ±2 SD shades about 95%
- ±3 SD shades about 99.7%
- 07Between 1 and 2 SDsquizPracticeApply
Use the known 68% and 95% totals to estimate the leftover area between one and two standard deviations above the mean.
- 68% lies within ±1 SD
- 95% lies within ±2 SDs
- Find the leftover area on one side
- 08Where the Rule BreaksslideMisconception repairExplain
Clarify that the rule only applies to approximately normal data and reinforce that the percentages are produced by the curve's shape.
- The rule works only for approximately normal data
- Check a histogram for rough symmetry before using it
- The curve is tallest near the mean, so the first SD step holds far more data than later steps
- 09Use It or Not?quizApplicationApply
Apply the rule to realistic histograms and decide when it is appropriate to quote 68-95-99.7.
- A right-skewed salary distribution does not qualify
- An approximately normal histogram does
- Estimate the share within 2 SDs from normal data
- 10Why It Works in One SentenceslideSynthesisExplain
Restate the full causal chain and invite learners to articulate the explanation in their own words.
- Total area under any density curve is 100%
- Standard deviations are the measuring tape along the x-axis
- The normal curve's fixed shape assigns fixed areas: 68%, 95%, and 99.7%
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