The Bell and Its Rule
The normal curve is a symmetric, single-peaked distribution whose 68–95–99.7 rule fixes what fraction of data falls within one, two, and three standard deviations of the mean. · 11 min
Many measurements pile up the same way: most values cluster near the middle, and fewer appear as you move out toward the extremes. Adult heights, standardized test scores, and repeated measurement errors all tend to follow this pattern. When a distribution is symmetric and single-peaked in this particular way, statisticians call it normal — and its most useful feature is that a single rule tells you how much of the data sits near the center.
Guess before you learn
A large group of adult men has a mean height of 70 inches with a standard deviation of 3 inches. Roughly what fraction of the men stand between 67 and 73 inches — that is, within one standard deviation of the mean?
Within one standard deviation of the mean lies about 68% of a normal distribution — close to two thirds. If you guessed higher, keep that pencil mark: the rule you are about to learn assigns a different, larger fraction to each additional step out from the center.
9–12
3–5
A normal distribution is a smooth, symmetric hill, with its peak at the average. Take one step of the usual spacing — one standard deviation — out from the peak in both directions, and you have already fenced in about 68 out of every 100 values. Take two steps and you capture about 95 out of 100. Three steps hold almost everyone, about 997 out of 1000. Near the middle, most of the data; far out, only a little.
6–8
A normal distribution is symmetric about its mean, with a single peak at the mean and tails that thin out smoothly on both sides; the standard deviation sets its width. The empirical rule, also called the 68–95–99.7 rule, gives the fraction of data near the center: about 68% of values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. Because the curve is symmetric, each band splits evenly — 68% within one standard deviation means about 34% on each side.
9–12
A normal distribution is fixed by two numbers: its mean μ, which locates the peak, and its standard deviation σ, which sets its width. Measured in units of σ, every normal curve has the same shape, so the same rule always applies: about 68% of the data lies in μ ± 1σ, 95% in μ ± 2σ, and 99.7% in μ ± 3σ. The bands are nested, so you can read between them by subtracting: between 1σ and 2σ from the mean lies about 95% − 68% = 27%, or 13.5% on each side. Beyond 3σ lies only about 0.3% — which is why a value that far out counts as genuinely unusual.
K–2
Line up your class by height. A few are short, a few are tall, and most stand near the middle. Draw their heads as a bump: low at the sides, high in the middle. That bump is the bell.
Undergrad
The normal density is f(x) = (1 / (σ√(2π))) · exp(−(x − μ)² / (2σ²)), a two-parameter family. The empirical rule is just the integral of this density over μ ± kσ for k = 1, 2, 3, which evaluates to 0.6827, 0.9545, and 0.9973. These areas depend only on k, not on μ or σ, because the substitution z = (x − μ)/σ collapses every normal curve onto the same standard one. That scale-invariance is what makes the rule portable: IQ (σ = 15) and adult height (σ ≈ 3 in) obey the same three percentages. The rule is exact for a true normal and only as good as the data's normality otherwise.
Postgrad
The three coverage probabilities are values of the error function: P(|Z| ≤ k) = erf(k/√2) for the standard normal Z. The normal law arises as the attractor in the central limit theorem, which is why aggregated measurements approximate it — and why the empirical rule earns its ubiquity rather than merely asserting it. Two cautions define the mature view. The tails are thin: P(|Z| > 3) ≈ 0.0027, so the normal model badly understates genuine outliers in heavy-tailed data such as finance or seismology. And 'approximately normal' is a hypothesis to be checked with a normal quantile plot, never assumed — the rule is a consequence of a model, and the model is a claim about the data.
the empirical rule
The 68–95–99.7 rule: for a normal distribution, about 68%, 95%, and 99.7% of the data lie within 1, 2, and 3 standard deviations of the mean.
What fraction of data lies between 64 and 76 on a normal distribution with mean 70 and standard deviation 3? — the steps fade as you master them
(64 − 70)/3 = −2 and (76 − 70)/3 = +2
about 95%
About 95% of values fall between 64 and 76
The rule is only as good as the assumption behind it: the data must be roughly normal. A strongly skewed distribution — household incomes, say — breaks the symmetry the rule depends on, and the percentages no longer hold. So before you reach for 68–95–99.7, look at the shape you learned to read in the previous folio.
The bell and its rule give you a fast reading of any roughly normal distribution: name the mean, name the standard deviation, and you can already say where about 68, 95, and 99.7 percent of the data must lie. Next, you will turn that same standard-deviation ruler on a single value, to say exactly how unusual it is.
Practice — new ink and old, interleaved
1.In a right-skewed distribution, which is typically larger?
2.Which measure of spread is least disturbed by a single extreme outlier?
3.Order these bands from the smallest to the largest fraction of data captured.
- within 2 standard deviations
- within 1 standard deviation
- within 3 standard deviations
4.Household incomes are strongly right-skewed. Which center should a report use for the typical household?
5.A histogram has one tall bar on the left and a long tail stretching to the right. Its shape is:
6.IQ scores are normal with mean 100 and standard deviation 15. About what percent score between 85 and 115? Enter a whole number.
7.For the sorted data 3, 5, 6, 8, 10, 11, 14, compute the IQR (Q3 − Q1).
8.Find the median of 5, 2, 9, 4, 12, 7.
9.You roll a fair die 6000 times and make a histogram of the results 1 to 6. What shape do you expect?