Empirical Rule — The 68-95-99.7 Rule
The empirical rule says that for a normal distribution, about 68% of values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. In z score language: P(−1 < Z < 1) ≈ 0.68, P(−2 < Z < 2) ≈ 0.95, P(−3 < Z < 3) ≈ 0.997. It is the fastest sanity check in statistics — and this page shows exactly where the three numbers come from.
The rule read off the z table
Nothing here is a coincidence or a convention: each figure is two table reads and a subtraction, at whole-number z values.
- Within ±1: Φ(1.00) − Φ(−1.00) = 0.8413 − 0.1587 = 0.6826 — the exact figure is 68.27%. Each side band (mean to +1) holds 0.3413.
- Within ±2: Φ(2.00) − Φ(−2.00) = 0.9772 − 0.0228 = 0.9544 — exactly 95.45%. The band between 1 and 2 on each side holds 0.1359.
- Within ±3: Φ(3.00) − Φ(−3.00) = 0.9987 − 0.0013 = 0.9974 — exactly 99.73%. The 2-to-3 band holds just 0.0214 per side.
- Beyond ±3: what remains — 0.27%, split as 0.135% in each tail. This is the number behind the quality-control habit of treating anything beyond 3σ as a signal rather than noise.
The four-decimal table gives 68.26 / 95.44 / 99.74 against the exact 68.27 / 95.45 / 99.73 — a rounding wobble in the fourth place, worth knowing about so a strict marker's "discrepancy" never rattles you. The rule is usually quoted as 68–95–99.7, and for estimation that rounding is a feature: three memorable numbers that are never off by more than half a percentage point.
The full band map (the version worth memorising)
Most teaching stops at the three totals, but the individual bands are what estimation questions actually use:
- Mean to ±1σ: 34.13% each side
- ±1σ to ±2σ: 13.59% each side
- ±2σ to ±3σ: 2.14% each side
- Beyond ±3σ: 0.13% each side
With the band map you can answer questions the totals cannot: "what percentage score between 1 and 2 standard deviations above the mean?" — 13.59%, instantly. "What percentage are more than 2 SD below the mean?" — 2.14 + 0.13 = 2.27%, essentially the 2.28% tail the z score table gives at −2.00. Estimation fluency is just these four bands plus symmetry.
Worked examples
Example 1 — heights by the rule
Heights average 170 cm with SD 7 cm, roughly normal. About 95% of people fall within ±2σ: 170 ± 14, that is 156 cm to 184 cm. About 68% fall between 163 and 177 cm. And only about 0.13% — roughly 1 person in 770 — is taller than 191 cm (mean + 3σ). If a dataset of heights shows 5% of people beyond ±3σ, the distribution is not normal, whatever its owner claims.
Example 2 — exam bands
A test with mean 70 and SD 10: the rule predicts about 13.59% of students score between 80 and 90 (the +1 to +2 band), and about 2.28% score above 90. In a class of 200 that is roughly 27 students in the band and 4 or 5 above 90 — a quick forecast a teacher can hold the actual results against. Large departures mean the marks are not following a normal shape (a bimodal class, a mis-set paper), which is itself useful information.
Example 3 — when the rule must NOT be used
Income distributions are right-skewed: a long tail of very high earners pulls the mean up. Apply the rule blindly and you would predict about 16% of people earn below μ − 1σ — often a negative income, an impossibility that exposes the misuse. The empirical rule is a property of the normal curve, not of the phrase "standard deviation": check the shape first (histogram, or known behaviour of the variable), then reach for 68–95–99.7.
Empirical rule vs the exact table
The rule is an estimation instrument. It answers "roughly how many?" in seconds, with no table, and it powers quick plausibility checks everywhere from lab work to Six Sigma. What it cannot do is precision: it knows nothing about z = 1.37 or z = 2.45. Professional work moves between the two without embarrassment — the rule for the corridor conversation and the first sanity check, the calculator or table for the number that goes in the report. Knowing both, and knowing which situation you are in, is most of practical statistics. The rule also generalises one step further than most courses mention: it is the discrete warm-up for Chebyshev's inequality, which guarantees at least 1 − 1/k² of any distribution within k standard deviations — weaker numbers (75% at k = 2, against the normal's 95.45%), but with no shape assumptions at all.
Frequently asked questions
Why is it called the empirical rule?
"Empirical" because it describes what is observed to happen across measurements that follow a normal pattern — and, underneath the name, it is simply the geometry of the normal curve at whole-number z values, verifiable against the table in three subtractions.
What percentage of values lie within 1.5 standard deviations?
The rule does not cover half-steps; the table does: Φ(1.50) − Φ(−1.50) = 0.9332 − 0.0668 = 0.8664, about 86.6%. This is the general pattern — the rule for whole numbers, the table for everything else.
Does the empirical rule apply to every dataset?
No — only to distributions that are approximately normal (bell-shaped, symmetric, single-peaked). Skewed or bimodal data breaks it, sometimes badly, as the income example shows. Chebyshev's inequality is the fallback that works for any shape, with weaker guarantees.
What is the 95% in the empirical rule, exactly?
95.45% within ±2 standard deviations, conventionally rounded to 95%. Do not confuse it with the middle-95% interval between z = −1.96 and +1.96, which is exactly 95.00% — two slightly different "95%s" that both appear in textbooks, each correct in its own sentence.
See the bands come from the table yourself on the standard normal table page, and drill the numbers in the practice set.