Once a measurement is roughly normal, two numbers answer almost any question about it. The 68–95–99.7 rule chops the bell into eight slices whose sizes never change — so you can convert a mean and a standard deviation into real percentages, and real head counts, without a table or a calculator. Pick a measurement, build a question, and check the answer against actual simulated data.
IQ scores — μ = 100 points, σ = 15 points
34% slices13.5% slices2.35% slices0.15% tailsyour selectionsimulated data
IQ scoresμ = 100 points · σ = 15 points
Pick a real measurement
Build a question
85 points
115 points
± 2σ
The eight slices, memorised once: 0.15% | 2.35% | 13.5% | 34% || 34% | 13.5% | 2.35% | 0.15%, running from the far left tail to the far right. They add to 100%, and every empirical-rule answer is just a sum of some of them. Notice the halves: 34 + 34 = 68, then add 13.5 twice to get 95, then 2.35 twice to get 99.7.
Three limits worth stating out loud. First, the rule is a rounding — the exact areas are 68.27%, 95.45% and 99.73%, and the "2.35%" slice is really 2.14%; the panel above shows both so you can see the gap. Second, it only holds when the distribution is approximately normal and roughly symmetric; on skewed data it can produce answers that are not merely imprecise but impossible. Third, it describes what fraction of a group falls in a range — it never tells you where any one individual will land.
Check it against real data
Each draw generates fresh observations from this measurement's actual distribution, then counts how many really landed inside 1, 2 and 3 standard deviations of the mean. Nothing here is looked up — the percentages below are counted from the sample you just made.