The Empirical Rule and Where It Stops

Bands on a mean of 100 and a spread of 15

A placement test has a mean of 100 and a standard deviation of 15, and the scores are close to bell-shaped. The empirical rule sketches three bands. Within one standard deviation, from 85 to 115, you expect about 68 percent of scores. Within two, from 70 to 130, about 95 percent. Within three, from 55 to 145, about 99.7 percent. The complements are part of the rule. About 32 percent sit outside the first band, about 5 percent outside the second, and about 0.3 percent outside the third. Those are model shares, not a count you have already made.

On 200 students, 68 percent is 136 students inside 85 to 115, 95 percent is 190 inside 70 to 130, and 99.7 percent is about 199 inside 55 to 145. Expect roughly one student beyond the three-spread band in a group this size, and do not expect one in a class of 30, where 0.3 percent of 30 is about a tenth of a student. If a class of 30 contains two scores past 145 or below 55, trust the papers enough to recheck the arithmetic, then trust the count over the sketch. The rule is a description of a shape, and the roster is a description of these people.

A histogram that refuses the sketch

Suppose 200 incomes are piled near a low center with a long right tail. The mean might be 45 thousand and the standard deviation 20 thousand, so the one-spread band runs from 25 to 65 thousand. You will not find 68 percent of households in that band. A large share sit below the mean, and a thin tail runs far above 65. Applying the 68 percent figure would overstate how many people are near the average and understate the tail. The Empirical Rule Calculator will still draw the bands from whatever mean and spread you type. The percentages are a claim about shape that you have to bring yourself.

Check the claim with a count before you publish it. Sort the 200 scores and tally how many fall from 85 to 115. If you count 150, that is 75 percent, a bit tighter than 68. If you count 100, that is 50 percent, and the rule is the wrong sentence for this list. Report the count. You can still mention the rule as a benchmark: this list put 50 percent inside one spread, against a normal sketch of about 68. That comparison is informative. Quoting 68 when you counted 100 is not a simplification. It is a different list.

The thin edge past three spreads

About 0.3 percent beyond three standard deviations is 3 per 1,000 under the bell sketch, split into both tails, so roughly 1 or 2 per 1,000 in a single tail. In a school of 1,000, a bell-shaped test would put a handful of students outside 55 to 145 if the mean is 100 and the spread is 15. Finding 40 students outside that band is not a handful, and it is evidence that the scores are heavier-tailed or that the spread you used is not the spread of this sitting. Recalculate the standard deviation from the sitting you are describing before you declare the students unusual.

The edge is also a bad place to round casually. A score of 146 is just outside the three-spread line of 145, and a score of 144 is just inside. Both are far from 100. Treating 146 as a different species from 144 because the rule's boundary fell between them overreads a round number. Use the band to describe the pile, not to write a policy that flips at exactly three standard deviations. If a policy needs a cutoff, choose it from the decision, state it as a raw score, and say what share of last year's list would have fallen past it.

One number from each side is not the rule

The rule is about both sides together. About 68 percent sit between the mean minus one spread and the mean plus one spread. It does not say that 68 percent sit above the mean, and it does not say that 34 percent sit in the slice from the mean to one spread above, as a separate claim you can detach without the matching lower slice. The upper slice from 100 to 115 holds about 34 percent only if the shape is symmetric. A skewed list can put most of its central mass on one side. Quote the full interval, from 85 to 115, unless you have actually counted one side.

A worked slice check keeps the habit honest. On a symmetric bell, the band from the mean to one spread above, 100 to 115, is about 34 percent, and the band from one to two spreads above, 115 to 130, is about 13.5 percent. Beyond two spreads on one side is about 2.5 percent. If you need the share above 130, say about 2.5 percent under the sketch, about 5 of 200 students. Count them. If the upper tail has 20 and the lower tail has 1, symmetry has already failed and the halved leftovers are the wrong tool.

Sample size and a shaky spread

The bands are only as good as the mean and the spread you plugged in. On 8 scores, a sample standard deviation can move a lot when one score changes, and the 68 percent band moves with it. Drawing a precise 85-to-115 style interval from a spread estimated on a handful of rows pretends the edges are known. With a small class, report the actual minimum and maximum, and treat the empirical percentages as a later lesson about shape, not as a prediction of how many students will land inside a line you drew from n equals 8.

Also match the divisor to the spread you think you have. A population standard deviation of 15 and a sample standard deviation of 15 are the same input to the band arithmetic, and they are not the same claim about where the 15 came from. If 15 was estimated from these 200 students, the band is a description of these 200, rebuilt from their own spread, which will fit a little too well. If 15 is a published reference from a national form, the band is a comparison of this room to that reference. Say which 15 you used. The percentages do not reveal the source.

Where to stop quoting 68, 95, and 99.7

Stop when the shape is the thing you are worried about. Incomes, waiting times, and test scores with a hard floor at zero are the usual refusals. A waiting time cannot be negative, so a mean of 4 minutes and a standard deviation of 6 minutes produces a lower band at negative 2, which is not a time. The rule has already left the building. Report a median and a pair of percentiles instead, or show the histogram. The sketch is for lists that pile in the middle and thin out on both sides without a wall.

Use the rule when a quick share is enough and the histogram looks roughly bell-shaped. It is the right tool for a sentence like this: about 95 percent of these placement scores should fall between 70 and 130 if this sitting resembles the reference mean of 100 and spread of 15. It is the wrong tool for setting a medical or hiring cutoff, and it is the wrong tool for a list you have not looked at. Recommend a count on the list you have, and use 68, 95, and 99.7 as the benchmark those counts are compared with, not as a substitute for the counts.

Open the worksheet

The matching page is the Empirical Rule Calculator. It opens with a worked example, names the formula, and keeps the steps next to the result. Use it when you want the arithmetic on your own numbers rather than the argument on this page.