Empirical Rule Calculator

What the 68, 95, 99.7 rule covers

The rule describes how a bell shaped curve piles up around its mean. About 68.27% of the values sit within one standard deviation of the mean, about 95.45% within two, and about 99.73% within three. Those shares come from the area under a normal curve, not from one particular data set, so they hold for any mean and any standard deviation. That is what makes the rule portable: a scale in pounds, a scale in test points and a scale in millimeters all follow the same three steps.

Read it as nested bands. A band is the mean plus or minus k standard deviations, where k is 1, 2 or 3. Each band sits over the one inside it, which is why the shares climb so fast. The first band catches the bulk of the crowd, the second picks up the shoulders, and the third leaves out roughly one value in 370.

In daily use the rule lets you skip a z table. If a process has a mean of 100 and a standard deviation of 15, you can say in your head that 85 to 115 holds about two thirds of the output and 70 to 130 holds about 19 values in 20. The NIST/SEMATECH e-Handbook entry on the normal distribution gives the density behind the areas, and the University of Houston STAT 2311 lecture slides walk through the same three percentages.

Each band written as numbers

The arithmetic is one line per band. The low end is the mean minus k standard deviations and the high end is the mean plus k. With a mean of 100 and a standard deviation of 15, the first band runs 100 - 15 = 85 to 100 + 15 = 115. The second runs 100 - 30 = 70 to 100 + 30 = 130, and the third runs 100 - 45 = 55 to 100 + 45 = 145. The full span of the third band is 90 points, which is 6 standard deviations wide.

Two columns are worth keeping apart: the share inside a band and the share left outside it. Outside is split evenly between the two tails, so the middle 95.45% at k = 2 leaves 2.275% below 70 and 2.275% above 130. When a headline quotes a one sided number, check whether someone already halved the tail.

The per 1,000 column is the one people reuse. Multiply the share by the group size and you have an expected count. At k = 1 the count is 0.6827 x 1,000 = 682.7; at k = 2 it is 0.9545 x 1,000 = 954.5; at k = 3 it is 0.9973 x 1,000 = 997.3. Round those to 683, 955 and 997 and the table reads like a head count instead of a probability.

Worked example: mean 100, standard deviation 15

This is the shape most people meet first, and the numbers are easy to carry around. A group of 1,000 puts 0.6827 x 1,000 = 682.7 people between 85 and 115, so about 683. At k = 2 the count is 0.9545 x 1,000 = 954.5 people between 70 and 130, so about 955. At k = 3 it is 0.9973 x 1,000 = 997.3 people between 55 and 145, so about 997. Only about 3 people in that group of 1,000 land outside the widest band.

The tails answer the questions people ask most. A score of 130 or higher is 2 standard deviations above the mean, and the outside share of 0.0455 splits into two tails of 0.02275 each. Multiply: 0.02275 x 1,000 = 22.75, so about 23 people out of 1,000. A score of 85 or lower is the mirror case: 0.15865 x 1,000 = 158.65, so about 159 people.

Keep the word expected in view. These are averages over many groups, not a promise about the next 1,000 people you test. A group of that size with a mean of 100 and a standard deviation of 15 will land near those counts most of the time, with sampling noise of a few dozen on the tail numbers. A group of 40 will swing much wider.

Worked example: 240 students, mean 72, SD 8

Same steps, smaller group. The first band is 72 - 8 = 64 to 72 + 8 = 80. The second is 72 - 16 = 56 to 72 + 16 = 88. The third is 72 - 24 = 48 to 72 + 24 = 96.

Now turn the shares into students. A class of 240 puts 0.6827 x 240 = 163.848 students between 64 and 80, so about 164. At two standard deviations the count is 0.9545 x 240 = 229.08, so about 229 students sit between 56 and 88. At three it is 0.9973 x 240 = 239.352, so about 239 of the 240 sit between 48 and 96. The 0.27% outside the widest band works out to 0.648 students in 240, so a class that size often sees none at all.

Above 88 is the upper tail at k = 2: 0.02275 x 240 = 5.46, so about 5 students. Below 56 is the lower tail, the same 5 or so. A score of 92 is a different question, because 92 sits 2.5 standard deviations out and the rule only marks 1, 2 and 3. Work out (92 - 72) / 8 = 2.5 and read the tail from a table: 0.62%, or about 1.5 students in 240. The rule is a shortcut for three cutoffs, not a substitute for the whole curve.

Turning shares into head counts

Shares are easy to quote and hard to picture. Multiply the share by the group size and you get an expected count. The two shares you will use most are 0.6827 and 0.9545, and both scale in a straight line, so one multiplication covers any group size.

Read the 2,500 row as an example. A plant checking 2,500 parts expects 0.9545 x 2,500 = 2,386.25 inside the 2 SD band, so about 2,386, and 0.0455 x 2,500 = 113.75 outside it, so about 114. Those 114 parts are the group an inspector opens first, because the rule flags them as unusual before anyone measures a single one.

Two cautions. A count is not a quota: you cannot force exactly 5 of 100 people outside a band, and small groups jump around. And the count inherits the shape assumption, so on skewed data it can be off by a wide margin.

Running the rule backwards: how wide is the band?

Sometimes the share is fixed and the width is the unknown. Say you need the middle 95% of a normal curve. The rule's k = 2 gives 95.45%, a little more than asked, so the exact k is 1.96. With a standard deviation of 15 the half width is 1.96 x 15 = 29.4, so the band runs 100 - 29.4 = 70.6 to 100 + 29.4 = 129.4.

Other common targets, same mean and standard deviation. The middle 90% needs k = 1.645, so 1.645 x 15 = 24.675 and the band runs 75.3 to 124.7. The middle 99% needs k = 2.576, so 2.576 x 15 = 38.64 and the band runs 61.36 to 138.64. The middle 99.7% is the rule's own k = 3, so the band is 55 to 145 with nothing left to compute.

The pattern is worth remembering: 2 standard deviations is a fast stand in for 95%, and it over covers by 0.45 points. For a rough plan that is fine. For a published tolerance or a test spec, use the exact k and show the work.

Three bands, three jobs

Each width has a natural job. One standard deviation is the everyday range, holding two thirds of a group, so it is where the typical case lives. Two standard deviations is the flag line: a value outside it is unusual enough to check, yet not rare, since about 1 value in 22 lands there. Three standard deviations is the alarm line: about 1 value in 370, which in a lab or on a line usually means something changed.

Two rules of thumb fall out of the table. A screening cutoff set at 2 standard deviations will pull in about 1 person in 22 for a closer look, which is why busy clinics do not use it as a diagnosis. An alarm set at 3 standard deviations fires about once in 370 checks, which is often too rare for a small shop: 100 checks a day means an alarm every four days or so.

Nothing in the rule says a process is good or bad. It says how often a bell shaped process lands in a window. Whether a 5% miss rate matters is a decision about costs, not about the curve.

Where the rule breaks, and when to open a z table

The rule assumes a normal shape, and that assumption fails in predictable places: incomes and home prices with long right tails, wait times and repair bills that cannot go below zero, test scores pressed against a ceiling, and any group that mixes two populations such as beginners and experts in one class. On data like that the middle band still holds most values, but the tail counts can be wrong by an order of magnitude.

When the bell shape is hard to defend, Chebyshev's inequality is the floor that always holds. At least 1 - 1/k squared of the values sit within k standard deviations of the mean, whatever the shape. At k = 2 the floor is 1 - 0.25 = 0.75, so at least 75%. At k = 3 it is 1 - 0.1111 = 0.8889, so at least 88.89%. Set those next to the rule's 95.45% and 99.73% and the trade is clear: the rule buys sharper numbers, and the price is the shape assumption.

Open a z table or a stats package when you need a share between two arbitrary cutoffs, a percentile for one score, a tail past 3 standard deviations, or a number you plan to publish. The rule marks three points on the ruler, and the table marks every point. The University of South Carolina STAT 110 slides put the rule and the table side by side for that reason.