Empirical rule calculator

Find the mean ± one, two and three standard deviations for a proposed normal model.

On your device · No account

Privacy details

Empirical rule calculator processes your input in this browser. Inputs and results are excluded from analytics. Recent tools save tool names, visit counts and last-visit times. Saved tools store only their names. Draft saving is optional and stays on this device; delete saved text with the draft controls. Loading text from a URL is optional and contacts that server; its URL and your IP address are visible to the server. An optional “Use in” action keeps a handoff in this tab. The next tool removes it on arrival and ignores it if more than a minute has passed.

Copy link includes the inputs you choose to share in the address. Anyone with that link can reopen them. Files and generated results are excluded. Password and key tools share settings only.

Mean

Standard deviation

Answer

…

Find the mean ± one, two and three standard deviations for a proposed normal model.

More results

Precision and calculation record

Decimal places in numeric results and text downloads

Rounding changes displayed numbers, not the calculation. Text, fractions, matrices and CSV schedules keep their own formatting. Automatic keeps up to 12 significant digits, with money shown to cents where applicable.

Saved scenario

Save the current answer, then change an input to compare. This clears when you leave.

Formula and assumptions

For an exact normal distribution these intervals contain about 68.27%, 95.45% and 99.73% of its probability. These percentages do not automatically apply to skewed or heavy-tailed data.

Automatic precision shows numbers to 12 significant digits. Thousands separators such as 1,000 are accepted; use a dot for decimals.

Scratch calculator

Check another calculation

Use +, −, ×, ÷, parentheses and %. A percentage is divided by 100. This does not change the main tool.

Reset reloads the tool with its starting values and releases open files and device controls. Saved drafts stay on this device until you delete them.

Links include your inputs and settings. Anyone with the link can see them, so leave out private text. Files and generated results are left out; random draws run again.

Keyboard shortcuts

Updated 2026-09-26

Using Empirical rule calculator

Find the mean ± one, two and three standard deviations for a proposed normal model.

For an exact normal distribution these intervals contain about 68.27%, 95.45% and 99.73% of its probability. These percentages do not automatically apply to skewed or heavy-tailed data.

Next: Normal distribution calculator

A worked example

Example inputs: mean = 100, standard deviation = 15. One-SD lower bound = 85.

Before you use the result

For an exact normal distribution these intervals contain about 68.27%, 95.45% and 99.73% of its probability. These percentages do not automatically apply to skewed or heavy-tailed data.

Example results for mean

These examples use Standard deviation: 15. They are reference calculations, not recommended settings.

Mean	One-SD lower bound
50	35
100	85
200	185

Cite this page

Citation style

ToolOctopus. “Empirical rule calculator.” Updated 2026-09-26. https://tooloctopus.com/empirical-rule-calculator.

Add an access date if your instructions require one.

Citation formatting uses citeproc-js by Frank Bennett and Citation Style Language styles. Licence and source code.