Calculator Form
Example Data Table
| Example | Mean | Standard Deviation | Reference Count | ±1σ Range | ±2σ Range | ±3σ Range |
|---|---|---|---|---|---|---|
| Test scores | 100 | 15 | 200 | 85 to 115 | 70 to 130 | 55 to 145 |
| Machine output | 50 | 4 | 500 | 46 to 54 | 42 to 58 | 38 to 62 |
| Service times | 30 | 6 | 120 | 24 to 36 | 18 to 42 | 12 to 48 |
Formula Used
Empirical rule: approximately 68% of values lie within μ ± 1σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.
Bounds: lower bound = μ − kσ, upper bound = μ + kσ.
Expected count: reference count × inside percentage.
Advanced comparison: exact normal coverage for any k uses P(|Z| ≤ k) = erf(k / √2).
Use the empirical values for quick estimates and the exact normal comparison when you want a more precise benchmark for a chosen sigma level.
How to Use This Calculator
- Choose summary mode if you already know the mean and standard deviation.
- Choose dataset mode if you want the page to compute those values.
- Enter an optional reference count to estimate how many observations fall inside each range.
- Select a custom sigma level when you want an exact normal coverage estimate beyond 1, 2, or 3 sigma.
- Press the calculate button to show the result block above the form.
- Use the CSV or PDF buttons to save the current result.
Frequently Asked Questions
1. What does the empirical rule describe?
It estimates how data are distributed around the mean in a bell-shaped, approximately normal dataset. The common pattern is 68%, 95%, and 99.7% within one, two, and three standard deviations.
2. When should I use this calculator?
Use it when your data are reasonably symmetric and close to normal. It helps with classroom examples, quality checks, process monitoring, test score reviews, and quick probability estimates.
3. Why does the page show both empirical and exact percentages?
The empirical rule uses rounded teaching values. The exact percentages come from the normal distribution curve, so they help you compare the quick rule with a more precise estimate.
4. What is the difference between sample and population deviation?
Sample deviation uses n − 1 in the denominator and suits data taken from a larger population. Population deviation uses n and suits complete populations.
5. Can I paste raw values instead of entering mean and deviation?
Yes. Dataset mode accepts numbers separated by commas, spaces, semicolons, or line breaks. The page then calculates the mean and the selected type of deviation automatically.
6. What does reference count mean?
Reference count is the number of observations you want to evaluate. The calculator multiplies the inside percentage by that count to estimate how many values fall within each interval.
7. Does this calculator prove my data are normal?
No. It assumes approximate normality. You should still inspect histograms, Q-Q plots, skewness, kurtosis, or formal tests before treating the empirical rule as a strong model.
8. What happens if my standard deviation is zero or negative?
The calculation stops and shows an error. A valid empirical rule setup needs a positive standard deviation because the sigma intervals depend on meaningful spread.