Calculator inputs
Enter your distribution settings below. The calculator accepts a mean, standard deviation, optional observed value, and either a custom sigma interval or manual bounds.
Example data table
This example uses a normal distribution with mean 100 and standard deviation 15. It shows how the empirical rule estimates expected coverage around the center.
| Band | Lower bound | Upper bound | Coverage | Expected count in 1,000 | Interpretation |
|---|---|---|---|---|---|
| ±1σ | 85 | 115 | About 68% | About 680 | Most observations cluster near the center. |
| ±2σ | 70 | 130 | About 95% | About 950 | Only a small share falls outside. |
| ±3σ | 55 | 145 | About 99.7% | About 997 | Extreme values become very rare here. |
Formula used
1) Empirical rule intervals
For a normal distribution:
±1σ interval: μ ± 1σ
±2σ interval: μ ± 2σ
±3σ interval: μ ± 3σ
2) Empirical rule percentages
Approximately 68% of values fall within ±1σ, 95% within ±2σ, and 99.7% within ±3σ.
3) Z-score for an observed value
z = (x − μ) / σ
4) Exact coverage from the normal distribution
P(a ≤ X ≤ b) = CDF(b) − CDF(a)
How to use this calculator
- Enter the mean of your dataset.
- Enter the standard deviation.
- Add an observed value if you want a z-score.
- Choose a custom sigma multiplier, or enter manual bounds.
- Set your preferred decimal precision.
- Click Calculate now.
- Review the interval table, custom coverage, and graph.
- Download the results as CSV or PDF when needed.
Frequently asked questions
1) What does the 68 95 99 rule describe?
It summarizes how values cluster in a normal distribution. About 68% lie within one standard deviation, 95% within two, and about 99.7% within three standard deviations from the mean.
2) When should I use this calculator?
Use it when your data is reasonably bell-shaped and you want quick interval estimates, z-scores, percentile context, or probability coverage around the mean.
3) Why does the third rule level usually mean 99.7%?
The third interval is commonly rounded from the exact normal probability. Many people shorten the name to 68 95 99, but the more precise value is about 99.7%.
4) What is the difference between theoretical and exact coverage?
Theoretical coverage is the well-known empirical rule approximation. Exact coverage is calculated from the normal distribution CDF using your entered mean, standard deviation, and interval bounds.
5) What does the observed value section tell me?
It shows the z-score, percentile, probability below the value, probability above the value, and which sigma band the observation belongs to.
6) Should I use manual bounds or custom sigma?
Use custom sigma when you want symmetric intervals around the mean. Use manual bounds when your interval is fixed by business rules, limits, or testing thresholds.
7) Can I use this for skewed data?
Be careful. The empirical rule works best for approximately normal data. Strong skewness, heavy tails, or multimodal patterns can make these percentages misleading.
8) Why include CSV and PDF downloads?
They make it easier to save, share, document, and review your statistical summaries outside the calculator, especially for reports, audits, or classroom work.