Analyze normal distributions with practical sigma bands. Check tails, central coverage, and custom score ranges. See charts, exports, and examples for faster data decisions.
Use this tool for approximate probabilities on near-normal data. The full page stays single-column, while the input grid adapts across screen sizes.
The curve is a visual normal reference. The reported probabilities follow the empirical-rule approximation.
About 68% of observations fall within one standard deviation of the mean.
About 95% of observations fall within two standard deviations of the mean.
About 99.7% of observations fall within three standard deviations of the mean.
Example distribution: mean = 100, standard deviation = 15.
| Scenario | Value Range | Z-Range | Approximate Probability | Approximate Percentage |
|---|---|---|---|---|
| Central 1σ coverage | 85 to 115 | -1σ to +1σ | 0.6800 | 68.00% |
| Central 2σ coverage | 70 to 130 | -2σ to +2σ | 0.9500 | 95.00% |
| Central 3σ coverage | 55 to 145 | -3σ to +3σ | 0.9970 | 99.70% |
| Lower tail | Below 85 | Below -1σ | 0.1600 | 16.00% |
| Upper tail | Above 130 | Above +2σ | 0.0250 | 2.50% |
The empirical rule estimates probabilities for near-normal data using standard deviation bands around the mean.
Z-score conversion: z = (x - μ) / σ
Custom intervals: For custom cut points, the calculator linearly interpolates within empirical-rule checkpoints at -4σ, -3σ, -2σ, -1σ, 0σ, +1σ, +2σ, +3σ, and +4σ.
Segment probabilities used: 34% from the mean to ±1σ on each side, 13.5% from 1σ to 2σ on each side, 2.35% from 2σ to 3σ on each side, and 0.15% from 3σ to 4σ on each side.
It is the 68-95-99.7 guideline for bell-shaped distributions. About 68% lie within one standard deviation, 95% within two, and 99.7% within three.
Use it when your data is approximately normal and you need quick probability estimates for ranges, tails, or sigma-based coverage.
No. This tool gives an empirical-rule approximation. It is fast and practical, but exact normal calculators use a more precise cumulative distribution function.
Z-scores convert raw values into standard deviation units. That makes it easy to map values onto empirical-rule probability bands.
Yes. Choose the lower-tail or upper-tail mode, enter a target value, and the calculator estimates the probability on that side.
The outer 3σ to 4σ bands smooth the approximation for custom intervals. They help produce cleaner estimates near the extremes.
It multiplies the estimated probability by your sample size. That gives an approximate count of values expected in the selected region.
Yes. After calculating, use the CSV or PDF buttons to download a concise report of your current inputs and results.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.