Calculate one, two, and three sigma coverage instantly. Download results, inspect examples, and compare intervals. Use clean inputs for quick normal-distribution checks and reporting.
This sample dataset is suitable for testing raw input mode. It produces a mean near 50.50 and a sample standard deviation near 5.34.
| Observation | Value |
|---|---|
| 1 | 42 |
| 2 | 45 |
| 3 | 47 |
| 4 | 48 |
| 5 | 50 |
| 6 | 51 |
| 7 | 53 |
| 8 | 55 |
| 9 | 56 |
| 10 | 58 |
Mean: μ = (Σx) / n
Sample standard deviation: s = √[ Σ(x - x̄)² / (n - 1) ]
Population standard deviation: σ = √[ Σ(x - μ)² / n ]
Empirical rule intervals:
68.27% band = mean ± 1 × standard deviation
95.45% band = mean ± 2 × standard deviation
99.73% band = mean ± 3 × standard deviation
Z-score for an observation: z = (x - mean) / standard deviation
The 68-95-99 rule is a normal-distribution shortcut. It estimates how much data falls close to the center. One standard deviation captures most common observations. Two and three standard deviations show increasingly wider ranges.
1. Choose Mean and standard deviation if you already know both summary values.
2. Choose Raw dataset if you want the page to compute the mean and standard deviation for you.
3. Select Sample or Population depending on how your dataset should be interpreted.
4. Set the decimal places to control output precision.
5. Add an optional observation value if you want its z-score and sigma classification.
6. Submit the form. The result appears above the form, directly below the header.
7. Review the interval table, graph, and export the result as CSV or PDF.
This calculator works best when data is approximately normal. If the data is strongly skewed or contains heavy outliers, the empirical rule percentages may not reflect the real distribution well.
It estimates how much normally distributed data falls within one, two, and three standard deviations from the mean. The common approximations are 68%, 95%, and 99.7%.
Use it when your data is approximately bell-shaped and you want quick interval estimates, sigma bands, or a simple interpretation of spread around the mean.
No. It is most reliable for distributions that are close to normal. Strong skewness, outliers, or multimodal patterns can make the percentage bands misleading.
Population standard deviation uses all values in the full population. Sample standard deviation uses a subset and applies Bessel’s correction, dividing by n minus 1.
Yes. Paste comma-separated numbers, choose sample or population mode, and the calculator will compute the mean, standard deviation, sigma intervals, and the graph.
The z-score tells how many standard deviations an observation sits above or below the mean. It helps classify whether a value is typical, unusual, or extreme.
Those are the more precise normal-distribution percentages. The shorter 68-95-99 wording is a rounded teaching rule used for fast interpretation.
Yes. Negative observations are valid whenever the measured variable allows them. The calculator handles negative means, values, and lower bounds normally.
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.