Calculator Inputs
Plotly Graph
The bell curve shows the mean, standard deviation bands, and the observed value marker.
Formula Used
- Z-score: z = (x − μ) / σ
- Within one deviation: μ − σ to μ + σ ≈ 68%
- Within two deviations: μ − 2σ to μ + 2σ ≈ 95%
- Within three deviations: μ − 3σ to μ + 3σ ≈ 99.7%
- Percentile estimate: Normal CDF converts the observed value into cumulative probability.
This calculator combines the empirical rule with a normal distribution percentile estimate. That gives quick interval checks and a more detailed location estimate.
How to Use This Calculator
- Enter the dataset mean.
- Enter the standard deviation.
- Type the observed value you want to evaluate.
- Select the decimal precision you prefer.
- Click calculate to view ranges and percentile estimates.
- Review the table, metric cards, and the bell curve.
- Export the result as CSV or PDF if needed.
Example Data Table
| Mean | Standard Deviation | Observed Value | Z-Score | Estimated Percentile |
|---|---|---|---|---|
| 50 | 10 | 64 | 1.4 | 91.92% |
| 72 | 8 | 80 | 1 | 84.13% |
| 100 | 15 | 130 | 2 | 97.72% |
Frequently Asked Questions
1. What does the empirical rule explain?
It estimates how much normal data falls within one, two, and three standard deviations from the mean. The classic percentages are 68%, 95%, and 99.7%.
2. When should I use this calculator?
Use it when your dataset is approximately bell-shaped. It is helpful for test scores, measurements, quality checks, and other values that resemble a normal distribution.
3. What is a z-score here?
A z-score shows how many standard deviations the observed value is from the mean. Positive values sit above the mean. Negative values sit below it.
4. Does the percentile come from the empirical rule?
No. The percentile is estimated from the normal cumulative distribution. The interval coverage values come from the empirical rule percentages.
5. Why must standard deviation be greater than zero?
A zero or negative standard deviation breaks the distribution model. The spread must be positive to calculate z-scores, density values, and interval boundaries correctly.
6. Is the 68-95-99.7 rule exact?
It is a practical approximation for normal distributions. Exact probabilities can differ slightly, especially when data are skewed or when a sample is not truly normal.
7. What does it mean if a value is outside ±3σ?
That value is unusually far from the mean in a normal setting. Only about 0.3% of values are expected beyond three standard deviations.
8. What do the export buttons save?
The CSV file saves summary rows and interval values. The PDF file saves the key outputs, making reporting and sharing much easier.