Calculator Inputs
Formula Used
f(x) = 1 / (σ√(2π)) × e-0.5((x - μ) / σ)²
z = (x - μ) / σ
P(a ≤ X ≤ b) = Φ((b - μ) / σ) - Φ((a - μ) / σ)
The density function gives curve height, not direct probability at one point. Interval probability comes from the cumulative distribution function, which integrates area under the curve.
How to Use This Calculator
- Enter the target x value where you want the density.
- Provide the normal distribution mean and standard deviation.
- Choose the number of decimal places for output formatting.
- Leave graph limits blank for automatic bell-curve range selection.
- Add optional interval bounds to estimate probability over a range.
- Enter multiple x values to create a batch comparison table.
- Press Calculate Density to show results above the form.
- Use the CSV and PDF buttons to export the generated report.
Example Data Table
Standard normal example with μ = 0 and σ = 1.
| x | z | Density f(x) | CDF Φ(x) |
|---|---|---|---|
| -2.0000 | -2.0000 | 0.053991 | 0.022750 |
| -1.0000 | -1.0000 | 0.241971 | 0.158655 |
| 0.0000 | 0.0000 | 0.398942 | 0.500000 |
| 1.0000 | 1.0000 | 0.241971 | 0.841345 |
| 2.0000 | 2.0000 | 0.053991 | 0.977250 |
Frequently Asked Questions
1. What does the density value mean?
The density value is the height of the normal curve at a selected x. It is useful for comparing likelihood concentration, but it is not the probability of one exact point.
2. Why is the density sometimes greater than 1?
A probability density can exceed 1 when the distribution is narrow enough. Total probability is still valid because probability comes from curve area across an interval, not from height alone.
3. What happens if standard deviation is very small?
A smaller standard deviation makes the curve tighter and taller near the mean. That raises peak density and increases sensitivity to small x changes around the center.
4. What is the difference between PDF and CDF?
The PDF gives the curve height at x. The CDF gives the accumulated probability from negative infinity up to x, which is why it is used for tail and interval probabilities.
5. Why does the calculator include z-score?
The z-score standardizes x relative to the mean and spread. It helps compare positions across different normal distributions and is central to many statistical interpretations.
6. Can I estimate probability between two values?
Yes. Enter lower and upper interval bounds. The calculator estimates the probability between them by subtracting the lower cumulative probability from the upper cumulative probability.
7. What are batch x values useful for?
Batch values let you compare multiple points under the same distribution. This is helpful for threshold reviews, anomaly screening, quality limits, and quick reporting.
8. Is this useful for data science work?
Yes. It supports distribution checks, score standardization, tail analysis, threshold evaluation, and quick report exports, which are common needs in analytics and modeling workflows.