Calculator
Formula Used
The normal (Gaussian) distribution is defined by mean μ and standard deviation σ. The probability density function (PDF) is:
f(x) = 1 / (σ √(2π)) · exp( - (x − μ)² / (2σ²) )
The cumulative distribution function (CDF) gives the probability up to x:
F(x) = P(X ≤ x) = 1/2 · [ 1 + erf( (x − μ) / (σ √2) ) ]
Interval probabilities come from CDF differences: P(a ≤ X ≤ b) = F(b) − F(a).
How to Use This Calculator
- Enter the mean μ and standard deviation σ.
- Pick a mode: density, cumulative, tails, interval, or z-score.
- Provide x for x-based modes, or bounds a and b for interval modes.
- Click Calculate. The result appears above the form.
- Use the CSV or PDF buttons to export your summary.
Example Data Table
| μ | σ | x | a | b | Mode | Output (approx.) |
|---|---|---|---|---|---|---|
| 0 | 1 | 1.0000 | - | - | Cumulative P(X ≤ x) | 0.841345 |
| 0 | 1 | 1.9600 | - | - | Above x | 0.024998 |
| 10 | 2 | - | 8 | 12 | Between a and b | 0.682689 |
| 5 | 0.5 | 5.2000 | - | - | Density at x | 0.736540 |
Example outputs are rounded and may vary slightly by environment.
Article
1) Why Gaussian probability matters in physics
Many measured quantities fluctuate because countless small effects add together. Under broad conditions, that sum approaches a normal distribution, making Gaussian probability a practical model for noise, timing jitter, and thermal variations. This calculator helps quantify those likelihoods for lab decisions.
2) Typical physics use cases and datasets
Sensor output errors are often reported as μ ± σ. For example, a temperature probe may have μ = 25 °C and σ = 0.2 °C, while a photodiode voltage might show σ in millivolts. In timing experiments, sub‑nanosecond spreads can still be Gaussian‑like after averaging. With these inputs, you can estimate the chance of a reading falling outside a tolerance band.
3) Inputs: mean and standard deviation
The mean μ represents the expected value, and σ measures dispersion. Roughly 68.27% of outcomes lie within μ ± 1σ, 95.45% within μ ± 2σ, and 99.73% within μ ± 3σ. These reference percentages are widely used for error budgets.
4) Density versus probability
The PDF value f(x) is a density, not a direct probability. A probability comes from area: P(x_1 \u2264 X \u2264 x_2) equals the integral of the PDF between the bounds. This tool reports both density and cumulative/interval probabilities so you can interpret results correctly.
5) CDF for thresholds and acceptance testing
The CDF F(x) provides P(X \u2264 x). In calibration, you might ask: “What fraction of readings fall below a cutoff?” Tail probabilities like P(X \u2265 x) are useful for false‑alarm rates when a threshold triggers a detection event.
6) Interval probability for tolerance windows
Many specifications are two‑sided, such as keeping a beam position within \u00B10.5 mm or a frequency within \u00B15 Hz. Choose “Between a and b” to compute P(a \u2264 X \u2264 b). The “Outside [a, b]” mode estimates out‑of‑spec probability for yield.
7) Z-scores for comparability across units
The z‑score converts to a standardized scale: z = (x-\u03bc)/\u03c3. This makes comparisons consistent whether your variable is volts, meters, or seconds. For example, z = 2 indicates a value two standard deviations above the mean.
8) Reporting and reproducibility
Exporting results supports traceable reporting. CSV is convenient for spreadsheets and lab notebooks, while PDF provides a clean, printable summary for reports or QA reviews. Keep the same input set to reproduce probability statements alongside instrument metadata and sampling conditions.
FAQs
1) What is the difference between PDF and CDF?
The PDF is a density at a point, while the CDF is an accumulated probability up to x. Use CDF for “below threshold” questions and PDF for curve shape or relative likelihood near x.
2) Why can the PDF be greater than 1?
A density can exceed 1 when σ is small because probability depends on area, not height. The total area under the PDF over all x remains exactly 1.
3) How do I compute the probability within a tolerance band?
Select “Between a and b,” enter your lower and upper limits, and compute. The calculator evaluates F(b) − F(a), giving the fraction expected inside the band.
4) What does a z-score tell me physically?
A z-score expresses how far x is from μ in units of σ. It helps compare deviations across different measurements and units, and it maps directly to common tail probabilities.
5) When should I use “Above x” mode?
Use it for exceedance risk, such as overload probability, false-alarm rates, or “greater than threshold” detection. It computes 1 − F(x), the right-tail probability.
6) What if my data are not perfectly Gaussian?
Gaussian models are often good approximations, especially for averaged noise. If you see strong skew, heavy tails, or outliers, consider robust statistics or a different distribution model.
7) How accurate are the results?
The CDF uses a standard error-function approximation that is typically very accurate for practical engineering and physics calculations. For extreme tails, increase decimals and validate against a trusted reference if needed.