Example data table
| Scenario | μ | σ | Lower | Upper | Interval probability |
|---|---|---|---|---|---|
| IQ between 85 and 115 | 100 | 15 | 85 | 115 | ~0.6827 |
| Heights ≤ 170 cm | 175 | 7 | — | 170 | ~0.2389 |
| Defects ≥ 2.5 (standardized) | 0 | 1 | 2.5 | — | ~0.0062 |
| Central coverage 95% | 50 | 10 | ~30.4 | ~69.6 | 0.9500 |
| Outside 2σ band | 0 | 1 | -2 | 2 | ~0.0455 |
For a normal random variable X ~ N(μ, σ²):
- z = (x − μ) / σ (standardization)
- Φ(x) = 0.5 · (1 + erf((x − μ)/(σ√2))) (normal CDF)
- P(a ≤ X ≤ b) = Φ(b) − Φ(a) (interval probability)
- P(X ≥ x) = 1 − Φ(x) and P(X ≤ x) = Φ(x) (tails)
- a = μ + σ·Φ⁻¹(α), b = μ + σ·Φ⁻¹(1 − α) for a central interval with α=(1−p)/2
- Enter μ and σ for your distribution.
- Select whether you want a probability from bounds or bounds from a target probability.
- Pick the interval type (between, outside, left tail, right tail, or central).
- Fill the relevant fields: bounds for interval mode, or p for target-coverage mode.
- Press Calculate to view the probability, z-scores, and shaded graph.
- Use CSV/PDF buttons to export the current result, or save runs to build a log.
Interval probabilities in quality control
Manufacturing teams often track a dimension with μ and σ from recent samples. If a shaft diameter has μ=25.00 mm and σ=0.08 mm, the probability inside 24.90–25.10 mm drives yield. A 0.20 mm window equals 2.5σ total width, typically producing coverage near 0.9876. With n=200 parts per shift, that implies about 2–3 out-of-band parts, not 13. This calculator helps compare tolerance changes before tooling adjustments.
Risk bands for forecasting and finance
Forecast errors are frequently modeled as normal after bias correction. With μ=0 and σ=1.2%, a central 90% interval sets bounds at roughly ±1.645σ, or ±1.97%. Analysts can translate an error budget into practical bands, then quantify tail risk such as P(|X|≥2.5%) to decide hedging thresholds. For stricter monitoring, 99% uses z≈2.326, giving a ±2.79% band under the same σ.
Model residual coverage checks
Residual diagnostics often ask, “How much mass sits in a reasonable range?” Suppose residuals have μ=0.3 and σ=2.1. Coverage within −3 to +3 is not symmetric about μ, so computing P(−3≤X≤3) exposes calibration issues. If the result is 0.78 instead of an expected 0.86, you may need feature revisions or variance stabilization. Many teams benchmark the 68–95–99.7 rule to check whether σ estimates drift.
Service levels and operational targets
Operations teams use one-sided cutoffs to enforce service levels. If delivery time is normal with μ=46 minutes and σ=9 minutes, a left-tail cutoff for p=0.95 yields x≈μ+σ·1.645≈60.8 minutes. That becomes a defensible SLA. Alternatively, compute P(X≤60) to validate whether current performance meets the 95% target. If process improvements reduce σ to 7, the 95% cutoff drops to about 57.5 minutes, improving customer promises.
Experiment ranges in product analytics
A/B testing summaries sometimes need probability of exceeding a practical effect. If effect estimates are normal with μ=0.6% and σ=0.4%, then P(X≥0.0%) is 1−Φ((0−0.6)/0.4)=1−Φ(−1.5)≈0.9332. Using “greater than” mode quickly converts business thresholds into interpretable probabilities. You can also test ambitious targets: P(X≥1.0%) uses z=(1.0−0.6)/0.4=1.0, giving roughly 0.1587 probability of clearing 1% lift.
Interpreting tails and outside intervals
Outside-interval probability is useful for anomaly detection. For standardized scores (μ=0, σ=1), the mass outside −2 to 2 is about 0.0455, meaning ~45 out of 1,000 points may exceed that band by chance. Pair the number with your sample size and alert policy to avoid over-triggering while still catching meaningful outliers. Moving to a 3σ rule drops outside mass to ~0.0027, or about 27 points in 10,000.
1) What does “interval probability” mean?
It is the chance that a normally distributed value falls between two bounds. The calculator computes it using the normal CDF difference: Φ(b) − Φ(a), after standardizing with μ and σ.
2) When should I use “outside” instead of “between”?
Use “outside” when you care about extremes on either side of a band, such as outliers beyond a control range. It returns Φ(a) + (1 − Φ(b)).
3) How is the central interval from probability computed?
The calculator splits the uncovered probability into two equal tails. With target p, it uses α=(1−p)/2, then bounds are μ+σ·Φ⁻¹(α) and μ+σ·Φ⁻¹(1−α).
4) Why do I see z-scores in the results?
z-scores convert your value to the standard normal scale. They make probabilities comparable across different μ and σ and are required for using Φ and Φ⁻¹ accurately.
5) What if my data is not normal?
Results may misstate probabilities when the distribution is skewed or heavy-tailed. Consider transforming the data, using an empirical distribution, or switching to a distribution that matches your process.
6) How should I interpret the shaded Plotly area?
The shaded area equals the computed probability under the density curve for your chosen interval. Wider shaded regions mean higher coverage; thin tails usually indicate rare events.