Calculated Result
The summary appears here after submission.
Interpretation
Calculator Inputs
Choose a probability mode, enter distribution values, and calculate interval, tail, exact point, or inverse normal results.
Example Data Table
| Scenario | Mean | Standard Deviation | Mode | Input | Expected Insight |
|---|---|---|---|---|---|
| Exam scores | 70 | 10 | P(X ≤ x) | x = 82 | Find the share scoring at or below 82. |
| Manufacturing weight | 500 | 12 | P(a ≤ X ≤ b) | a = 490, b = 515 | Measure the acceptable production band. |
| Delivery time | 35 | 6 | P(X ≥ x) | x = 42 | Estimate the late delivery probability. |
| Quality control | 100 | 15 | Find x from cumulative probability | p = 0.95 | Locate the 95th percentile threshold. |
Formula Used
- Z-score: z = (x - μ) / σ
- Standard normal cumulative probability: Φ(z) = P(Z ≤ z)
- Below a value: P(X ≤ x) = Φ((x - μ) / σ)
- Above a value: P(X ≥ x) = 1 - Φ((x - μ) / σ)
- Between two values: P(a ≤ X ≤ b) = Φ((b - μ) / σ) - Φ((a - μ) / σ)
- Outside two values: P(X ≤ a or X ≥ b) = 1 - P(a ≤ X ≤ b)
- Inverse normal: x = μ + zσ, where z = Φ-1(p)
The calculator uses a numerical approximation for the standard normal cumulative distribution and an inverse approximation for percentile estimates.
How to Use This Calculator
- Select the probability mode that matches your question.
- Enter the distribution mean and standard deviation.
- Provide x, interval bounds, or cumulative probability as needed.
- Click Calculate Probability to display the result above the form.
- Review the probability, z-scores, percentile details, and interpretation.
- Download the result in CSV or PDF format for records.
Why This Calculator Helps
Supports tail, interval, outside, and inverse modes Shows z-scores and percentile context Exports analysis to CSV and PDF Useful for teaching, testing, and reportingFrequently Asked Questions
1. What does this calculator measure?
It estimates probabilities and related z-scores for values following a normal distribution. It also finds percentiles from cumulative probabilities.
2. When should I use the below mode?
Use it when you want the cumulative share at or below a value. This is common for scores, quality levels, or service times.
3. What is the difference between between and outside?
Between returns the probability inside two bounds. Outside returns the combined probability in both tails beyond those bounds.
4. Why must standard deviation be positive?
Standard deviation represents spread. A zero or negative value would not describe a valid normal distribution for probability analysis.
5. Can I use decimals for all inputs?
Yes. The calculator accepts decimal values for means, deviations, cutoffs, bounds, and cumulative probabilities.
6. What does inverse probability return?
It returns the x value associated with a chosen cumulative probability. For example, p = 0.95 gives the 95th percentile cutoff.
7. Are the results exact?
Results use strong numerical approximations for the normal cumulative function and inverse function. They are suitable for most educational and analytical needs.