Enter Distribution Inputs
Choose a calculation mode, enter the distribution parameters, and estimate areas, percentiles, cutoff values, and standardized z positions.
Example Data Table
| Scenario | Mean μ | Standard Deviation σ | Input Values | Expected Result Type |
|---|---|---|---|---|
| Exam score below cutoff | 70 | 10 | X = 82 | Left-tail probability |
| Manufacturing tolerance band | 50 | 4 | A = 46, B = 54 | Between probability |
| Top 5 percent threshold | 100 | 15 | Percentile = 0.95 | Inverse normal value |
| Service time unusually high | 12 | 3 | X = 18 | Right-tail probability |
Formula Used
The calculator uses the normal distribution with mean μ and standard deviation σ. A raw value is standardized with the z-score formula:
z = (x - μ) / σ
Left-tail probability uses the cumulative distribution function:
P(X ≤ x) = Φ(z)
Right-tail probability uses:
P(X ≥ x) = 1 - Φ(z)
Between two values uses:
P(a ≤ X ≤ b) = Φ(zb) - Φ(za)
Outside two values uses:
P(X ≤ a or X ≥ b) = 1 - P(a ≤ X ≤ b)
For percentile mode, the inverse normal function estimates the cutoff value:
x = μ + zpσ
How to Use This Calculator
- Select the probability mode you need.
- Enter the mean and standard deviation of the distribution.
- Provide one value, two bounds, or a percentile.
- Press the calculate button to generate the result summary.
- Review the z-scores, probabilities, percentages, and interpretation.
- Export the results as CSV or PDF if needed.
Frequently Asked Questions
1. What does this calculator measure?
It measures probabilities under a normal distribution. You can estimate left tails, right tails, middle ranges, outside areas, percentiles, and z-scores from your inputs.
2. When should I use the left-tail option?
Use left-tail mode when you need the probability that a value is less than or equal to a specified point, such as scores below a cutoff.
3. What is the difference between between and outside modes?
Between mode finds the area within two bounds. Outside mode finds the combined area below the lower bound and above the upper bound.
4. Why must the standard deviation be positive?
Standard deviation represents spread. A zero or negative spread does not define a valid normal distribution, so the calculator blocks those entries.
5. What does the z-score tell me?
The z-score shows how many standard deviations a value sits above or below the mean. It helps compare values on a common standardized scale.
6. Can I use decimal values?
Yes. The calculator accepts decimal means, standard deviations, bounds, and percentiles, which is useful for scientific, financial, and quality-control applications.
7. What percentile format should I enter?
Enter percentiles as probabilities between 0 and 1. For example, use 0.90 for the 90th percentile and 0.025 for 2.5 percent.
8. Are the results exact?
The results are highly accurate numerical approximations based on standard statistical formulas. They are suitable for analysis, teaching, planning, and general decision support.