Z Probability Calculator
Use direct z values, raw scores, interval bounds, or inverse percentile targets. The calculator supports cumulative, tail, central, and interval-based standard normal results.
Example Data Table
These rounded values show common z scores and their related cumulative, tail, middle, and two-tailed probabilities.
| z | Φ(z) | Right Tail | Area Between 0 and z | Two-Tailed p |
|---|---|---|---|---|
| 0.5 | 0.691462 | 0.308538 | 0.191462 | 0.617075 |
| 1 | 0.841345 | 0.158655 | 0.341345 | 0.317311 |
| 1.28 | 0.899727 | 0.100273 | 0.399727 | 0.200545 |
| 1.64 | 0.949497 | 0.050503 | 0.449497 | 0.101005 |
| 1.96 | 0.975002 | 0.024998 | 0.475002 | 0.049996 |
| 2.58 | 0.99506 | 0.00494 | 0.49506 | 0.00988 |
Formula Used
1) Standardization Formula
Convert a raw value into a z score using:
z = (x - μ) / σ
Here, x is the raw score, μ is the mean, and σ is the standard deviation.
2) Cumulative Probability Formula
The standard normal cumulative distribution is:
Φ(z) = 0.5 × [1 + erf(z / √2)]
This returns the area to the left of the chosen z score.
3) Tail and Interval Formulas
Right Tail = 1 - Φ(z)
P(z₁ ≤ Z ≤ z₂) = Φ(z₂) - Φ(z₁)
P(-z ≤ Z ≤ z) = 2Φ(z) - 1
These formulas support right-tail, between-two-z, and central-area lookups.
4) Inverse Percentile Formula
For a chosen cumulative percentage p, the calculator finds z = Φ⁻¹(p).
This is useful for critical values, cutoffs, limits, and confidence-style thresholds.
How to Use This Calculator
- Choose the calculation mode that matches your problem.
- Enter one z score, two z scores, a raw score set, or a percentile target.
- Select the decimal precision you want in the result table.
- Press Calculate Probability to display the result below the header and above the form.
- Review the summary metrics, detailed table, and interpretation note.
- Use the CSV or PDF buttons to export the current result or the example table.
FAQs
1) What does a z table probability represent?
It represents area under the standard normal curve. That area can mean cumulative probability, tail probability, or the probability between two z values.
2) When should I use left-tail mode?
Use left-tail mode when you need the probability that a value is less than or equal to a chosen z score.
3) When is right-tail probability useful?
Right-tail probability is useful for exceedance analysis, upper-side quality limits, hypothesis testing, and rare-event checks above a threshold.
4) Why convert a raw score into a z score?
Standardizing lets you compare values from different normal distributions. After conversion, the same z table can be applied consistently.
5) What does central area mean?
Central area is the probability captured between a negative z bound and its positive mirror. It is often used for interval coverage.
6) Can I enter a percentile instead of a probability?
Yes. Enter 95 for 95% or 0.95 for the same cumulative probability. The calculator converts the input and finds the matching z value.
7) Does the calculator assume a normal distribution?
Yes. These results are based on the standard normal model. Raw score mode also assumes your source data follows a normal distribution.
8) Why are exported values rounded?
Exported values follow the decimal setting chosen in the form or the fixed rounding used in the example table for readable reporting.