Understand z scores with fast percentage conversions quickly. Compare left, right, middle, and two-tail areas. Learn distributions using examples, charts, exports, and guided steps.
Enter a z score, choose the percentage view, and update the graph range if needed.
| Metric | Value | Percentage |
|---|---|---|
| Scenario label | Standard Normal Case | — |
| Z value | 1.9600 | — |
| Selected mode | Left-tail cumulative area | 97.5002% |
| Left-tail cumulative area | 0.9750 | 97.5002% |
| Right-tail area | 0.0250 | 2.4998% |
| Area between the mean and z | 0.4750 | 47.5002% |
| Central area within ±|z| | 0.9500 | 95.0004% |
| Two-tail area outside ±|z| | 0.0500 | 4.9996% |
| Percentile rank | 97.5002 | 97.5002% |
The shaded region changes with the selected percentage mode.
| Z Value | Left-tail % | Right-tail % | Central within ±z % |
|---|---|---|---|
| 0.00 | 50.000% | 50.000% | 0.000% |
| 0.50 | 69.146% | 30.854% | 38.292% |
| 1.00 | 84.134% | 15.866% | 68.269% |
| 1.645 | 95.000% | 5.000% | 90.000% |
| 1.96 | 97.500% | 2.500% | 95.000% |
| 2.00 | 97.725% | 2.275% | 95.450% |
| 2.58 | 99.506% | 0.494% | 99.012% |
| 3.00 | 99.865% | 0.135% | 99.730% |
This calculator converts a z value into one or more percentages from the standard normal distribution. The cumulative distribution function is:
Φ(z) = 0.5 × [1 + erf(z / √2)]
Left % = Φ(z) × 100
Right % = [1 - Φ(z)] × 100
Middle % = |Φ(z) - 0.5| × 100
Central % = [2Φ(|z|) - 1] × 100
Outside % = 2 × [1 - Φ(|z|)] × 100
The code uses a standard approximation for erf(), which is accurate for practical educational and reporting use.
A z value of 0 sits at the mean. Its left-tail cumulative percentage is 50%. Its right-tail percentage is also 50%. The area between the mean and z is 0%.
A negative z value lies below the mean. Its left-tail cumulative percentage becomes smaller than 50%, while its right-tail percentage becomes larger than 50%.
Left-tail percentage measures the area to the left of the z value. Right-tail percentage measures the area to the right. Together, they always sum to 100%.
It measures the probability between the center of the distribution and the chosen z value. This is useful when tables or test questions ask for middle-region probability.
It shows how much of the normal distribution lies inside symmetric bounds. This is commonly used in confidence intervals, coverage percentages, and empirical rule interpretation.
This value is useful for hypothesis testing and p-value interpretation. It measures the combined probability in both tails beyond the same absolute z distance.
Yes, but convert the raw score first. Use z = (x - μ) / σ. After that, enter the z value here to obtain the needed percentage.
The graph makes the probability easier to understand visually. It shows exactly which region is being counted, which helps reduce mistakes during study, reporting, and interpretation.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.