Measure probability between lower and upper z scores easily. See area, tails, and distribution behavior. Save outputs, compare scenarios, and verify statistics with clarity.
The curve below shows the standard normal distribution and the interval selected between your z scores.
| Example | Lower z | Upper z | Area Between | Percent Between |
|---|---|---|---|---|
| 1 | -1.00 | 1.00 | 0.682689 | 68.268947% |
| 2 | -2.00 | 0.50 | 0.668712 | 66.871240% |
| 3 | -0.50 | 1.96 | 0.666465 | 66.646464% |
Area between two z scores = Φ(zupper) − Φ(zlower)
Here, Φ(z) is the cumulative distribution function of the standard normal distribution.
Φ(z) = 0.5 × [1 + erf(z / √2)]
The calculator also reports left tail probability, right tail probability, interval width, midpoint, and density values at both z scores.
The area under a normal curve between z scores is a core probability concept in mathematics, statistics, quality control, and test analysis. This calculator helps you find the probability that a value from a standard normal distribution lies between two selected z-score boundaries. Because z scores are standardized, the same method works for many different datasets after raw values are converted into standard units.
This page does more than return one probability. It also reports cumulative probability at each boundary, the left tail below the lower z score, the right tail above the upper z score, interval width, midpoint, and boundary density values. Those details are useful when checking symmetry, understanding tail behavior, or comparing intervals across different statistical problems.
The included graph helps visualize the selected region under the bell curve. The example table shows sample inputs and results, while the history table keeps recent calculations ready for export. This makes the tool useful for homework, exam preparation, classroom demonstrations, and quick verification during data analysis tasks.
If you start with raw observations instead of z scores, first standardize them using the mean and standard deviation. After that, enter the two z values here, and the calculator will estimate the probability between them. The result is shown as both a decimal and a percentage, making interpretation easier for both technical and nontechnical use cases.
It represents the probability that a standard normal value falls between the two selected z-score boundaries. The calculator shows that probability as both a decimal and a percentage.
The calculator sorts the values automatically. It always evaluates the interval from the lower z score to the upper z score, so the final probability remains correct.
Yes. First convert each raw score into a z score using the distribution mean and standard deviation. Then enter those z values here to find the probability between them.
They help you see how much probability remains outside your chosen interval. This is useful when comparing central regions with tail regions in probability and hypothesis work.
A negative z score means the value lies below the distribution mean. A positive z score means it lies above the mean. Zero is exactly at the mean.
Symmetry helps you quickly understand whether the selected interval is centered around the mean. Symmetric bounds around zero often create balanced areas on both sides of the bell curve.
The result is a very close numerical approximation based on the standard normal cumulative function. It is suitable for most academic, educational, and practical probability tasks.
Use more decimal places when you need finer precision for homework checking, report writing, or statistical comparison. Fewer decimals are usually enough for quick 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.