Calculator
Formula Used
The standard normal variable is Z. Its mean is 0. Its standard deviation is 1.
Density: f(z) = e-z²/2 / √(2π)
Cumulative probability: Φ(z) = P(Z ≤ z)
Right tail: P(Z ≥ z) = 1 - Φ(z)
Interval: P(a ≤ Z ≤ b) = Φ(b) - Φ(a)
Outside interval: P(Z ≤ a or Z ≥ b) = 1 - [Φ(b) - Φ(a)]
Central area: P(-z ≤ Z ≤ z) = Φ(z) - Φ(-z)
Inverse modes solve for the z value after a probability is entered.
How to Use This Calculator
- Select the probability type from the dropdown.
- Enter the required z value or z interval.
- For inverse modes, enter a probability between 0 and 1.
- Choose the number of decimal places.
- Press Calculate to display results below the header.
- Use CSV or PDF export to save the calculation.
Example Data Table
| Case | Input | Formula | Approximate Result | Meaning |
|---|---|---|---|---|
| Left tail | z = 1.96 | Φ(1.96) | 0.975002 | Area left of 1.96 |
| Right tail | z = 1.645 | 1 - Φ(1.645) | 0.049984 | Upper five percent cutoff |
| Between | -1.96 to 1.96 | Φ(1.96) - Φ(-1.96) | 0.950004 | Common confidence area |
| Central inverse | 0.95 | ±Φ-1((1 + 0.95) / 2) | ±1.959964 | Two sided critical values |
About Standard Normal Probability
What This Tool Measures
A standard normal probability describes area under the bell curve. The curve is centered at zero. Its spread is fixed at one standard deviation. Because the total area is one, each area can be read as a probability. This calculator handles common z tasks in one place. It finds left tails, right tails, intervals, outside areas, central coverage, and inverse cutoffs.
Why Z Values Matter
A z value tells how far a point sits from the mean. Positive values are above the mean. Negative values are below it. A value near zero is ordinary. A large absolute value is less common. This makes z scores useful in hypothesis tests, quality checks, exams, surveys, and risk summaries.
Using Tail Areas
Left tail probability is the area before a chosen z value. Right tail probability is the area after that value. These tail areas are important in significance testing. A small tail area can show that an observed result is unusual under a model. The calculator also reports the complement when useful, so the result can be checked quickly.
Working With Intervals
Many problems ask for probability between two z values. The method is direct. Find the cumulative area at the upper value. Then subtract the cumulative area at the lower value. The remaining area is the interval probability. Outside probability is the complement. It combines the two tails beyond the interval limits.
Inverse Probability Uses
Inverse calculation works backward. You enter an area. The calculator returns the matching z cutoff. This is useful when building confidence limits or finding rejection regions. For central areas, it gives symmetric cutoffs around zero. For example, a central area near 0.95 gives cutoffs near -1.96 and 1.96.
Good Practice
Use enough decimal places for your context. Four to six decimals are usually enough for study work. More decimals can help with audit trails. Always match the selected mode to the wording of the question. Check whether the problem asks for below, above, between, outside, or a critical value. That choice controls the final answer.
FAQs
What is a standard normal random variable?
It is a normal random variable with mean 0 and standard deviation 1. It is often written as Z.
What does Φ(z) mean?
Φ(z) means the cumulative probability to the left of z. It equals P(Z ≤ z).
How do I find a right tail probability?
Choose the right tail option. Enter z. The calculator uses 1 - Φ(z) to return the upper area.
How is interval probability calculated?
The calculator subtracts the lower cumulative area from the upper cumulative area. The formula is Φ(b) - Φ(a).
What is an inverse z calculation?
It finds the z value that matches a given probability. This is useful for critical values and confidence limits.
Can I enter z values in any order?
Yes. For interval and outside modes, the calculator sorts the two z values before computing the probability.
Why must inverse probability be between 0 and 1?
Normal probabilities are areas under the curve. Valid areas must be greater than 0 and less than 1.
What exports are available?
You can download the current result as a CSV file or a simple PDF summary for records.