Calculator Input
Example Data Table
| Z | e-z²/2 | φ(z) | Left Tail | Right Tail | Central Area |
|---|---|---|---|---|---|
| -2.00 | 0.1353 | 0.0540 | 0.0228 | 0.9772 | 0.9545 |
| -1.00 | 0.6065 | 0.2420 | 0.1587 | 0.8413 | 0.6827 |
| 0.00 | 1.0000 | 0.3989 | 0.5000 | 0.5000 | 0.0000 |
| 1.00 | 0.6065 | 0.2420 | 0.8413 | 0.1587 | 0.6827 |
| 1.96 | 0.1465 | 0.0584 | 0.9750 | 0.0250 | 0.9500 |
| 2.58 | 0.0359 | 0.0143 | 0.9951 | 0.0049 | 0.9901 |
Formula Used
The calculator starts with the standard normal kernel:
kernel = e^(-z² / 2)
The standard normal density is:
φ(z) = e^(-z² / 2) / √(2π)
If a raw value is used, the z score is:
z = (x - μ) / σ
The polar radius is:
r = |z|
The coordinate conversion is:
x = r cos(θ) and y = r sin(θ)
The radial probability from the polar Gaussian integral is:
P(R ≤ r) = 1 - e^(-r² / 2)
How to Use This Calculator
- Select whether you want to enter a direct z score or a raw value.
- Enter the z score, or enter X, mean, and standard deviation.
- Add a polar angle if you want coordinate conversion.
- Choose the probability type, such as left tail or two tail.
- Select decimal places for the final output.
- Press the calculate button to view results above the form.
- Use the CSV or PDF button to save the report.
Understanding Polar Normal Probability
Why the Kernel Matters
The expression e raised to negative z squared over two is the core of the normal curve. It controls how fast values fade as a z score moves away from zero. A small z score gives a large kernel. A large absolute z score gives a small kernel. This is why extreme values have low density. The calculator first finds this kernel. It then converts it into useful probability measures.
How Polar Coordinates Help
Polar coordinates are helpful when a normal model is viewed through radius and angle. The radius measures distance from the center. The angle describes direction. In a circular normal setting, the radial probability becomes one minus the same exponential kernel. This creates a direct link between z based distance and polar area. The tool shows radius, x coordinate, y coordinate, and recovered radius. These values help students compare linear and radial views.
Using Tail Results
Many probability tasks need tail areas. A left tail result shows the probability below the z score. A right tail result shows the probability above it. A two tail result is useful for symmetric tests. A central area shows the probability between negative and positive distance. The calculator includes all these outputs together. This reduces table lookup work. It also makes checking easier.
Practical Uses
This calculator can support statistics lessons, engineering checks, conversion pages, and quality control examples. It is also useful when normal density and polar radius must be compared. You can enter a direct z score for fast work. You can also enter a raw value with mean and standard deviation. The output can be saved as a spreadsheet file or a formatted document. That makes the result easy to record, share, and review.
Frequently Asked Questions
1. What does e^(-z²/2) mean?
It is the exponential kernel inside the standard normal density formula. It measures how quickly density falls as the z score moves away from zero.
2. Is the kernel the same as probability?
No. The kernel is not a full probability by itself. It becomes density after division by square root of two pi.
3. What is the polar radius here?
The calculator uses the absolute z score as the radius. This connects distance from the center with the polar Gaussian integral.
4. What is radial probability?
Radial probability is one minus e raised to negative radius squared over two. It represents circular area under a two dimensional normal model.
5. When should I use left tail probability?
Use left tail probability when you need the area below a z score. It is common in percentile and cutoff problems.
6. When should I use two tail probability?
Use two tail probability when extreme values on both sides matter. It is common in hypothesis testing and significance checks.
7. Can I use a raw value instead?
Yes. Select raw value mode. Then enter X, mean, and standard deviation. The calculator converts the value into a z score.
8. Why include CSV and PDF downloads?
Downloads help save the result for reports, records, lessons, or audits. They also make repeated statistical checks easier to document.