Calculator Form
Use z values, interval limits, chart range, and optional sample data to analyze the standard normal distribution in one place.
Example Data Table
Reference values for common z scores| Z Score | CDF | Right-Tail Probability | |
|---|---|---|---|
| -3 | 0.004432 | 0.001350 | 0.998650 |
| -2 | 0.053991 | 0.022750 | 0.977250 |
| -1 | 0.241971 | 0.158655 | 0.841345 |
| 0 | 0.398942 | 0.500000 | 0.500000 |
| 1 | 0.241971 | 0.841345 | 0.158655 |
| 2 | 0.053991 | 0.977250 | 0.022750 |
| 3 | 0.004432 | 0.998650 | 0.001350 |
Formula Used
μ = 0
σ² = 1
σ = √1 = 1
f(z) = (1 / √(2π)) × e-z²/2
Φ(z) = P(Z ≤ z)
P(a ≤ Z ≤ b) = Φ(b) − Φ(a)
Because this page is specifically for the standard normal distribution, the theoretical mean and variance remain fixed at 0 and 1.
How to Use This Calculator
- Enter a z value to evaluate PDF, CDF, and tail probabilities.
- Enter lower and upper z limits to calculate the shaded interval area.
- Choose your preferred decimal precision for displayed results.
- Set chart minimum and maximum x values for a wider or narrower graph.
- Optionally paste standardized sample data to compare its mean and variance with the theoretical standard normal values.
- Click Calculate Now to show the result section above the form.
- Use the CSV and PDF buttons to export the generated outputs.
Frequently Asked Questions
1. What is the mean of a standard normal distribution?
The mean is always 0. This means the distribution is centered exactly at zero, with equal spread on both sides of the horizontal axis.
2. What is the variance of a standard normal distribution?
The variance is always 1. This fixed variance gives the standard normal curve its standard scale and makes comparison across z scores easy.
3. Why does this calculator still ask for a z value?
The z value helps calculate density, cumulative probability, and tail areas. Mean and variance stay fixed, but z-based probabilities change with the selected point.
4. What does the interval probability represent?
It represents the area under the standard normal curve between the lower and upper z values. This equals the probability that Z falls within that range.
5. What is the difference between PDF and CDF?
PDF shows the curve height at a single z value. CDF shows the total accumulated probability from negative infinity up to that z value.
6. Why can I enter sample data here?
Optional sample data let you compare entered standardized values against the theoretical standard normal targets. This is useful for quick validation or teaching demonstrations.
7. Is the graph useful for interpretation?
Yes. The graph helps you see the bell shape, the chosen z position, and the shaded interval area. Visual context makes probability interpretation much easier.
8. What do the CSV and PDF downloads include?
They include the generated results table from your current calculation. This makes it easier to save, share, review, or attach outputs to reports.