Calculator Input
Example Data Table
This example shows common z-scores with their probability density, cumulative probability, and upper tail probability.
| Z-Score | PDF φ(z) | CDF Φ(z) | Upper Tail 1 - Φ(z) |
|---|---|---|---|
| -2.00 | 0.053991 | 0.022750 | 0.977250 |
| -1.00 | 0.241971 | 0.158655 | 0.841345 |
| 0.00 | 0.398942 | 0.500000 | 0.500000 |
| 1.00 | 0.241971 | 0.841345 | 0.158655 |
| 2.00 | 0.053991 | 0.977250 | 0.022750 |
Formula Used
Probability density function:
φ(z) = e-z²/2 / √(2π)
Cumulative distribution function:
Φ(z) = P(Z ≤ z)
Upper tail probability:
P(Z ≥ z) = 1 - Φ(z)
Range probability:
P(z1 ≤ Z ≤ z2) = Φ(z2) - Φ(z1)
Two tailed probability:
P(|Z| ≥ |z|) = 2 × [1 - Φ(|z|)]
Inverse percentile:
z = Φ-1(p), where p is the left tail probability.
How to Use This Calculator
- Select the calculation mode that matches your data science or statistics task.
- Enter one z-score for left tail, upper tail, or two tailed analysis.
- Enter both lower and upper z-scores for interval probability.
- Enter a percentile between 0 and 100 for inverse mode.
- Choose how many decimal places you want in the output table.
- Press the calculate button to show results above the form.
- Review the shaded Plotly chart to see the exact probability region.
- Use the CSV or PDF buttons to save the generated result table.
FAQs
1. What does a standard normal distribution mean?
A standard normal distribution is a bell-shaped distribution with mean 0 and standard deviation 1. Every z-score tells how many standard deviations a value sits above or below the mean.
2. When should I use this calculator?
Use it when your variable is already standardized or when you have converted a raw value into a z-score. It helps with probabilities, critical thresholds, p-values, and interval areas.
3. What is the difference between PDF and CDF?
The PDF gives density at a specific z-score, not a direct probability at one point. The CDF gives the cumulative probability from negative infinity up to that z-score.
4. What does the two tailed result represent?
It measures the combined probability of values equally far from zero in both tails. This is commonly used in hypothesis testing when deviations on either side matter.
5. Why is my inverse percentile result negative?
If the percentile is below 50%, the corresponding z-score lies left of the mean. Because the standard normal mean is zero, lower percentiles naturally produce negative z-values.
6. Can I use raw values instead of z-scores?
Yes, but you must standardize them first. Convert raw values with z = (x - μ) / σ before using this calculator, where μ is the mean and σ is the standard deviation.
7. How accurate are the results?
The calculator uses reliable numerical approximations for the normal CDF and inverse CDF. For most academic, analytics, and applied data science work, the displayed precision is more than sufficient.
8. Why does the shaded graph matter?
The graph helps confirm what probability region you actually calculated. It reduces interpretation mistakes, especially when comparing left tail, right tail, range, and two tailed results.