Plotly graph
Use the form below and submit a calculation to highlight the selected standard normal region on the curve.
Calculator
Choose a calculation mode, enter the needed values, and submit. Results will appear above this form under the page header.
Example data table
These sample scenarios show the kinds of outputs produced by the calculator.
| Scenario | Inputs | Example result |
|---|---|---|
| Left tail probability | Mode: Left Tail, Z = 1.25 | P(Z ≤ 1.25) = 0.894350 |
| Right tail probability | Mode: Right Tail, Z = 1.96 | P(Z ≥ 1.96) = 0.024998 |
| Interval probability | Mode: Between, Lower Z = -1, Upper Z = 1 | P(-1 ≤ Z ≤ 1) = 0.682689 |
| Inverse normal | Mode: Inverse, Probability = 0.975 | Z = 1.959964 |
| Standardization | Mode: Raw X to Z, X = 85, Mean = 70, SD = 10 | Z = 1.500000 and P(Z ≤ 1.5) = 0.933193 |
Formula used
1) Standard normal density
This gives the curve height at a specific z score.
f(z) = exp(-z² / 2) / √(2π)
2) Standard normal cumulative probability
This gives the total area to the left of z.
Φ(z) = 0.5 × [1 + erf(z / √2)]
3) Right tail probability
This gives the area to the right of z.
P(Z ≥ z) = 1 - Φ(z)
4) Probability between two z scores
This gives the area inside an interval.
P(z₁ ≤ Z ≤ z₂) = Φ(z₂) - Φ(z₁)
5) Central probability
This gives the area between symmetric limits.
P(-z ≤ Z ≤ z) = Φ(z) - Φ(-z)
6) Standardization from a raw value
This converts a raw observation into a z score.
z = (x - μ) / σ
How to use this calculator
- Select the calculation mode that matches your task.
- Enter the required value fields for that mode.
- Choose how many decimal places you want displayed.
- Press Calculate to show the result above the form.
- Review the result table, summary note, and shaded Plotly graph.
- Use the CSV or PDF buttons to save the generated output.
- For raw observations, switch to the standardization mode.
- For percentiles or cutoffs, use the inverse normal mode.
Frequently asked questions
1) What is a standard normal calculator?
A standard normal calculator works with the Z distribution, where the mean is 0 and the standard deviation is 1. It estimates density values, cumulative probabilities, tail areas, interval probabilities, and inverse Z values from a probability.
2) When should I use a z score?
Use a z score when you want to measure how far a value sits from the mean in standard deviation units. After standardization, you can compare observations from different scales using one common probability model.
3) What is the difference between PDF and CDF?
The PDF gives curve height at a specific z value, not a direct probability at one exact point. The CDF gives accumulated probability from negative infinity up to that z score.
4) What does right tail probability mean?
Right tail probability is the area under the curve to the right of a chosen z score. It answers questions like the chance of observing a value at least that extreme.
5) Why do interval probabilities matter?
Interval probabilities show how much total probability lies between two z scores. They are useful for confidence coverage, acceptance regions, process limits, and risk thresholds.
6) What does inverse normal calculation do?
Inverse normal starts with a probability and returns the z score that leaves that amount of area on the left side. It is useful for percentiles, cutoffs, and critical values.
7) Can I use raw x values here?
Yes. Choose the standardization option, then enter x, mean, and standard deviation. The calculator converts the raw value into a z score and also reports standard normal probabilities for that standardized position.
8) How accurate are the results?
The calculator uses established numerical approximations for the error function and inverse normal function. They are accurate enough for most educational, analytical, and engineering style calculations, though specialized statistical software may show tiny rounding differences.
Notes
- This page uses numerical approximations for the error function and inverse normal function.
- The standard normal model assumes mean 0 and standard deviation 1 after standardization.
- For very high precision statistical work, cross-check with dedicated statistical software.