Standard Normal Probability Calculator
Use this calculator to find probabilities from a z score. Choose the probability type, enter your z value(s), and click calculate.
What this probability calculator from z score does
This tool computes probabilities from the standard normal distribution (mean 0, standard deviation 1). If you already have a z score, it gives the area under the normal curve for common cases like left-tail, right-tail, interval probability, and two-tailed probability.
In plain language, it answers questions like:
- What is the probability of getting a value below a z score?
- What is the probability of getting a value above a z score?
- What is the probability of being between two z scores?
- What is the two-sided tail probability for hypothesis tests?
How to use it
- Select the probability type from the dropdown.
- Enter one z score (or two z scores for the “between” mode).
- Click Calculate Probability.
- Read the probability as a decimal and percent.
Quick z score refresher
A z score tells you how far a value is from the mean in units of standard deviation. A z score of +1 means one standard deviation above the mean. A z score of -2 means two standard deviations below the mean.
If you start with a raw score x, population mean μ, and standard deviation σ, convert to z first, then use this calculator.
Probability formulas used
Let Φ(z) be the standard normal CDF.
- Left-tail: P(Z ≤ z) = Φ(z)
- Right-tail: P(Z ≥ z) = 1 - Φ(z)
- Between: P(a ≤ Z ≤ b) = Φ(b) - Φ(a)
- Two-tailed: P(|Z| ≥ |z|) = 2 × (1 - Φ(|z|))
The calculator evaluates Φ(z) with a standard numerical approximation to the error function, which is accurate for practical use.
Worked examples
Example 1: Left-tail probability
If z = 1.00, then P(Z ≤ 1.00) ≈ 0.8413. Interpretation: about 84.13% of observations in a normal distribution fall below one standard deviation above the mean.
Example 2: Right-tail probability
If z = 1.96, then P(Z ≥ 1.96) ≈ 0.0250. Interpretation: only 2.5% of observations lie above 1.96 standard deviations.
Example 3: Between two z scores
For z₁ = -1 and z₂ = 1, P(-1 ≤ Z ≤ 1) ≈ 0.6827. That is the familiar 68% region around the mean.
Common mistakes to avoid
- Using this tool with non-normal data without checking assumptions.
- Mixing up left-tail and right-tail probabilities.
- Forgetting to convert from raw score to z score first.
- Using sample statistics where population values are required without acknowledging approximation.
When this calculator is useful
- Statistics homework and exam prep
- Hypothesis testing and p-value estimation
- Quality control and process analysis
- Percentile and probability interpretation in research
FAQ
Does this calculator replace a z table?
Yes for most purposes. It gives the same type of probabilities that a z table provides, but instantly and with flexible modes.
Can I enter very large z values?
Yes. For very large positive or negative z scores, probabilities approach 1 or 0 as expected.
Is this for t-distributions?
No. This tool is for the standard normal distribution only. Use a t-distribution calculator when degrees of freedom matter.