Statistics & Probability

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Inverse Normal Distribution Calculator.

Calculate this statistic instantly with validated formulas.

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Result: z for p=0.975: 1.95996399

Result

z for p=0.975: 1.95996399

Use Cases

Statistical Hypothesis Testing

Find critical z-values for confidence intervals or hypothesis tests. For a given significance level (e.g., 0.05), the inverse normal gives the z-score that defines the rejection region.

Example: For a two-tailed test with alpha = 0.05, enter 0.975 to get z = 1.96.

Data Normalization and Outlier Detection

Convert probabilities to z-scores to standardize data or identify outliers. This helps in comparing values from different normal distributions.

Example: If a value corresponds to the 90th percentile, enter 0.9 to get z ≈ 1.28.

Frequently Asked Questions

What is the inverse normal distribution?
The inverse normal distribution, also known as the inverse Gaussian or normal quantile function, returns the z-score corresponding to a given cumulative probability under the standard normal distribution. For example, a probability of 0.975 gives a z-score of approximately 1.96.
How do I use this calculator?
Enter one or more probabilities (values between 0 and 1) separated by commas or spaces. The calculator will compute the corresponding z-scores using the standard normal distribution. Ensure each value is a valid probability; otherwise, the result may be undefined.
What is the range of input values?
The input values must be probabilities between 0 and 1 (exclusive). Values outside this range are not valid for the inverse normal distribution. For example, 0.5 is valid, but 1.5 is not.

Tips & Common Mistakes

Tips

  • Ensure your input values are between 0 and 1. If you have percentages, convert them to decimals (e.g., 95% becomes 0.95).
  • For a two-tailed test, use the probability 1 - (alpha/2) to get the critical z-score. For example, alpha = 0.05 gives 0.975.
  • You can enter multiple probabilities at once to get a list of z-scores, which is useful for comparing different percentiles.
  • Remember that the inverse normal distribution assumes a standard normal distribution (mean = 0, standard deviation = 1).

Common Mistakes to Avoid

  • Entering values outside the 0-1 range, such as 1.5 or -0.2, which are not valid probabilities.
  • Confusing the inverse normal with the normal CDF: the inverse normal takes a probability and returns a z-score, not the other way around.
  • Using percentages without converting to decimals, e.g., entering 95 instead of 0.95.

Last updated: August 13, 2026