What is the inverse normal distribution calculator used for?+
The inverse normal calculator finds the x value (or z-score) such that the cumulative probability P(X ≤ x) equals a given target. It is used in hypothesis testing to find critical z-values, in confidence interval construction to find z_alpha/2, in percentile calculations for standardized tests (IQ, SAT, GRE), in quality control to set defect thresholds, and in finance to compute Value at Risk. Essentially any time you need to go from a probability to a value on a normal distribution.
What is the inverse normal formula?+
The inverse normal formula is x = μ + Φ−1(p) × σ, where μ is the mean, σ is the standard deviation, p is the target left-tail probability, and Φ−1(p) is the standard normal quantile (probit). For right-tail probability q, substitute p = 1 − q. Common quantiles: Φ−1(0.90) = 1.2816, Φ−1(0.95) = 1.6449, Φ−1(0.975) = 1.9600, Φ−1(0.99) = 2.3263.
How do I find the critical z value for a 95% confidence interval?+
For a 95% confidence interval, use the Critical Values mode with significance level α = 5% and test type = two-tailed. The critical z-value is z = 1.9600, because each tail holds 2.5% of the area. The 95% confidence interval is x-bar ± 1.9600 × SE, where SE is the standard error. For a 99% CI, the critical value is z = 2.5758 (α = 1%, two-tailed).
What is the difference between left-tail and right-tail in the inverse normal?+
Left-tail finds x such that P(X < x) = p, meaning p fraction of the distribution is to the left of x. Right-tail finds x such that P(X > x) = p, meaning p fraction is to the right. For left-tail p = 95%: x is the 95th percentile. For right-tail p = 5%: x is the 95th percentile (same value). Internally, right-tail probability p is converted to left-tail 1 − p before applying the formula.
What is the 99th percentile of the standard normal distribution?+
The 99th percentile of the standard normal (mean 0, SD 1) is z = 2.3263. This means P(Z < 2.3263) = 99%. The 1st percentile is z = −2.3263. Other key percentiles: 90th = 1.2816, 95th = 1.6449, 97.5th = 1.9600, 99.5th = 2.5758, 99.9th = 3.0902. Enter mean = 0 and SD = 1 in this calculator to confirm any of these values.
How do I convert a percentile to a score for a standardized test?+
Enter the test mean and standard deviation in Find X Value mode, set the percentile as the left-tail probability, and read the score. For example, the SAT (mean 1050, SD 217) score at the 75th percentile: z = Φ−1(0.75) = 0.6745, so score = 1050 + 0.6745 × 217 = 1050 + 146.4 = 1196. For GRE Verbal (mean 151, SD 9): 90th percentile = 151 + 1.2816 × 9 = 162.5.
What critical z value corresponds to a 99% confidence level?+
For a 99% confidence interval (two-tailed), the critical z-value is 2.5758, because α = 1% is split equally: 0.5% in each tail. The exact value is Φ−1(0.995) = 2.5758. For a 90% CI: z = 1.6449 (α = 10%, 5% each tail). For a 95% CI: z = 1.9600 (α = 5%, 2.5% each tail). For a 99.9% CI: z = 3.2905 (α = 0.1%, 0.05% each tail).
How is the inverse normal related to the probit function?+
The probit function is exactly the inverse standard normal CDF: probit(p) = Φ−1(p). The term "probit" (probability unit) was coined by Chester Bliss in 1934. Probit regression transforms a probability into a linear predictor using this function. The inverse normal for a general N(μ, σ) distribution is then just x = μ + probit(p) × σ. This calculator computes both the general inverse normal (Find X Value mode) and the probit for hypothesis testing (Critical Values mode).
What is the rejection region for a one-tailed z-test at alpha = 0.05?+
For a right-tailed test at α = 5%, the rejection region is z > 1.6449, which is Φ−1(0.95). For a left-tailed test at α = 5%, the rejection region is z < −1.6449 = Φ−1(0.05). Reject the null hypothesis H0 if the calculated test statistic falls in the rejection region. Use the Critical Values mode in this calculator and select the appropriate tail type.
Can I find Value at Risk using the inverse normal?+
Yes. If daily portfolio returns follow a normal distribution with mean μ and standard deviation σ, the VaR at confidence level c is VaR = −(μ + Φ−1(1 − c) × σ). For example, with μ = 0.1% and σ = 2%, VaR at 95% confidence: z = Φ−1(0.05) = −1.6449, VaR = −(0.001 + (−1.6449) × 0.02) = −(0.001 − 0.0329) = 3.19%. This means a 5% chance of losing more than 3.19% on any given day.
What numerical method does this calculator use for the inverse normal?+
This calculator uses the Beasley-Springer-Moro (BSM) rational approximation. It partitions the probability range into three regions: the far-left tail (p < 0.02425), the central region (0.02425 ≤ p ≤ 0.97575), and the far-right tail (p > 0.97575). Each region uses a separate rational polynomial fit to Φ−1(p). The maximum absolute error across all three regions is less than 4.5 × 10−4. The forward verification (CDF check) uses the Abramowitz and Stegun error function approximation, accurate to 1.5 × 10−7.
How does the inverse normal differ from the inverse t-distribution?+
The inverse t-distribution finds critical values for the t-distribution, which has heavier tails than the normal and depends on degrees of freedom (df). For large df (above 30), the t and z critical values converge: at df = 120, the t critical value for 95% two-tailed is 1.980, very close to z = 1.960. For small samples (df < 30), the t critical values are noticeably larger than z, reflecting greater uncertainty. Use the inverse normal (this calculator) when the population standard deviation is known or the sample is large. Use the inverse t (Critical Value Calculator) for small samples.