Inverse Normal Distribution

Created:April 14, 2025

This calculator helps you compute the values of a normal distribution given probabilities and parameters. Unlike the regular normal distribution calculator that finds probabilities given values, this inverse calculator finds values given probabilities. You can find the value x such that P(X < x) = p, P(X > x) = p, P(|X - μ| < x) = p, or P(|X - μ| > x) = p.

Calculator

Parameters

Mean (μ)

Standard Deviation (σ)

Important:If you have variance (σ² = 25), enter standard deviation (σ = 5)

Calculation Type

Probability (p)

Distribution Chart

Click Calculate to view the distribution chart

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Inverse Normal Distribution

Definition: The inverse normal distribution calculation involves finding the value(s) in a normal distribution that correspond to a given probability. This is the opposite of the regular normal distribution calculation, which finds probabilities given values.

Formula:For a normal distribution with mean

\mu

and standard deviation

\sigma

, if

p = P(X \leq x)

, then:

x = F^{-1}(p) = \mu + \sigma \Phi^{-1}(p)

Where

\Phi^{-1}

is the inverse of the standard normal cumulative distribution function.

Common cases:

To find $x$ such that $P(X < x) = p$ : Use $x = \mu + \sigma \Phi^{-1}(p)$
To find $x$ such that $P(X > x) = p$ : Use $x = \mu + \sigma \Phi^{-1}(1-p)$
To find $x$ such that $P(|X - \mu| < x) = p$ : Use $x = \sigma \Phi^{-1}(0.5 + p/2)$
To find $x$ such that $P(|X - \mu| > x) = p$ : Use $x = \sigma \Phi^{-1}(1 - p/2)$

Example: Let

X \sim N(10, 4)

, find the value

x

such that

P(X < x) = 0.95

$x = 10 + 2 \Phi^{-1}(0.95) = 10 + 2(1.645) = 13.29$

So the value $x$ where $P(X < x) = 0.95$ is 13.29.

How to Calculate Inverse Normal in R

library(tidyverse)

# P(X < x) = 0.95, X ~ N(10, 4)
x_value <- qnorm(0.95, mean = 10, sd = 2)
print(x_value) # 13.28971

# P(X > x) = 0.05, X ~ N(10, 4)
x_value <- qnorm(1 - 0.05, mean = 10, sd = 2)
print(x_value) # 13.28971

# P(|X - μ| < x) = 0.95
mean_val <- 10
sd_val <- 2
prob <- 0.95

z_critical <- qnorm(0.5 + prob/2)
x_value <- z_critical * sd_val
lower_bound <- mean_val - x_value
upper_bound <- mean_val + x_value
print(c(x_value, lower_bound, upper_bound)) # 3.919928  6.080072 13.919928

#P(|X - μ| > x) = 0.05
z_critical <- qnorm(1 - 0.05/2)
x_value <- z_critical * sd_val
lower_bound <- mean_val - x_value
upper_bound <- mean_val + x_value
print(c(x_value, lower_bound, upper_bound)) # 3.919928  6.080072 13.919928

How to Calculate Inverse Normal in Python

Python

import numpy as np
from scipy import stats

# P(X < x) = 0.95, X ~ N(10, 4)
x_value = stats.norm.ppf(0.95, loc=10, scale=2)
print(x_value)  # 13.28971

# P(X > x) = 0.05, X ~ N(10, 4)
x_value = stats.norm.ppf(1 - 0.05, loc=10, scale=2)
print(x_value)  # 13.28971

# P(|X - μ| < x) = 0.95
mean_val = 10
sd_val = 2
prob = 0.95

z_critical = stats.norm.ppf(0.5 + prob/2)
x_value = z_critical * sd_val
lower_bound = mean_val - x_value
upper_bound = mean_val + x_value
print([x_value, lower_bound, upper_bound])  # [3.919928, 6.080072, 13.919928]

# P(|X - μ| > x) = 0.05
z_critical = stats.norm.ppf(1 - 0.05/2)
x_value = z_critical * sd_val
lower_bound = mean_val - x_value
upper_bound = mean_val + x_value
print([x_value, lower_bound, upper_bound])  # [3.919928, 6.080072, 13.919928]