Normal Distribution Probability Calculator

Calculator

Parameters

Mean (μ)

Standard Deviation (σ)

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

Calculation Type

Lower bound (x1)

Upper bound (x2)

Distribution Chart

Click Calculate to view the distribution chart

Learn More

Normal Distribution: Definition, Formula, and Applications

Normal Distribution

Definition: The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric about its mean and follows a characteristic "bell-shaped" curve.

Formula:The probability density function (PDF) and cumulative density function (CDF) are given by:

f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

F(x) = P(X \leq x) = \Phi \left({\frac {x-\mu }{\sigma }}\right)

Where:

$\mu$ is the mean (location parameter)
$\sigma$ is the standard deviation (scale parameter)
$\sigma^2$ is the variance

Also,

P(a < X \leq b) = \Phi \left({\frac {b-\mu }{\sigma }}\right) - \Phi \left({\frac {a-\mu }{\sigma }}\right)

Example: Let

X \sim N(-5, 4)

, find

P(-7 < X < -3)

P(-7 < X < -3) = \Phi \left(\frac{-3+5}{2}\right) - \Phi \left(\frac{-7+5}{2}\right) = \Phi(1) - \Phi(-1) = 0.8413 - 0.1587 = 0.6826

Properties

Symmetric about the mean
Bell-shaped curve
Mean, median, and mode are all equal
68-95-99.7 rule:
- 68% of data falls within 1 standard deviation of the mean
- 95% of data falls within 2 standard deviations
- 99.7% of data falls within 3 standard deviations

Z-Scores

Definition: A z-score represents how many standard deviations away from the mean a data point is.

z = \frac{x - \mu}{\sigma}

Where:

$x$ is the data point
$\mu$ is the mean
$\sigma$ is the standard deviation

Applications

1. Quality Control

In manufacturing, normal distributions are used to monitor product specifications and ensure consistent quality standards. By tracking production outputs against a normal curve, companies can identify variations, detect defects, and maintain high levels of efficiency. Statistical process control often relies on this to minimize variability and avoid costly recalls or production downtimes.

2. Biology and Medicine

Normal distributions are key in analyzing biological measurements, drug effectiveness, and patient outcomes. Whether assessing the spread of a disease or evaluating clinical trials, researchers use these models to predict outcomes, understand variability in populations, and improve healthcare decision-making through data-driven insights.

3. Finance

In finance, the normal distribution is crucial for risk assessment, portfolio management, and pricing financial derivatives such as options. By modeling asset returns with a normal curve, financial analysts can estimate potential losses, optimize investment strategies, and calculate probabilities for various market scenarios, aiding in more informed and strategic financial decisions.

4. Social Sciences

In the social sciences, normal distributions are used to analyze standardized test scores, survey results, and demographic data. This helps researchers understand how populations behave, identify trends, and make predictions. From education to psychology, normal curves enable the effective measurement and comparison of human behaviors and societal changes.

5. Engineering

Engineering disciplines, especially reliability engineering, use normal distributions to predict the lifespan of products and systems. By modeling the time until failure of components, engineers can ensure optimal maintenance schedules, minimize downtime, and improve the design and durability of products based on probabilistic models of failure rates.

R Code Example

library(tidyverse)

# Calculate the probability of X between -1 and 1
P_between <- pnorm(1) - pnorm(-1)
print(str_glue("P(-1 < X < 1) = {round(P_between, 4)}"))

# X ~ N(-5, 4)
# Calculate the probability of X between -7 and -3
P_between <- pnorm(-3, mean = -5, sd = 2) - pnorm(-7, mean = -5, sd = 2)
print(str_glue("P(-7 < X < -3) = {round(P_between, 4)}"))

# plot the standard normal distribution
x <- seq(-3, 3, length.out = 1000)
pdf <- dnorm(x)
df <- tibble(x = x, pdf = pdf)

ggplot(df, aes(x = x, y = pdf)) +
  geom_line(color = "blue") +
  geom_area(data = subset(df, x >= -1 & x <= 1), aes(x = x, y = pdf), fill = "blue", alpha = 0.2) +
  labs(title = "Standard Normal Distribution",
       x = "x",
       y = "Probability Density") +
  annotate("text", x = 2, y = 0.3, label = str_glue("P(-1 < X < 1) = {round(P_between, 4)}"), hjust = 0) +
  theme_minimal()

Python Code Example

import numpy as np
import scipy.stats as stats
import pandas as pd
import matplotlib.pyplot as plt

# Calculate the probability of X between -1 and 1
P_between = stats.norm.cdf(1) - stats.norm.cdf(-1)
print(f"P(-1 < X < 1) = {round(P_between, 4)}")

# X ~ N(-5, 4)
# Calculate the probability of X between -7 and -3
P_between = stats.norm.cdf(-3, loc=-5, scale=2) - stats.norm.cdf(-7, loc=-5, scale=2)
print(f"P(-7 < X < -3) = {round(P_between, 4)}")

# Plot the standard normal distribution
x = np.linspace(-3, 3, 1000)
pdf = stats.norm.pdf(x)

# Create a pandas DataFrame
df = pd.DataFrame({'x': x, 'pdf': pdf})

# Plotting
plt.plot(df['x'], df['pdf'], color="blue")
plt.fill_between(df['x'], df['pdf'], where=(df['x'] >= -1) & (df['x'] <= 1), color="blue", alpha=0.2)
plt.title("Standard Normal Distribution")
plt.xlabel("x")
plt.ylabel("Probability Density")
plt.annotate(f"P(-1 < X < 1) = {round(P_between, 4)}", xy=(2, 0.3), ha='left')
plt.grid(True)
plt.show()

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