StatsCalculators

Normal Distribution

Created:November 22, 2024

Last Updated:March 29, 2025

This calculator helps you compute the probabilities of a normal distribution given the mean and standard deviation. You can calculate the probability of a value being less than, greater than, or between certain values. The distribution chart shows the probability density function (PDF) and cumulative density function (CDF) of the normal distribution.

Calculator

Parameters

Mean (μ)

Standard Deviation (σ)

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

Calculation Type

Tip:P(X≤x) = P(X<x) since the probability of any exact value is zero.

Lower bound (x1)

Upper bound (x2)

Distribution Chart

Click Calculate to view the distribution chart

Learn More

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) = 0.6827

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

Explore the Normal Distribution

Adjust the mean and standard deviation to see how they affect the shape of the normal curve.

Mean (average)50

Standard Deviation (spread)10

Observe how:

The mean shifts the center of the curve left or right
The standard deviation makes the curve wider or narrower
The total area under the curve always remains the same

R Code Example

R

library(tidyverse)

# P(X < 1) - P(X < -1)
P_between <- pnorm(1) - pnorm(-1)
print(p_between) # 0.6826895

# X ~ N(-5, 4)
# P(-7 < X < -3)
p_between <- pnorm(-3, mean = -5, sd = 2) - pnorm(-7, mean = -5, sd = 2)
print(p_between) # 0.6826895

# 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 = 1, y = 0.3, label = str_glue("P(-1 < X < 1) = {round(P_between, 4)}"), hjust = 0) +
  theme_minimal()

Python Code Example

Python

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

# P(X < 1) - P(X < -1)
p_between = stats.norm.cdf(1) - stats.norm.cdf(-1)
print(p_between)

# X ~ N(-5, 4)
# P(-7 < X < -3)
p_between = stats.norm.cdf(-3, loc=-5, scale=2) - stats.norm.cdf(-7, loc=-5, scale=2)
print(p_between)

# 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=(1, 0.3), ha='left')
plt.grid(True)
plt.show()

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