Normal Distribution Calculator

Distributions and Statistical Models

This tool helps you calculate probabilities and values associated with the normal (Gaussian) distribution. You can find the probability density (PDF), cumulative probability (CDF) for a given value, or the probability of a value falling within a specific range.

Examples

Explore some common use cases for the Normal Distribution Calculator.

IQ Score Probability

lessThan

Calculate the probability of a person having an IQ score less than 120, given the mean is 100 and standard deviation is 15.

μ: 100, σ: 15

x: 120

Stock Price Movement

greaterThan

Find the probability of a stock price exceeding $55 next month, assuming the mean price is $50 with a standard deviation of $5.

μ: 50, σ: 5

x: 55

Student Exam Scores

between

Determine the percentage of students who scored between 75 and 90 on an exam where the average score was 80 and the standard deviation was 5.

μ: 80, σ: 5

a: 75, b: 90

Standard Normal Distribution

lessThan

Using the standard normal distribution (μ=0, σ=1), find the cumulative probability for a Z-score of 1.96.

μ: 0, σ: 1

x: 1.96

Other Titles
Understanding the Normal Distribution: A Comprehensive Guide
Dive deep into the concepts, applications, and mathematics behind the most important distribution in statistics.

What is the Normal Distribution?

  • The Bell Curve
  • Key Characteristics
  • The Standard Normal Distribution
The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It is the most prominent distribution in statistics and is fundamental to many statistical theories and methods. A vast number of natural phenomena, from human heights to measurement errors, tend to follow this distribution.
Key Characteristics
The shape and position of the normal distribution are determined by two parameters: the mean (μ) and the standard deviation (σ). The mean represents the central point of the distribution, where the peak of the bell curve is located. The standard deviation measures the spread or dispersion of the data points around the mean. A smaller standard deviation results in a narrower, taller curve, while a larger standard deviation leads to a wider, shorter curve.
The Standard Normal Distribution
A special case of the normal distribution is the standard normal distribution, which has a mean of 0 and a standard deviation of 1. Any normal distribution can be converted to a standard normal distribution using the Z-score formula: Z = (X - μ) / σ. This standardization allows for the comparison of scores from different normal distributions and simplifies probability calculations.

Key Properties:

  • Symmetrical around the mean (μ).
  • Mean, median, and mode are all equal.
  • The total area under the curve is equal to 1 (or 100%).

Step-by-Step Guide to Using the Normal Distribution Calculator

  • Selecting Calculation Type
  • Entering Parameters
  • Interpreting the Results
This calculator is designed for ease of use. Follow these steps to perform your calculations:
1. Select Your Calculation Type
Choose what you want to calculate from the dropdown menu: P(X < x) for the probability of a value being less than x, P(X > x) for the probability of a value being greater than x, or P(a < X < b) for the probability of a value falling between two points.
2. Enter the Distribution Parameters
Input the Mean (μ) and the Standard Deviation (σ) of your dataset. The standard deviation must be a positive number.
3. Provide the Value(s) of Interest
Depending on your selection in step 1, enter the value for 'x', or the lower and upper bounds 'a' and 'b'.
4. Interpret the Results
The calculator will provide the Z-score(s) for your input values, the probability density function (PDF) value, and the calculated cumulative probability. The Z-score tells you how many standard deviations your value is from the mean, while the probability result gives you the likelihood according to your chosen calculation type.

Real-World Applications of Normal Distribution

  • Quality Control in Manufacturing
  • Finance and Economics
  • Natural and Social Sciences
The normal distribution's applicability spans numerous fields.
Quality Control
In manufacturing, the dimensions of products (like the diameter of a screw) often follow a normal distribution. Companies use this to set tolerance limits and determine the probability of a product being defective.
Finance
Asset returns are often assumed to be normally distributed, which is a cornerstone of models like the Black-Scholes option pricing model. It helps in assessing risk and modeling future price movements.
Natural and Social Sciences
Many biological measurements (height, weight, blood pressure) and psychological metrics (IQ scores, test results) are approximately normally distributed. This allows researchers to define 'normal' ranges and identify outliers.

Example Scenario: Exam Scores

  • If exam scores in a large class are normally distributed with a mean of 75 and a standard deviation of 10, a teacher can calculate the percentage of students who will score above 90 (A grade) or below 60 (fail).

Common Misconceptions and Correct Methods

  • Not All Data is Normal
  • The Meaning of PDF
  • Symmetry is Key
Assumption of Normality
A common mistake is to assume all datasets are normally distributed. While many are, it's crucial to test for normality first using methods like histograms or statistical tests (e.g., Shapiro-Wilk). Applying normal distribution principles to non-normal data can lead to incorrect conclusions.
Interpreting the PDF
The value of the Probability Density Function (PDF) is not a probability itself. For a continuous distribution, the probability of any single exact point is zero. The PDF represents the relative likelihood of a value. Probability is found by calculating the area under the curve (the integral) over a range, which is what the Cumulative Distribution Function (CDF) does.
Importance of Symmetry
The symmetry of the normal distribution is a core property. For a standard normal distribution, P(Z < -z) is equal to P(Z > z). This symmetry simplifies many calculations.

Mathematical Derivation and Formulas

  • The PDF Formula
  • The Z-Score Transformation
  • The Cumulative Distribution Function (CDF)
Probability Density Function (PDF)
The formula for the PDF of a normal distribution is: f(x) = (1 / (σ √(2π))) e^(-(x - μ)² / (2σ²)), where μ is the mean, σ is the standard deviation, and e is Euler's number (≈2.718). This calculator uses this formula to determine the height of the curve at a specific point x.
Z-Score Formula
To standardize a value from any normal distribution, we use: Z = (X - μ) / σ. This gives us a Z-score that can be used with the standard normal distribution (μ=0, σ=1).
Cumulative Distribution Function (CDF)
The CDF, which gives P(X ≤ x), does not have a simple closed-form expression. It is calculated by integrating the PDF. Computationally, it's often approximated using the error function (erf): Φ(z) = 0.5 * (1 + erf(z / √2)), where z is the Z-score. Our calculator uses a precise numerical method to compute this value.