What is the Normal Distribution Equation?
The normal distribution equation is the mathematical formula that defines the probability density function (PDF) of a normal distribution, also known as a Gaussian distribution. This bell-shaped curve describes how values are distributed around the mean in many natural and statistical processes.
The normal distribution formula gives the probability density for a given value of a continuous random variable (X).
The normal distribution equation is the following:
This formula returns the probability density, or height of the curve, at any given value of X, not a probability directly. However, the total area under the curve equals 1, and you can find the probability of a range of values by calculating the area under the curve for that range.
Learn more in-depth about the Normal Distribution in Statistics.
Components of the Normal Distribution Equation
- f(x): The value of the probability density function at a given x-value. It represents the height of the curve at that point.
- x: The value of the random variable being evaluated.
- μ (mu): The mean of the distribution. This is the center of the curve.
- σ (sigma): The standard deviation, which controls how spread out the distribution is.
- π (pi): The mathematical constant pi, approximately equal to 3.1416.
- e: The base of the natural logarithm, approximately equal to 2.718.
How the Normal Distribution Formula Works
- The term (x − μ)² measures the squared distance between a specific value and the mean. This tells you how far x is from the center of the distribution.
- Dividing by 2σ² scales that distance based on how spread out the data are. A larger σ (standard deviation) leads to a slower drop-off in density.
- The negative exponent in e^(−(x − μ)² / 2σ²) causes values farther from the mean to have much smaller density values. This creates the bell shape of the curve.
- The constant in front, 1 / (σ√(2π)), ensures that the total area under the curve equals 1, so the function represents a valid probability distribution.
- The height produced by the function, f(x), is a probability density, not a direct probability. Probabilities come from the area under the curve between two values.
This normal distribution equation defines the shape of the curve, which is symmetric around the mean and characterized by its bell-like appearance. It is the basis for many statistical methods, including hypothesis testing, confidence intervals, and z-scores.
Example Calculation using the Normal Distribution Formula
Ok, let’s use the normal distribution equation to find a probability density. We’ll need to follow the PEMDAS Order of Mathematical Operations to get the correct answer!
Suppose a normal distribution has a mean (μ) of 100 and a standard deviation (σ) of 15. What is the value of the probability density function at x = 120?
We’ll calculate this using the normal distribution formula:
Step 1: Plug the values into the normal distribution formula
Step 2: Calculate the squared distance from the mean
(120 − 100)² = 400
Step 3: Multiply the standard deviation squared by 2
2 × 15² = 450
Step 4: Divide the squared distance by that result
400 / 450 ≈ 0.8889
Step 5: Take the negative and raise e to that power
e-0.8889 = 0.4111
Step 6: Calculate the front constant
Step 7: Multiply both parts together
0.0266 × 0.4111 ≈ 0.0109
Final Answer Using the Normal Distribution Formula
According to the normal distribution equation, the value of the normal distribution at x = 120 is approximately 0.0109.
This value is the height of the curve at x = 120. It tells you the relative likelihood of observing a value near 120, assuming the data follow a normal distribution with μ = 100 and σ = 15.
The graph below illustrates how this one point fits in with the entire curve that the normal distribution formula produced.
