Normal Distribution Probability Density Function Calculator
Result:
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The Normal Distribution, often referred to as the Gaussian distribution or bell curve, is one of the most important probability distributions in statistics. It describes how the values of a variable are distributed, with most values clustering around a central mean and tapering off symmetrically as they move away from the mean.
What is a Probability Density Function (PDF)?
For continuous random variables, we use a Probability Density Function (PDF) to describe the likelihood of the variable taking on a given value. Unlike discrete probability distributions where we can find the probability of an exact value, for continuous variables, the probability of any single exact value is technically zero. Instead, the PDF gives us the relative likelihood of the variable being near a specific value. The area under the PDF curve between two points represents the probability that the variable falls within that range.
The Normal Distribution PDF Formula
The formula for the Probability Density Function of a Normal Distribution is:
f(x) = (1 / (σ * sqrt(2 * π))) * exp(-((x - μ)^2) / (2 * σ^2))
x(Value): This is the specific point on the x-axis for which you want to calculate the probability density.μ(Mean): This is the average or central value of the distribution. It determines the peak of the bell curve.σ(Standard Deviation): This measures the spread or dispersion of the distribution. A smaller standard deviation means the data points are clustered closely around the mean, resulting in a taller, narrower curve. A larger standard deviation indicates data points are spread out, leading to a flatter, wider curve.π(Pi): A mathematical constant, approximately 3.14159.exp: The exponential function, e raised to the power of the expression in parentheses.sqrt: The square root function.
How to Use the Calculator
Our Normal Distribution PDF Calculator allows you to quickly find the probability density for a given value (x) based on the distribution's mean (μ) and standard deviation (σ).
- Value (x): Enter the specific data point or value for which you want to find the density.
- Mean (μ): Input the mean of your normal distribution. This is the average value of your dataset.
- Standard Deviation (σ): Enter the standard deviation of your normal distribution. This indicates how much variation or dispersion there is from the average.
- Click "Calculate Probability Density" to see the result.
Interpreting the Results
The output of the calculator is the probability density f(x) at the specified value x. A higher density value at a particular x means that values around x are more likely to occur within that distribution. Remember, this is not a direct probability but a density. To find the actual probability of a value falling within a range, you would need to integrate the PDF over that range.
Examples:
Let's consider a few examples to illustrate:
Example 1: Standard Normal Distribution
The standard normal distribution has a mean (μ) of 0 and a standard deviation (σ) of 1.
- Value (x): 0
- Mean (μ): 0
- Standard Deviation (σ): 1
- Result: Approximately 0.39894228 (This is the peak of the standard normal curve).
Example 2: Value One Standard Deviation Away
Using the standard normal distribution again, let's see the density at x = 1.
- Value (x): 1
- Mean (μ): 0
- Standard Deviation (σ): 1
- Result: Approximately 0.24197072 (Lower than at the mean, as expected).
Example 3: A Different Distribution
Suppose we have a distribution of adult male heights in a certain population, with a mean height of 175 cm and a standard deviation of 7 cm.
- Value (x): 180 cm
- Mean (μ): 175 cm
- Standard Deviation (σ): 7 cm
- Result: Approximately 0.05086338. This indicates the relative likelihood of an adult male having a height of 180 cm in this population.
This calculator is a useful tool for students, statisticians, and anyone working with data analysis to understand and apply the Normal Distribution PDF.