Normal Distribution PDF
f(x) = (1/ฯโ2ฯ) ร e^(-(x - ฮผ)ยฒ / 2ฯยฒ)
The probability density function of the normal (Gaussian) distribution describes the bell-shaped curve that arises throughout statistics. It is fully characterized by its mean (ฮผ) and standard deviation (ฯ). The area under the curve between any two values gives the probability of an observation falling in that range.
Variables
The height of the curve at value x (not a probability itself)
The center of the distribution
Controls the spread of the distribution
The mathematical constant approximately equal to 2.71828
Example Calculation
Scenario
IQ scores follow a normal distribution with ฮผ = 100 and ฯ = 15. What is the probability density at x = 115?
Given Data
Calculation
f(115) = (1/(15โ(2ฯ))) ร e^(-(115-100)ยฒ/(2ร15ยฒ)) = (1/37.60) ร e^(-225/450) = 0.02660 ร e^(-0.5) = 0.02660 ร 0.6065
Result
f(115) = 0.01613
Interpretation
The density at IQ = 115 is 0.01613. This is not a probability; it is the height of the curve at that point. To find the probability of scoring between, say, 110 and 120, you would integrate the PDF over that interval (or use a z-table). By the 68-95-99.7 rule, about 68% of IQ scores fall between 85 and 115.
When to Use This Formula
- โModeling continuous data that clusters symmetrically around the mean
- โServing as the basis for z-scores, confidence intervals, and hypothesis tests
- โApproximating other distributions (e.g., binomial) when sample sizes are large
- โUnderstanding the theoretical foundation of the Central Limit Theorem
Common Mistakes
- โInterpreting the PDF value f(x) as a probability (it is a density; probabilities come from areas)
- โAssuming all data is normally distributed without checking with plots or tests
- โForgetting that the normal distribution extends from negative infinity to positive infinity
- โConfusing the PDF with the cumulative distribution function (CDF)
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Common questions about this formula
For any normal distribution, approximately 68% of values fall within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This is also known as the empirical rule and provides a quick way to assess the spread of data.
The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's shape. This makes the normal distribution central to confidence intervals, hypothesis tests, and many other statistical methods.