Normal Approximation
- Uses the normal (bell curve) distribution to approximate more complex or unknown probability distributions.
- Lets you make population-level predictions from sample statistics when the full population distribution is unavailable or hard to work with.
- Commonly applied to estimate means, standard deviations, or probabilities (including rare events) from sample data.
Definition
Section titled “Definition”Normal approximation is a statistical method that approximates complex probability distributions using the normal (bell curve) distribution.
Explanation
Section titled “Explanation”Normal approximation is applied when the exact distribution of data is unknown or too complex to work with directly. By assuming the underlying distribution is approximately normal, practitioners use sample statistics (for example, sample mean and sample standard deviation) to estimate corresponding population parameters. These estimates are then used with the normal distribution to make predictions or calculate probabilities about the population. The method acknowledges sampling uncertainty and relies on statistical techniques to account for imperfect samples.
Examples
Section titled “Examples”Example 1: Estimating the mean and standard deviation of a population
Section titled “Example 1: Estimating the mean and standard deviation of a population”Suppose we study heights of students in a high school but cannot measure every student. We take a sample of 100 students and measure their heights. Using the sample mean and sample standard deviation as estimates, and assuming the population distribution of heights is normal, we use the normal distribution to approximate the population mean and standard deviation and make predictions about the population.
Example 2: Estimating the probability of a rare event
Section titled “Example 2: Estimating the probability of a rare event”Suppose we study failure rates of a particular type of car battery where failures within the first year are rare and direct calculation is impractical. We collect data on a large sample of batteries and compute the sample mean and sample standard deviation of failure rates. Assuming the population distribution of failure rates is normal, we use those estimates with the normal distribution to approximate the probability that a battery will fail within the first year.
Use cases
Section titled “Use cases”- Making predictions or calculations about a population when the full population distribution is unknown.
- Approximating probabilities for events (including rare events) using sample-based estimates.
Notes or pitfalls
Section titled “Notes or pitfalls”- A sample may not perfectly represent the population; statistical techniques are needed to account for this sampling uncertainty.
- The method relies on the assumption that the population distribution can be reasonably approximated by a normal distribution.
Related terms
Section titled “Related terms”- Normal distribution (bell curve)