Standard normal distribution
Introduction
The standard normal distribution is a specific type of probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In graph form, standard normal distribution will appear as a bell curve.
Definition
The standard normal distribution is a normal distribution with a mean of zero and standard deviation of one. It is often denoted using the Greek letter Φ (phi). The standard normal distribution, also known as the z-distribution, is a special case of the normal distribution where the mean (μ) is 0 and the standard deviation (σ) is 1. The standard normal distribution is used in statistics to represent real-valued random variables of unknown distributions but with known mean and variance.
Properties
The standard normal distribution has several properties that make it extremely useful in statistics and probability theory. These include:
- Symmetry: The standard normal distribution is symmetric about the mean. This means that the left and right halves of the distribution are mirror images of each other.
- Mean and Median: The mean and median of a standard normal distribution are both zero.
- Standard Deviation: The standard deviation of a standard normal distribution is one. This means that about 68% of values drawn from a standard normal distribution lie within one standard deviation away from the mean (between -1 and 1), about 95% lie within two standard deviations, and about 99.7% lie within three standard deviations.
- Kurtosis: The kurtosis of a standard normal distribution is 3. This means that the distribution has heavier tails and a sharper peak than the normal distribution.
- Skewness: The skewness of a standard normal distribution is zero. This means that the distribution is symmetric and does not lean in any direction.
Applications
The standard normal distribution is widely used in the field of statistics and in the natural and social sciences as a simple model for complex random variables. It is used in hypothesis testing, confidence intervals, and regression analysis. The central limit theorem states that the sum of a large number of independent and identically distributed random variables, each with finite mean and variance, will approximately follow a normal distribution, regardless of the shape of the original distribution.
Transformation to Standard Normal Distribution
Any normal distribution can be transformed into a standard normal distribution. This process is called standardization. The transformation is done by subtracting the mean and dividing by the standard deviation. The resulting z-score tells us how many standard deviations an element is from the mean.