Nonparametric Statistics

Nonparametric statistics is the field of statistical inference where no assumption is made about the probability distribution of the population under study. While most sample tests assume that the parent population(s) of the sample data is (are) normally distributed, the statistical methods of nonparametric statistics are valid for any probability distribution that the population may have (with certain exceptions).

Sampling Theory

Sampling theory is the field of statistics that is involved with the collection, analysis and interpretation of data gathered from random samples of a population under study. The application of sampling theory is concerned not only with the proper selection of observations from the population that will constitute the random sample; it also involves the use of probability theory, along with prior knowledge about the population parameters, to analyze the data from the random sample and develop conclusions from the analysis. The normal distribution, along with related probability distributions, is most heavily utilized in developing the theoretical background for sampling theory.

Normal Distribution

The normal distribution with parameters m and s , denoted as N(m,s₂) , is the continuous probability distribution with the following probability density function:

where p = 3.14159..... and e = 2.71828..... .

A random variable that has a normal distribution is called a normal random variable and is said to be normally distributed. Also known as the Gaussian distribution, the normal distribution is the cornerstone of the field of statistical inference, and definitely the most important continuous probability distribution in the study of statistics.

Discrete Probability Distribution

A discrete probability distribution is a function with a domain whose elements are the discrete values that a discrete random variable can assume, and a range whose elements are the probabilities associated with the values in the domain. The domain of a discrete probability distribution consists of the sample points of a discrete sample space. The sum of all the probability values in the range is equal to 1. The mean and variance of a discrete probability distribution are the mean and variance, respectively, of all elements in its range.

Probability Distributions

The probability distribution of a random variable is a function whose domain contains the values that the random variable can assume, and whose range is a set of values associated with the probabilities of the elements of the domain. The probability distribution of a discrete random variable is called a discrete probability distribution, while the probability distribution of a continuous random variable is called a continuous probability distribution.

Random Variables

A random variable is a function that can take on values corresponding to a sample point in a sample space. As each sample point is associated with a probability value, random variables assumes its values with a certain probability that depends on the sample point on which the value is based. A random variable that is defined over a discrete sample space has a finite or countable number of possible values and is called a discrete random variable. A random variable that is defined over a continuous sample space has an infinite set of possible values and is called a continuous random variable.

Statistical Representation of Data

The statistician is often faced with the task of summarizing large amounts of data in a compact format that yields meaningful information concerning the data. Without displaying the values for each observation taken from the population, it is possible to present the data concisely and meaningfully using certain procedures. Such procedures often involve frequency distributions or graphs of the data.