Distributions

A measurable single valued function of the observations in a sample is called a statistic. If x1, x2,...xn  are the given n number of observations then the given sample can be considered as a measure of a single valued function and refers to a statistic distribution. So A measure obtained from a sample is called a sample statistic.

Example:

Sample mean evaluated from the sample is a sample statistic. It is denoted by x.

Sample Distribution

Sample statistic is a random variable. As every random variable has a probability distribution, sample statistic also has a probability distribution. The probability distribution of a sample statistic is called the sampling distribution of that statistic.

Example:

Sample mean is a sample statistic and the distribution of the sample mean is a sampling distribution.

Sampling distributions plays a very important role in the study of statistical inference Standard Error

Standard deviation of a sampling distribution is called standard error.

Standard error of the sample mean x is $\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and n is the sample size.

Types of Distributions

There are mainly two types of distributions
(1) Discrete distributions

(2) Continuous distributions
Most of the sampling distributions are continuous. Following are the important sampling distributions

1. Normal Distribution

When the population has Normal distribution or when the sample is large, the sample statistic will have a standard Normal distribution. We denote this statistic by z.

Uses of sampling distribution of z

  • To test the given population mean
  • To test the significance of difference between two population means
  • To test the given proportion
  • To test the difference between two population proportions
  • To test the given standard deviation
  • To test the difference between two population standard deviations

2. Chi Square Distribution

Following are the cases where we get a chi square distribution

  • If z follows a standard normal distribution, then z2 follows chi square distribution with one degree of freedom
  • Let s and $\sigma$ be the standard deviations of sample and population respectively. Let n be the size of the sample, then $\frac{ns^{2}}{\sigma^{2}}$ follows a chi square distribution

Uses of chi square distribution

  • To test the given population when sample is small
  • To test the goodness of fit between the observed and expected frequencies
  • To test the independence of two attributes
  • To test the homogeneity of data.

3. Students t Distribution

Let x and s be the mean and standard deviation of a sample drawn from a normal population with size n (n is small) then -μ)/s/√n-1 follows a t distribution with n - 1 degrees of freedom.

Uses of students t distribution

  • To test the given population mean when the sample is small
  • To test whether the two populations have same mean when the samples are small.
  • To test whether there is a difference in the observations of the two dependent samples
  • To test the significance of population correlation coefficient.

4. F Distribution

If n1 and n2 are the sized and s12 and s22 are the variances of the two independently drawn samples from a normal population having common standard deviation σ, then (n1s12/(n1-1)/(n2s22/(n2-1) follows F distribution with n1 - 1 , n2 - 1 degrees of freedom.

Uses of F distribution

  • To test the equality of variances of two populations when samples are small.
  • To test the equality of means of three or more populations.

Statistics Distribution

Select a number of random samples of definite size from a given population and calculate some statistic, from each of these samples. We find they are different in value. So sample statistic is a variable and therefore has a distribution. The distribution this obtained for a sample statistic is called a sampling distribution. It is a probability distribution. These distributions play an important role in the study of inferences. There are different types of distribution such as t-distribution, chi square distribution and so on.