Friday, December 10, 2010

Student's t Distribution

Student's t Distribution
According to the central limit theorem, the sampling distribution of a statistic (like a sample mean) will follow a normal distribution, as long as the sample size is sufficiently large. Therefore, when we know the standard deviation of the population, we can compute a z-score, and use the normal distribution to evaluate probabilities with the sample mean.
But sample sizes are sometimes small, and often we do not know the standard deviation of the population. When either of these problems occur, statisticians rely on the distribution of the t statistic (also known as the t score), whose values are given by:
t = [ x - μ ] / [ s / sqrt( n ) ]
where x is the sample mean, μ is the population mean, s is the standard deviation of the sample, and n is the sample size. The distribution of the t statistic is called the t distribution or the Student t distribution.
Degrees of Freedom
There are actually many different t distributions. The particular form of the t distribution is determined by its degrees of freedom. The degrees of freedom refers to the number of independent observations in a set of data.
When estimating a mean score or a proportion from a single sample, the number of independent observations is equal to the sample size minus one. Hence, the distribution of the t statistic from samples of size 8 would be described by a t distribution having 8 - 1 or 7 degrees of freedom. Similarly, a t distribution having 15 degrees of freedom would be used with a sample of size 16.
For other applications, the degrees of freedom may be calculated differently. We will describe those computations as they come up.
Properties of the t Distribution
The t distribution has the following properties:
  • The mean of the distribution is equal to 0 .
  • The variance is equal to v / ( v - 2 ), where v is the degrees of freedom (see last section) and v > 2.
  • The variance is always greater than 1, although it is close to 1 when there are many degrees of freedom. With infinite degrees of freedom, the t distribution is the same as the standard normal distribution.
When to Use the t Distribution
The t distribution can be used with any statistic having a bell-shaped distribution (i.e., approximately normal). The central limit theorem states that the sampling distribution of a statistic will be normal or nearly normal, if any of the following conditions apply.
  • The population distribution is normal.
  • The sampling distribution is symmetric, unimodal, without outliers, and the sample size is 15 or less.
  • The sampling distribution is moderately skewed, unimodal, without outliers, and the sample size is between 16 and 40.
  • The sample size is greater than 40, without outliers.
The t distribution should not be used with small samples from populations that are not approximately normal.
Probability and the Student t Distribution
When a sample of size n is drawn from a population having a normal (or nearly normal) distribution, the sample mean can be transformed into a t score, using the equation presented at the beginning of this lesson. We repeat that equation below:
t = [ x - μ ] / [ s / sqrt( n ) ]
where x is the sample mean, μ is the population mean, s is the standard deviation of the sample, n is the sample size, and degrees of freedom are equal to n - 1.

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