T Distribution

Definition:

The t-distribution, also known as Student’s t-distribution, is a probability distribution that describes the sampling distribution of the t-statistic, which is a standardized measure of the difference between a sample mean and the population mean, divided by the sample standard deviation.

Key Properties:

• Central tendency: The t-distribution has a mean of 0.
• Symmetry: The t-distribution is symmetrical about 0.
• Unimodality: The t-distribution is unimodal, meaning it has only one mode at 0.
• Skewness: The t-distribution has a skewness of 0.
• Kurtosis: The t-distribution has a kurtosis of 0.
• Degrees of freedom: The t-distribution is defined by a number of degrees of freedom, which is the number of observations in the sample minus 1.
• Tail behavior: The t-distribution has heavy tails, meaning that it has a high probability of extreme values.

Uses:

• T-tests: The t-distribution is used to perform t-tests, which are used to compare the means of two or more groups.
• Confidence intervals: The t-distribution is used to calculate confidence intervals for the population mean.
• Hypothesis testing: The t-distribution is used to test hypotheses about the population mean.

Formula:

The t-distribution is given by the following formula:

t = (x - μ) / sqrt(s/n)

where:

• t is the t-statistic
• x is the sample mean
• μ is the population mean
• s is the sample standard deviation
• n is the number of observations

Note:

The t-distribution is a special case of the Student’s t-distribution, which is a more general family of probability distributions. The Student’s t-distribution is used when the sample size is large and the population standard deviation is unknown.