What is a 'T Distribution'

A T distribution is a type of probability distribution that is appropriate for estimating population parameters for small sample sizes and unknown population variances. A T distribution resembles a normal distribution, but it differs from the normal distribution by its degrees of freedom. A T distribution has a smaller number of independent observations. However, the higher the degrees of freedom, the closer that distribution will resemble a standard normal distribution with a mean of 0, and a standard deviation of 1. The T distribution is also known as the "Student's T Distribution."

BREAKING DOWN 'T Distribution'

When a sample of a population is sufficiently large, the sample will follow a normal distribution. A z-score can be calculated with a known population standard deviation. But when a sample is not large enough, or the population standard deviation is unknown, a t-score must be calculated as a basis for the T distribution.

The t-score (or t-statistic or t-value) formula is the sample mean minus the population mean divided by the quotient of the standard deviation of the sample and the square root of the sample size.

Application Example of a T Distribution

The Centers for Disease Control and Prevention (CDC) regularly conducts studies on the prevalence of diseases or chronic conditions. The methodology is consistently applied for each study. According to the current methodology, "when sample sizes ... are less than 50, as they are in some of the subgroup tables, the standard error may be large relative to the prevalence value, leading to a wide 95% confidence interval," in which case "it is appropriate to use values of the t-distribution to compute confidence intervals." A confidence interval is the probability that a value will fall between an upper and lower bound of a probability distribution. For example, the CDC calculated that at a 95% confidence interval 6.5% to 11.6% of children in New York suffered from asthma in 2014. The sample size was only 752 children, which made use of the T distribution suitable.

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