t Distribution

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Student t Distribution

References: Dr. Mikel J. Harry, Juran's Quality Control Handbook


The t distribution is a symmetric, bell shaped distribution that resembles the standardized normal (Z) distribution, but with more area in its tails. That is, with more variability than the Z distribution. The t-test is used to test population means when small samples are involved.

The t distribution (aka, Student's t-distribution) is a probability distribution that is used to estimate population parameters when the sample size is small and/or when the population variance is unknown.

The t distribution is a family of distributions that look almost identical to the normal distribution curve, only a bit shorter and wider. The t distribution is primarily used to calculate probabilities for small sample sizes. The larger the sample size, the more the t distribution looks like the normal distribution.

In fact, for sample sizes larger than 25, the distribution is almost identical to the normal distribution. When you look at distribution tables, you’ll see that you need to know the degrees of freedom“df.” This is just the sample size minus one.


t Distributiont Distribution


A. Vertical axis - Scale to measure the probability at different values of the t statistic.

B. Horizontal axis - Scale of measure of the t statistic.

C. Area under the curve representing the probability that the t-static will take on specific values.

D. Curve of the t Distribution for various degrees of freedom. The total area under each of the curves is equal to one (1). The t distribution has the following major properties:

– It is centered around zero and is symmetric about its mean.

– Its variance is greater than 1, but as the sample size (n) increases, the variance approaches 1.

– The t distribution has less area in the middle and more in the tails than the Z distribution.

– The t distribution approaches the Z distribution as the number of degrees of freedom (df) increases. The number of degrees of freedom equals the sample size (n), minus the number of population parameters which must be estimated from sample observations. In this case we must estimate mu, therefore df = n – 1.

Major Considerations

The t distribution is used when sample size (n) is small and the population’s standard deviation is unknown. The parent population from which the sample is taken follows a normal distribution with mean.

Application Cookbook

1. Use Excel functions to calculate the t distribution and the inverse of the t distribution. For example, there is a 0.05 probability that a sample with 10 degrees of freedom would have t = 1.812 or smaller.

Note: When using the function TINV the probability is multiplied by 2 since Excel returns the value associated to a two-tail curve.

2. In Minitab use the following menu to generate a t distribution CALC>PROBABILITY DISTRIBUTIONS> T

3. Alternatively, the tables printed at the end of most statistics can be used.

From t Distribution to Six Sigma Tools.

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