0

For a given normal distribution, figuring out what percentage of scores fall between two bounds is straight forward. Calculate the z score and look it up on a z table. Or one can also evaluate the integral:

$\int\frac{1}{\sigma \sqrt{2\pi}}e^{\frac{-(\mu-x)^2}{2\sigma^2}}dx$

from the given bounds, where sigma is the standard deviation and mu is the mean. In fact this is a way to calculate the values on a given z-table. You can use this for sampling distributions, but it often doesn't always estimate the true confidence interval correctly. Hence t-statistics.

My question was where do t-tables come from? Is there an adjusted equation for them? Or is there a correction factor?

Jean Valjean
  • 101
  • 1
  • The t distribution has a known density that depends on the sample size (degrees of freedom are a function of the sample size). Tables are obtained in the same way as with the normal distribution. It is done by numerically integrating its own density. – Michael R. Chernick Mar 13 '18 at 18:19
  • @MichaelChernick the v parameter of t-distribution in general terms has nothing to do with sample size. It is just a specific application you are talking about. I train a lot of models where I estimate the v parameter as part of the maximum likelihood estimation, and it has nothing to do with sample size. – Cagdas Ozgenc Mar 13 '18 at 18:22
  • 1
    The degrees of freedom in the one sample case where the variance is unknown and estimated is n-1. The sample size -1. In other cases It is roughly the sample size minus the number of parameters estimated. The only case where it is a non-integer that I can think of is for the Behren's Fisher problem and it is an approximation to the "true" distribution and the parameter is estimated. In that case interpolation from the table between adjacent integers is used. – Michael R. Chernick Mar 13 '18 at 18:37
  • Wikipedia is a good resource for basic information about statistical distributions. See [its article on the t distribution](https://en.wikipedia.org/wiki/Student%27s_t-distribution). Your remarks at the end about "adjusted" and "correction" are mysterious because you haven't clearly described the application you have in mind or what needs "correcting." – whuber Mar 13 '18 at 19:04
  • I think my first comment answers the question "where do the t-tables come from". I also really object to @CagdasOzgenc that the t distribution has nothing to do with sample size. I agree with whuber that the degrees of freedom depends on the application and the Wikipedia article probably gives a detailed explanation about the t distribution. – Michael R. Chernick Mar 13 '18 at 19:13
  • So are t table values derived empirically through simulations and there actually isn't a formal definition? @Mi – Jean Valjean Mar 14 '18 at 15:07
  • @JeanValjean no. There’s a formal definition. t-distribution is a distribution same as normal distribution is one. You don’t have tables for it, the cumulative probability can be calculated directly, same as for normal distribution. In fact, outside of statistics classes, nobody uses the tables since you can just calculate the probabilities and every statistical software can do this for you. – Tim Oct 13 '20 at 06:45

0 Answers0