How does one find the p-value for the T-statistic? Got kinda lost there…

Hey Jamie!

This is a very interesting question (if I understood it correctly).

I believe your question is: ‘it is clear with the **z-table**, but it doesn’t work with the **t-table**‘.

Correct?

If so, think about this:

The z-table is very detailed – most values are there. The rows and columns work together to describe all possible cells. Note that it is not REALLY needed to be a table. You can just have one very, very long row (or column) of values.

**As ONE column (300 rows):**

z = 0.01 -> value

z = 0.02 -> value

z = 0.03 -> value

…

z = 2.99 -> value

z = 3.00 -> value

**As ONE row (300 columns):**

z = 0.01 ; 0.02 ; 0.03 ; … ; 2.99 ; 3.00

value = a ; b ; c ; … ; d ; e

Alternatively, we can reverse those (it won’t make a difference). For my purposes I need this representation:

**value = a ; b ; c ; … ; d ; e
z = 0.01 ; 0.02 ; 0.03 ; … ; 2.99 ; 3.00**

(remember that!)

We have a z-**TABLE**, which is usually (30×10) so it takes less space and can be ‘carried arround’ (before computer were good enough).

***

The t-table is not as detailed.

Why?

Because there is **another dimension **– the degrees of freedom. But a table can only have two dimensions. For the t-table, they are: degrees of freedom and level significance.

Looking at the (first 5 rows of the) t-table:

We realize that it is actually (from above) …

**value = a ; b ; c ; … ; d ; e
t = 0.01 ; 0.02 ; 0.03 ; … ; 2.99 ; 3.00**

Right?

But this is done for **EACH degree of freedom.**

So, We don’t have **one row, **but have as many rows as there are degrees of freedom. And the T values are different for **each degree of freedom.** We have 30 degrees of freedom in most tables so we have 30 rows.

Usually we also include the z-row, because after df=30 for t, we can assume normality:

With this last row, we are basically summarizing t>30 (infinitely many rows).

***

Okay, but what about the columns?

Didn’t we say that when there is a single row, we need 300 columns?

Well, we already have 30 rows for the degrees of freedom. And we can’t pull of the ‘trick’ from the z-table to make it 30×10, instead it will be 30×300. Which is a table which we cannot really carry around or investigate manually (doesn’t make sense when we have p-value calculators).

There are several solutions to this problem:

1) Make 30 t-tables. T-table for 1 degree of freedom, T-table for 2 df, 3df, etc., where we pull of the z-table trick, so we get 30 t-tables, each 30×10.

2) Make a 3-dimensional table. 30x30x10.

That’s cool, but not very useful per se.

3) Cut most of the table and keep only the most improtant values. The result?

**The t-table you know.**

Obviously that’s the best decision if we want a **paper copy.**

***

After this long explanation we can answer your question:

The information you are looking for **is NOT in the t-table**, because it was intentionally cut out.

The solution we’ve opted for is an online p-value calculator. You can find a pdf with detailed instructions with this lecture: https://www.udemy.com/the-data-science-course-complete-data-science-bootcamp/learn/v4/t/lecture/10764560?start=15

In practice, usually, you’d have a software calculate it for you!

Hope this helps!

Best,

The 365 Team