# Finding $p$ value corresponding to $t$ value, using the table with critical $t$ values

Assuming a table with a row per degrees of freedom and a column per upper tail probability
Two sided
##### $p$ value is the probability of finding the observed $t$ value or more extreme, given that the null hypothesis is true.
If you have found a positive $t$ value ($t \geq 0$):
1. Find the row with the appropriate degrees of freedom (df)
2. Search for the two $t$ values in this row, that enclose the $t$ value you found
3. Find the upper tail probabilities corresponding to these two $t$ values. You can find them at the top of each column
4. Multiply each of the upper tail probabilities by 2; the two sided $p$ value corresponding to the $t$ value you found is between these two values
If you have found a negative $t$ value ($t < 0$):
1. Multiply the $t$ value you found by -1 (since the table only works with positive $t$ values), resulting in a positive value $t_{pos}$
2. Find the row with the appropriate degrees of freedom (df)
3. Search for the two $t$ values in this row, that enclose the positive $t_{pos}$ value you found
4. Find the upper tail probabilities corresponding to these two $t$ values. You can find them at the top of each column
5. Multiply each of the upper tail probabilities by 2; the two sided $p$ value corresponding to the negative $t$ value you found is between these two values
Right sided
##### $p$ value is the probability of finding the observed $t$ value or larger, given that the null hypothesis is true.
If you have found a positive $t$ value ($t \geq 0$):
1. Find the row with the appropriate degrees of freedom (df)
2. Search for the two $t$ values in this row, that enclose the $t$ value you found
3. Find the upper tail probabilities corresponding to these two $t$ values. You can find them at the top of each column. The right sided $p$ value corresponding to the $t$ value you found is between these two values
If you have found a negative $t$ value ($t < 0$):
1. Multiply the $t$ value you found by -1 (since the table only works with positive $t$ values), resulting in a positive value $t_{pos}$
2. Find the row with the appropriate degrees of freedom (df)
3. Search for the two $t$ values in this row, that enclose the positive $t_{pos}$ value you found
4. Find the upper tail probabilities corresponding to these two $t$ values. You can find them at the top of each column
5. The upper tail probabilities for the positive $t$ values are the same as the lower tail probabilities for the negative $t$ values. Since you want the upper tail probability for the negative $t$ value you found (you are testing right sided), compute 1 minus the upper tail probabilities you found for the positive $t$ values that enclose $t_{pos}$. The right sided $p$ value corresponding to the negative $t$ value you found is between these two values
Left sided
##### $p$ value is the probability of finding the observed $t$ value or smaller, given that the null hypothesis is true.
If you have found a positive $t$ value ($t \geq 0$):
1. Find the row with the appropriate degrees of freedom (df)
2. Search for the two $t$ values in this row, that enclose the $t$ value you found
3. Find the upper tail probabilities corresponding to these two $t$ values. You can find them at the top of each column
4. Compute 1 - uppertail probability1 and 1 - uppertail probability2; the left sided $p$ value corresponding to the $t$ value you found is between these two values
If you have found a negative $t$ value ($t < 0$):
1. Multiply the $t$ value you found by -1 (since the table only works with positive $t$ values), resulting in a positive value $t_{pos}$
2. Find the row with the appropriate degrees of freedom (df)
3. Search for the two $t$ values in this row, that enclose the positive $t_{pos}$ value you found
4. Find the upper tail probabilities corresponding to these two $t$ values. You can find them at the top of each column. The left sided $p$ value corresponding to the negative $t$ value you found is between these two values