I apologize if I didn't structure the question good but english isn't my first language so I hope you understand but I will try my best to be very clear. I wanted to know how can I use a chi squared distribution table with right tail probability to find the probabilities/quantiles. For example, I am trying to calculate '$a$' in: $P(X < a) = 0.95$ for a chi squared distribution with $5$ degrees of freedom.
If someone could explain how to find the probability above it could really help me a lot.
| d.f. | .995 | .99 | .975 | .95 | .9 | .1 | .05 | .025 | .01 |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.02 | 2.71 | 3.84 | 5.02 | 6.63 |
| 2 | 0.01 | 0.02 | 0.05 | 0.10 | 0.21 | 4.61 | 5.99 | 7.38 | 9.21 |
| 3 | 0.07 | 0.11 | 0.22 | 0.35 | 0.58 | 6.25 | 7.81 | 9.35 | 11.34 |
| 4 | 0.21 | 0.30 | 0.48 | 0.71 | 1.06 | 7.78 | 9.49 | 11.14 | 13.28 |
| 5 | 0.41 | 0.55 | 0.83 | 1.15 | 1.61 | 9.24 | 11.07 | 12.83 | 15.09 |
| 6 | 0.68 | 0.87 | 1.24 | 1.64 | 2.20 | 10.64 | 12.59 | 14.45 | 16.81 |
The table you have included works like this: The row is the number of degrees of freedom (d.f.). The column is a probability. The cell corresponding to a given row and column is the quantile for which the probability of a chi-square random variable with the respective number of degrees of freedom being greater than the quantile equals the probability specified in the column. In other words, if $\nu$ is the degrees of freedom (row), and $\alpha$ is the probability (column), and the quantile is $x$, then $$\Pr[X > x] = \alpha$$ for $$X \sim \chi^2_\nu.$$
Let us try this in practice: Suppose I want to find the $99^{\rm th}$ percentile of a chi-square distribution with $\nu = 2$ degrees of freedom. That means I want $x$ such that $$\Pr[X \le x] = 0.99,$$ where $X \sim \chi^2_2$, or equivalently, $$\Pr[X > x] = 1 - 0.99 = 0.01.$$ So $\alpha = 0.01$ is the column we must reference in the table, and $\nu = 2$ is the row. The cell in that row and column is $x = 9.21$. So this is our desired quantile, meaning that for a chi-square random variable $X$ with $2$ degrees of freedom, the probability $X$ exceeds $9.21$ is approximately $0.01$.
Based on this example, can you now compute the quantile you asked for in your question?
Conversely, suppose we wanted to know the probability $\Pr[1.24 < X \le 2.20]$ for $X \sim \chi^2_6$. How would you use the table to calculate this?
Nowadays, we don't use statistical tables to do these calculations, since computers are much more convenient and precise. For example, the quantile $9.21$ that we obtained earlier, is actually more precisely $9.2103403719761827361\ldots$.