Finding Chi-Square Quantiles from Tables

What a Chi-Square Quantile Tells You

A chi-square table can answer two complementary questions. Given a cutoff, it can help find the probability in a tail; given a tail probability, it can help find the cutoff that leaves that probability beyond it. This tutorial focuses on the second task: finding a chi-square quantile from a table.

The table lookup depends on two pieces of information: the degrees of freedom and the requested tail area. Before reading a value, check which tail the table uses. Many chi-square tables are organized by right-tail areas, but a probability statement may instead describe the area to the left of a cutoff or the area between two cutoffs.

Definition: Let $X$ have a chi-square distribution with $df$ degrees of freedom. The right-tail chi-square quantile $\chi^2_{\alpha,df}$, for $0<\alpha<1$, is the cutoff satisfying $$ P(X>\chi^2_{\alpha,df})=\alpha. $$ Because a chi-square variable is continuous, using $X\geq \chi^2_{\alpha,df}$ gives the same probability. The probability to the left of this cutoff is $1-\alpha$.

For example, if a table gives $\chi^2_{0.05,8}=15.507$, then a chi-square variable with 8 degrees of freedom has probability 0.05 to the right of 15.507 and probability 0.95 to its left. The subscript 0.05 in this convention means right-tail area, not left-tail area.

A lookup assumes that the random variable of interest follows a chi-square distribution with the stated degrees of freedom. The table does not determine whether that model is appropriate; it supplies a cutoff for that model. Degrees of freedom select the row, and the area convention identifies the column.

Reading a Right-Tail Table

A common table has degrees of freedom in the first column and right-tail probabilities across the top. To find a quantile, locate the row for the required $df$, then move across to the column for the requested $\alpha$. The entry where the row and column meet is $\chi^2_{\alpha,df}$.

Information in the question	Conversion for a right-tail table	Lookup target
Right-tail area $\alpha$	No conversion	Column $\alpha$
Left-tail area $p=P(X\leq c)$	Right-tail area $1-p$	Column $1-p$
Central area $1-\alpha$, split equally between tails	Each tail has area $\alpha/2$	Lower cutoff: column $1-\alpha/2$; upper cutoff: column $\alpha/2$

The table heading matters. Some tables display left-tail cumulative probabilities instead of right-tail probabilities. The same cutoff can appear under different column labels in those tables. Read the heading and any note above the table rather than assuming that a column labelled with a probability refers to a particular tail.

Worked Example: Finding a Cutoff for a Given Right-Tail Area

Suppose $X\sim\chi^2_{8}$. Find the value $c$ such that $P(X>c)=0.05$.

The probability is already stated as a right-tail area, so no conversion is needed. In a table headed by right-tail probabilities, use the row $df=8$ and the column $0.05$. Their intersection is 15.507.

c=\chi^2_{0.05,8}=15.507

Thus, under the stated chi-square model, the probability that $X$ exceeds 15.507 is 0.05. The probability that $X$ is below this cutoff is $1-0.05=0.95$. The cutoff is in chi-square units; it is not itself a probability.

Converting a Left-Tail Probability

A question may specify a percentile, such as the value with 95% of the distribution below it. That is a left-tail statement. To use a right-tail table, subtract the left-tail probability from 1. The conversion follows because the areas on the two sides of a cutoff add to 1.

P(X\leq c)=p \quad\Longrightarrow\quad P(X>c)=1-p

This conversion is especially important for chi-square distributions. Unlike the standard normal distribution, a chi-square distribution is not symmetric about its center. One cannot find the opposite-tail cutoff by reflecting a value across a center; instead, convert the probability and look up the appropriate column.

Worked Example: Converting a Left-Tail Percentile

Let $X\sim\chi^2_{10}$. Find the cutoff $c$ with 95% of the distribution to its left, so that $P(X\leq c)=0.95$.

A right-tail table needs the area to the right of $c$. The two tail areas sum to 1, so the required right-tail area is $1-0.95=0.05$. Use the $df=10$ row and the right-tail $0.05$ column; the table entry is 18.307.

P(X\leq c)=0.95 \quad\Longrightarrow\quad P(X>c)=1-0.95=0.05 \quad\Longrightarrow\quad c=\chi^2_{0.05,10}=18.307

The requested 95th percentile is therefore about 18.307. It would be an error to use the right-tail $0.95$ column: that column would find a cutoff with only 5% of the distribution to its left.

Finding Endpoints for a Central Area

A central interval contains a specified probability with equal areas left outside its lower endpoint and right outside its upper endpoint. For a central area of $1-\alpha$, the total area outside the interval is $\alpha$, so each tail has area $\alpha/2$. A right-tail table then gives the lower endpoint from the $1-\alpha/2$ column and the upper endpoint from the $\alpha/2$ column.

Key result: For a continuous chi-square variable, an equal-tail central interval containing probability $1-\alpha$ has endpoints $$ \chi^2_{1-\alpha/2,df} \quad\text{and}\quad \chi^2_{\alpha/2,df}. $$ The first is the lower endpoint and the second is the upper endpoint under the right-tail convention.

The order of these endpoints can seem surprising. A larger right-tail area leaves less area to the left, so its cutoff is farther down the distribution. Thus the right-tail $1-\alpha/2$ value is the lower endpoint, while the right-tail $\alpha/2$ value is the upper endpoint.

Worked Example: A Central 90% Interval

Suppose $X\sim\chi^2_{12}$. Find an equal-tail central interval containing 90% of its probability.

Here $1-\alpha=0.90$, so $\alpha=0.10$. Split the remaining probability equally: each tail has area $0.10/2=0.05$. The lower endpoint is found using right-tail area $1-0.05=0.95$; the upper endpoint uses right-tail area 0.05.

In the $df=12$ row, the right-tail $0.95$ entry is 5.226, and the right-tail $0.05$ entry is 21.026. Therefore the interval is approximately 5.226 to 21.026.

P(5.226\leq X\leq 21.026) =1-0.05-0.05 =0.90

The result is an interval for values of $X$, not a statement that there is a 90% probability that a fixed parameter lies in this interval. The equal-tail construction refers to probability under the specified chi-square distribution.

When the Requested Column Is Missing

Printed tables usually show only selected tail areas. If the exact $\alpha$ is absent, do not silently substitute a nearby column and present its entry as exact. Instead, use the neighboring columns to bound the requested quantile, or state that a table with finer columns is needed for a more precise numerical value.

For a fixed $df$, the right-tail probability decreases as the cutoff increases. Consequently, a smaller right-tail area corresponds to a larger quantile. This ordering lets us bound a requested value from the neighboring entries without interpolation.

Worked Example: Bounding a Quantile Between Columns

Suppose $X\sim\chi^2_{8}$, and a table does not include right-tail area 0.04. Bound the cutoff $c$ satisfying $P(X>c)=0.04$, using the available columns 0.05 and 0.025.

For $df=8$, the table gives $\chi^2_{0.05,8}=15.507$ and $\chi^2_{0.025,8}=17.535$. Since $0.025<0.04<0.05$, the cutoff for area 0.04 must lie between those two cutoffs: the smaller tail area corresponds to the larger cutoff.

$$ 15.507 < c < 17.535 $$

The table supports this bound, not an exact value for $c$. Linear interpolation between the entries may provide an approximation in some settings, but it is not an exact table lookup and is not justified merely by knowing the two endpoints. When higher precision is required, a more detailed table or an appropriate computational method is needed.

Why the Lookup Works

The table’s quantiles are well-defined because the chi-square distribution has a continuous density that is positive for $x>0$. As a result, increasing a cutoff strictly decreases the probability to its right. At the same time, that right-tail probability changes continuously with the cutoff. These properties mean that for any specified tail probability strictly between 0 and 1, there is one corresponding cutoff.

The complement rule also explains the conversion between table conventions. At any cutoff $c$, the events $X\leq c$ and $X>c$ do not overlap and together cover all possible values. Therefore their probabilities add to 1. Continuity means there is no point probability at $c$, so the same complement statement applies whether the left event is written $X<c$ or $X\leq c$.

These facts justify the practical workflow: determine which probability the question gives, convert it to the table’s tail convention if needed, and then find the row and column. They also explain why the direction of a bound in a missing-column problem depends on the direction of the tail probability.

Common Mistakes

Reading a right-tail label as a percentile. A right-tail column of 0.05 means 5% is above the cutoff, not 5% below it. For a 95th percentile, use right-tail area 0.05.
Forgetting the degrees of freedom. The same tail area gives different cutoffs in different $df$ rows. Confirm the row before reading across.
Using symmetry to switch tails. Chi-square curves are not symmetric. Convert probabilities with $1-p$ and read the relevant column; do not reflect a cutoff around a center.
Reversing central-interval endpoints. With a right-tail table, the column $1-\alpha/2$ gives the lower endpoint and $\alpha/2$ gives the upper endpoint.
Reporting an unsupported precision. A printed table entry is rounded, and a missing column does not supply an exact quantile. Preserve the table’s precision or report a bound.

Key takeaway: First identify the table’s tail convention. For a right-tail table, look up $\alpha$ directly, convert a left-tail probability $p$ to $1-p$, and split the excluded area equally when finding an equal-tail central interval.

Check Your Understanding

For each question, assume the stated variable follows a chi-square distribution with the stated degrees of freedom. Use a right-tail table unless the question says otherwise.

What row and column would you use to find the cutoff with right-tail probability 0.10 for $df=6$?
A cutoff has 0.92 of the distribution to its left. What right-tail area should you look up?
For an equal-tail central interval containing 95% of the distribution, what right-tail areas determine the lower and upper endpoints?
A table gives right-tail $0.05$ and $0.025$ cutoffs for a fixed $df$, but not the $0.04$ cutoff. Which of the two listed cutoffs is smaller, and why?
Why is it incorrect to find a chi-square cutoff for a 95th percentile by using the right-tail $0.95$ column?

Information in the question	Conversion for a right-tail table	Lookup target
Right-tail area \(\alpha\)	No conversion	Column \(\alpha\)
Left-tail area \(p=P(X\leq c)\)	Right-tail area \(1-p\)	Column \(1-p\)
Central area \(1-\alpha\), split equally between tails	Each tail has area \(\alpha/2\)	Lower cutoff: column \(1-\alpha/2\); upper cutoff: column \(\alpha/2\)