Chi-Square Quantiles and Tail Areas

Why Tail Conventions Matter

A chi-square table can report a critical value for a specified upper-tail probability, a lower-tail probability, or a cumulative probability. The numerical value in the table does not mean much until its column heading is interpreted correctly. A reversed tail can change a small probability into a large one, or turn a lower critical value into an upper critical value.

This tutorial focuses on reading those probabilities and quantiles. It is aligned with the SOA VEE Mathematical Statistics syllabus. Let \(Q\) have a chi-square distribution with \(\nu\) degrees of freedom, written \(Q\sim\chi^2_\nu\), where \(\nu>0\). The distribution is continuous, so the probability of \(Q\) equalling any one fixed value is zero.

Definition: The lower-tail probability at \(x\) is \(F_\nu(x)=P(Q\le x)\), and the upper-tail probability is \(P(Q>x)=1-F_\nu(x)\). The lower-tail \(p\)-quantile, denoted \(q_{p,\nu}\), satisfies \(P(Q\le q_{p,\nu})=p\). The upper-tail \(\alpha\) critical value, denoted \(c_{\alpha,\nu}\), satisfies \(P(Q>c_{\alpha,\nu})=\alpha\). Thus \(c_{\alpha,\nu}=q_{1-\alpha,\nu}\).

The subscript \(\nu\) identifies the degrees of freedom. The two probability conventions are complementary: an upper-tail area of \(0.05\) corresponds to a lower-tail area of \(0.95\) at the same cutoff. Consequently, a table column headed “upper-tail probability \(0.05\)” gives the \(0.95\) lower-tail quantile, not the \(0.05\) lower-tail quantile.

Reading and Translating a Chi-Square Table

A table entry is determined by both the degrees-of-freedom row and the probability heading. Many common chi-square tables list upper-tail probabilities across the columns. For such a table, if the row is \(\nu=8\) and the column is \(\alpha=0.10\), the entry \(c_{0.10,8}\) is the value with 10% of the distribution to its right. It has 90% to its left.

Other tables may label columns with lower-tail probabilities or cumulative probabilities. Check the heading before using an entry. If the table gives a value for an upper-tail area \(\alpha\), then:

P(Q>c_{\alpha,\nu})=\alpha,\qquad P(Q\le c_{\alpha,\nu})=1-\alpha.

If the requested event is between two cutoffs, subtract the lower-tail probability at the lower cutoff from the lower-tail probability at the upper cutoff. Equivalently, subtract the upper-tail probability at the upper cutoff from the upper-tail probability at the lower cutoff. This works because the event between the cutoffs is the part of the distribution accumulated between their CDF values.

P(a<Q\le b)=F_\nu(b)-F_\nu(a) =P(Q>a)-P(Q>b),\qquad a<b.

For a continuous distribution, using \(<\) or \(\le\) at either endpoint gives the same probability. In computations, however, keep the endpoints in increasing order and use a consistent tail convention.

Worked Example: Converting an Upper-Tail Table into an Interval Probability

A hypothetical quality-monitoring procedure uses a reference variable \(Q\sim\chi^2_{10}\). An upper-tail table gives the approximate critical values \(3.940\) for upper-tail probability \(0.95\) and \(18.307\) for upper-tail probability \(0.05\). Find the probability that \(Q\) falls between these two displayed cutoffs.

Method. Translate each table heading into the corresponding lower-tail probability. Since the table is an upper-tail table, the lower-tail probability at a cutoff is one minus its upper-tail probability.

At the lower displayed cutoff, the upper-tail probability is approximately \(0.95\), so its lower-tail probability is approximately \(1-0.95=0.05\). At the upper displayed cutoff, the upper-tail probability is approximately \(0.05\), so its lower-tail probability is approximately \(1-0.05=0.95\).

Subtract the lower-tail probability at the lower endpoint from that at the upper endpoint: \(0.95-0.05=0.90\). Therefore, using the displayed table values, the probability between the cutoffs is approximately \(0.9000\), or 90.00% to four decimal places.

Check. The two excluded tails are each approximately \(0.05\), so the central probability is approximately \(1-0.05-0.05=0.90\), as obtained by subtraction. The table entries are rounded, so the probabilities at the displayed endpoints are approximate, not exact. The unrounded quantiles have exactly the stated tail areas; the displayed endpoints yield an interval probability approximately \(0.9000\).

Common mistake avoided. Treating \(18.307\) as a lower-tail \(0.05\) quantile would put only about 5% to its left. For this upper-tail table entry, about 5% is to its right and about 95% to its left.

Why Complements and Quantiles Work

For completeness, the chi-square density and its CDF can be written using the gamma function. This gives a precise meaning to a chi-square quantile even when a table is used to approximate it. For \(x>0\),

f_\nu(x)=\frac{x^{\nu/2-1}e^{-x/2}}{2^{\nu/2}\Gamma(\nu/2)}, \qquad F_\nu(x)=\int_0^x f_\nu(t)\,dt.

The density is nonnegative and integrates to one. The CDF is the accumulated area under that density from zero to \(x\); the area to the right is the remaining area. This establishes the complement relationship. A quantile is the cutoff that makes the accumulated area equal the requested lower-tail probability.

Start with the table heading.
Identify whether its probability is lower-tail, upper-tail, or cumulative. Do not infer the convention from the number of the critical value.

Write the probability statement.
For an upper-tail heading \(\alpha\), write \(P(Q>c_{\alpha,\nu})=\alpha\). This makes the direction explicit.

Convert only if needed.
Use \(P(Q\le c_{\alpha,\nu})=1-\alpha\) for a lower-tail statement, or subtract CDF values for an interval.

The CDF is increasing, so a larger lower-tail probability corresponds to a larger quantile. The upper-tail convention runs in the opposite direction: a smaller upper-tail probability corresponds to a larger critical value. These observations are useful checks when comparing entries in a table.

Finding a Quantile by Solving a Probability Equation

Most chi-square quantiles are obtained from a table or statistical software. For certain degrees of freedom, the CDF has a simple form that allows a quantile to be found directly. For example, when \(\nu=2\), \(Q\) has density \(f_2(x)=\tfrac12 e^{-x/2}\) for \(x>0\). Integrating from zero to \(x\) gives \(F_2(x)=1-e^{-x/2}\). A lower-tail quantile is found by setting this expression equal to the desired probability and solving for \(x\).

Worked Example: Finding a 90th Percentile from Its CDF

A hypothetical screening calculation uses \(Q\sim\chi^2_2\), and a decision threshold is to be set at the 90th percentile. Find the threshold, and state its upper-tail probability. Report the threshold to four decimal places.

Method. The 90th percentile is the lower-tail quantile \(q_{0.90,2}\), so solve \(F_2(x)=0.90\). For two degrees of freedom, \(F_2(x)=1-e^{-x/2}\).

Set the CDF equal to \(0.90\): \(1-e^{-x/2}=0.90\). Rearranging gives \(e^{-x/2}=0.10\). Taking natural logarithms gives \(-x/2=\ln(0.10)\), hence \(x=-2\ln(0.10)\).

Using \(\ln(0.10)=-2.302585093\), the threshold is \(x=4.605170186\), or \(4.6052\) to four decimal places. The upper-tail probability is \(1-0.90=0.10\).

Check. Substitution gives \(F_2(4.605170186)=1-e^{-2.302585093}=1-0.10=0.90\). The remaining area is \(0.10\), confirming that this is an upper-tail \(0.10\) critical value.

Common mistake avoided. A “90th percentile” means 90% of the distribution is at or below the cutoff, not 90% above it. Solving \(P(Q>x)=0.90\) would instead find the lower 10th percentile.

Computing Tail Areas from a Known Cutoff

A table may provide a critical value, while a problem asks for the probability at a different cutoff. In general, this requires evaluating the CDF or using a probability calculator. For even degrees of freedom, the CDF can be written as a finite sum, which is useful for hand calculations and checks. If \(\nu=2m\), where \(m\) is a positive integer, then

F_{2m}(x)=1-e^{-x/2}\sum_{k=0}^{m-1}\frac{(x/2)^k}{k!},\qquad x\ge0.

The upper-tail probability is the term subtracted from one. In the next example, the degrees of freedom are four, so \(m=2\); only the terms for \(k=0\) and \(k=1\) are needed.

Worked Example: Calculating Both Tails at a Given Value

A hypothetical diagnostic reference variable has distribution \(Q\sim\chi^2_4\). Calculate the lower- and upper-tail probabilities at \(x=6\). Give probabilities to four decimal places.

Method. With \(\nu=4=2m\), \(m=2\). The upper-tail formula from the finite sum is \(P(Q>x)=e^{-x/2}\left(1+x/2\right)\). Substitute \(x=6\): \(x/2=3\), so \(P(Q>6)=e^{-3}(1+3)=4e^{-3}\).

Using \(e^{-3}=0.0497870684\), the upper-tail probability is \(4(0.0497870684)=0.1991482736\), which rounds to \(0.1991\).

The lower-tail probability is its complement: \(P(Q\le6)=1-0.1991482736=0.8008517264\), which rounds to \(0.8009\).

Check. Before rounding, \(0.1991482736+0.8008517264=1.0000000000\). The lower-tail probability is greater than one-half at \(x=6\), consistent with a cutoff lying to the right of much of the distribution.

Common mistake avoided. The expression \(e^{-3}(1+3)\) is the upper-tail probability, not the CDF. Taking its complement is necessary to report the lower-tail probability.

Practical Table Checks and Exam Traps

A careful solution can often be checked without recomputing the entire probability. Use the degrees-of-freedom row, the table’s tail convention, and the direction of the cutoff together.

Check the row. A correct probability from the wrong degrees-of-freedom row is still the wrong answer.
Check the tail label. An upper-tail area of \(0.05\) and a lower-tail area of \(0.05\) do not describe the same cutoff.
Check the order of cutoffs. If \(a<b\), then \(F_\nu(a)\le F_\nu(b)\), so \(P(a<Q\le b)\) must be nonnegative.
Check the size of the quantile. For a fixed \(\nu\), increasing a lower-tail probability increases its quantile. Increasing an upper-tail probability decreases its critical value.
Respect rounding. A printed table value is an approximation. Do not claim an exact tail probability at a rounded cutoff when the table only provides a rounded quantile.

If a table gives only selected probability columns, a requested tail probability may fall between them. The table then brackets the quantile rather than identifying it exactly. For example, if the requested lower-tail probability lies between two listed lower-tail probabilities, the requested quantile lies between the corresponding two critical values. Do not report an interpolated value as exact; interpolation is an approximation unless a specified method justifies it.

Key takeaway: First identify whether a table column is a lower-tail or upper-tail probability. Then write the probability statement, convert tails with \(1-p\) when needed, and treat rounded critical values as approximations.

Check Your Understanding

For table-based questions, interpret the stated table convention exactly as given; reported chi-square critical values are approximate when rounded.

Let \(Q\sim\chi^2_2\). Find \(P(Q>4)\) and \(P(Q\le4)\), each to four decimal places. (Answer: \(P(Q>4)=0.1353\), \(P(Q\le4)=0.8647\).)
An upper-tail chi-square table gives \(5.989\) for \(\nu=4\) and upper-tail probability \(0.20\). State the approximate lower-tail probability at this cutoff. (Answer: \(P(Q\le5.989)\approx0.8000\).)
A table gives the approximate \(\chi^2_6\) cutoffs \(1.237\) and \(14.449\), corresponding respectively to lower-tail probabilities \(0.025\) and \(0.975\). Find the approximate probability between them. (Answer: approximately \(0.9500\).)
Find the upper-tail \(0.15\) critical value for \(Q\sim\chi^2_2\), to four decimal places. (Answer: \(c_{0.15,2}=-2\ln(0.15)\approx3.7942\).)
An upper-tail table lists the approximate \(\chi^2_{10}\) critical value \(23.209\) under the \(0.01\) column. State both tail probabilities at this cutoff. (Answer: \(P(Q>23.209)\approx0.0100\) and \(P(Q\le23.209)\approx0.9900\).)

What You'll Learn