Shape of Chi-Square Distributions

What a Chi-Square Distribution Looks Like

In “Degrees of Freedom for a Two-Way Table,” you learned that a table’s dimensions determine its degrees of freedom. Those degrees of freedom do more than label the calculation: they identify the chi-square reference distribution used to assess a test statistic. Understanding the distribution’s shape helps you see why a statistic far to the right can count as evidence against a null hypothesis.

A chi-square statistic is a sum of cell contributions \((O-E)^2/E\). Each contribution is zero or positive, so the statistic cannot be negative. Its reference distribution therefore begins at zero. But a statistic can be much larger than zero if one or more observed counts differ substantially from their expected counts. This creates a tail extending to the right.

“Right-skewed” means that most of the curve’s area is toward the lower values, while a thinner tail stretches toward larger values. It does not mean that every value is close to zero. Nor does it mean the curve is symmetric. For small degrees of freedom, the peak is near zero and the right tail is pronounced.

As the degrees of freedom increase, the distribution’s center moves to the right. The curve also spreads out in absolute units, but it becomes less skewed relative to its center: the peak is less pressed against zero, and the curve looks more nearly symmetric. Even at larger degrees of freedom, the chi-square distribution is still bounded below by zero and still has a right tail.

Center, Spread, and Peak

Three features help describe how a chi-square distribution changes with \(df\). Its mean is \(df\), its standard deviation is \(\sqrt{2df}\), and, when \(df\) is at least 2, its mode is \(df-2\). The mode is the value at the highest point of the curve. For \(df=1\), the mode is at zero. These facts are useful for comparing curves; they do not replace using the appropriate chi-square distribution when carrying out a test.

The mean and mode are not the same. A long right tail pulls the mean to the right of the peak. As \(df\) grows, the peak and center move farther from zero. Although the standard deviation increases, it increases more slowly than the mean. One way to describe this relative change is the ratio of standard deviation to mean, \(\sqrt{2/df}\): this ratio decreases as \(df\) increases.

These features describe the general shape, not the exact location of a particular test statistic in a test’s right tail. That depends on both the statistic and the distribution for the correct degrees of freedom. The next tutorial, “Chi-Square Distribution Table Versus Calculator Probabilities,” will address finding probabilities from that distribution.

Worked Example: Comparing One and Six Degrees of Freedom

How Increasing Degrees of Freedom Changes the Curve

Think of increasing \(df\) as changing the reference curve, not changing the meaning of the observed table. The degrees of freedom come from the table’s category structure, as described in the earlier tutorial on degrees of freedom. Once \(df\) is fixed, it selects the reference distribution against which the chi-square statistic is assessed.

For small \(df\), the curve’s peak is close to zero and a comparatively large portion of its right tail extends far beyond that peak. As \(df\) increases, the mean moves right by one for each additional degree of freedom. The standard deviation also increases, but the curve becomes more balanced around its center relative to its overall position.

“Less skewed” does not mean “less spread out” in absolute units. For example, the standard deviation for \(df=15\) is greater than the standard deviation for \(df=6\). The useful distinction is between absolute spread and spread relative to the center. The latter decreases as \(df\) grows, so the curve looks less strongly skewed.

At sufficiently large degrees of freedom, the chi-square curve can look approximately bell-shaped. It is still not exactly symmetric: it starts at zero, not at negative infinity, and its right tail remains. For AP Statistics, use this as a visual description rather than a reason to substitute a different distribution.

Worked Example: Comparing Table Shapes

A Statistic’s Position Depends on the Curve

A chi-square statistic should not be called “large” or “small” without considering its degrees of freedom. The same numerical value can occupy different positions on two curves. For instance, a statistic of 8 is well to the right of the mean for \(df=2\), whose mean is 2. For \(df=10\), the mean is 10, so 8 is to the left of the mean. Those comparisons do not give exact tail probabilities, but they show why \(df\) matters when interpreting a statistic.

In a chi-square test, evidence against the null hypothesis is associated with a statistic far into the right tail of the correct reference distribution. A value’s distance from zero alone is not enough to judge how unusual it is: the appropriate curve depends on the table’s degrees of freedom. A statistic must be compared with that curve, not with a different curve or a universal cutoff.

Common Mistakes and AP Exam Tip

• Calling the distribution symmetric: It may look more nearly symmetric as \(df\) increases, but it remains bounded below by zero and has a right tail.
• Confusing spread with skew: The standard deviation grows as \(df\) grows, so absolute spread increases. The curve can still become less skewed relative to its center.
• Assuming the peak is the mean: The mode marks the peak; the mean is farther right because of the right tail. For example, at \(df=6\), the mode is 4 and the mean is 6.
• Comparing a statistic without checking \(df\): A particular value’s position depends on the reference curve. Identify the correct degrees of freedom before judging whether the value lies far into the right tail.
• Saying that a larger \(df\) makes the distribution less spread out: That confuses relative skew with absolute spread. The standard deviation increases, even though the distribution becomes less skewed relative to its center.