Right-Tail Probabilities and Complements

Why Right-Tail Probabilities Need Care

A question about a normal variable may ask for the probability of an observation being greater than a specified cutoff: for example, the probability that a hypothetical package weighs more than a target amount, or that a waiting time exceeds a stated number of minutes. This is a right-tail probability: it is the area under the normal density to the right of the cutoff.

The complement rule gives a direct way to calculate this area from a cumulative left-tail probability. The key is to apply it to the correct standardized cutoff. In particular, a cutoff below the mean has a negative z score, but the right-tail probability above it is large, not small. Keeping track of which side of the cutoff is requested prevents a common reversal.

The Complement Rule for a Normal Variable

Suppose a variable follows the normal model \(X\sim N(\mu,\sigma^2)\), where \(\mu\) is its mean and \(\sigma\) is its standard deviation. The model should be appropriate for the quantity being studied, and the values of \(\mu\) and \(\sigma\) should be specified. For a cutoff \(c\), standardize it using \(z=(c-\mu)/\sigma\). The resulting standard normal variable is \(Z=(X-\mu)/\sigma\), with \(Z\sim N(0,1)\).

Write \(\Phi(z)=P(Z\leq z)\) for the cumulative standard normal probability to the left of \(z\). Standardizing the event \(X>c\) gives \(Z>z\). Since the events \(Z\leq z\) and \(Z>z\) are disjoint and together cover all possible values of \(Z\), their probabilities sum to one.

This is the same formula whether \(c\) is above or below the mean. What changes is the value of \(z\) and, consequently, the size of \(\Phi(z)\). A cutoff above the mean has \(z>0\), so its right tail is less than one-half. A cutoff below the mean has \(z<0\), so its right tail is greater than one-half. At the mean, \(z=0\), and symmetry gives a right-tail probability of one-half.

Worked Examples

Using a Mean-to-Z Table

Some standard normal tables report the area between zero and a z score rather than the cumulative area to its left. Let \(A(a)=P(0<Z<a)\) for \(a\geq 0\). By symmetry of the standard normal distribution, a table entry for a positive value \(a\) also gives the area between \(-a\) and zero. The area on either side of zero is one-half.

These facts yield two different-looking calculations for a right-tail probability, depending on whether the cutoff is above or below the mean. If \(z>0\), the right tail is the half-area to the right of zero minus the area from zero to \(z\). If \(z<0\), the right tail consists of the half-area to the right of zero plus the area between \(z\) and zero.

The absolute value in the negative-z case matters: a mean-to-z table is commonly arranged for nonnegative z scores, so look up the magnitude of the negative score. The plus sign is also essential. When the cutoff is below zero, adding the between-area to 0.5 gives the right tail; subtracting it would give the left tail.

Why the Complement Method Works

The complement rule is not a special property of the normal distribution. It follows from the basic structure of probability: an event and its complement are mutually exclusive and exhaustive. For a continuous normal variable, the event that \(X\) is at or below \(c\) is the complement of the event that \(X\) is above \(c\). Their probabilities therefore add to one.

In terms of the density, the cumulative area \(\Phi(z)\) includes all the area to the left of \(z\), while the right tail includes all the remaining area. The total area under the density is one, so the remaining area is \(1-\Phi(z)\). Because a continuous distribution assigns probability zero to a single point, it does not matter for these probabilities whether the cutoff itself is included in the left or right event.

Checks and Common Pitfalls

The sign and size of the answer offer useful checks. If \(c>\mu\), then \(z>0\) and the right-tail probability must be below 0.5. If \(c<\mu\), then \(z<0\) and the right-tail probability must be above 0.5. If \(c=\mu\), symmetry makes it exactly 0.5. These checks can reveal a mistaken sign or a subtraction in the wrong direction.

• Using the left tail as the answer. For a request for values greater than \(c\), \(\Phi(z)\) is the area on the wrong side. Use \(1-\Phi(z)\).
• Subtracting the mean-to-z area for a negative z score. For \(z<0\), the right tail is \(0.5+A(|z|)\), not \(0.5-A(|z|)\).
• Confusing a density height with an area. A normal-table probability is an area over an interval or tail, not the density’s height at one point.
• Forgetting the model and units. State the assumed normal model and standardize using the standard deviation in the same units as the cutoff and mean.

For instance, in the 34-minute example, a common mistake is to report \(0.1587\) as the probability of taking more than 34 minutes. That value is \(\Phi(-1.00)\), the probability of a time at or below 34 minutes. Since the question asks for a time above the cutoff, its complement is \(1-0.1587=0.8413\). The value exceeding one-half is also consistent with a cutoff below the mean.