Symmetry and Equal-Tail Areas

Why Symmetry Is Useful

A normal model describes a continuous random variable with a bell-shaped density centered at its mean. Its symmetry gives a useful shortcut: a point a fixed distance below the mean and a point the same distance above it cut off equal areas. Thus, if one tail probability or cutoff is known, its counterpart on the other side can be found without calculating that area again.

This relationship is especially useful when a question asks for equal-tail cutoffs: two endpoints that leave the same probability below the lower endpoint and above the upper endpoint. Such endpoints are often used to describe a central portion of a normal distribution. The symmetry tells us where the endpoints must lie relative to the mean; a probability calculation or an inverse normal calculator supplies their distance from it.

Definition: If \(X\sim N(\mu,\sigma^2)\), then \(\mu\) is the center of the normal model and \(\sigma>0\) is its standard deviation. A pair of cutoffs \(L\) and \(U\) are symmetric about the mean if they are the same distance from \(\mu\), one below and one above it. Equivalently, \(L=\mu-d\) and \(U=\mu+d\) for some \(d>0\), or \(L+U=2\mu\).

Where the Symmetry Comes From

A normal random variable has density \(f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)\). The density at a point depends on the squared distance from \(\mu\). Replacing a distance below the mean with the same distance above it does not change that square, and therefore does not change the density.

Theorem: For \(X\sim N(\mu,\sigma^2)\), the density is symmetric about \(\mu\). For every \(d>0\), \(f(\mu-d)=f(\mu+d)\), and the two tail areas match: \(P(X\leq \mu-d)=P(X\geq \mu+d)\). Because a normal variable is continuous, including or excluding a single endpoint does not change these probabilities.

The tail-area equality follows by integrating the density. In the left-tail integral, set \(x=\mu-t\). As \(x\) runs from negative infinity to \(\mu-d\), \(t\) runs from positive infinity down to \(d\); reversing the limits gives an integral from \(d\) to infinity. The density at \(\mu-t\) equals the density at \(\mu+t\), so this becomes exactly the right-tail area.

\begin{aligned} P(X\leq \mu-d) &=\int_{-\infty}^{\mu-d} f(x)\,dx\\ &=\int_d^\infty f(\mu-t)\,dt\\ &=\int_d^\infty f(\mu+t)\,dt\\ &=P(X\geq \mu+d). \end{aligned}

Taking \(d=0\) shows that the area on each side of the mean is one half. The total area under the density is 1, and the two sides have equal area, so \(P(X\leq\mu)=P(X\geq\mu)=0.5\). More generally, the probability between two cutoffs that are equally far from the mean can be found by subtracting the matching tail areas from 1.

Reflecting Cutoffs and Matching Tail Areas

Reflection across the mean turns a cutoff \(x\) into the point \(2\mu-x\). For example, a cutoff 7 units below the mean reflects to a cutoff 7 units above it. If a lower cutoff \(L\) is given, its symmetric upper cutoff is \(U=2\mu-L\); if the upper cutoff is given, the lower cutoff is \(L=2\mu-U\).

U=2\mu-L \qquad\text{and}\qquad L=2\mu-U.

The same reflection connects cumulative probabilities. For any \(d>0\), the probability to the left of \(\mu-d\) equals the probability to the right of \(\mu+d\). Consequently, if a cutoff leaves a specified small probability below it, its reflected cutoff leaves that same probability above it. The probability between the two cutoffs is what remains after both tails are removed.

Worked Example: Matching Opposite Tail Areas

Suppose a hypothetical measurement \(X\) follows the model \(X\sim N(50,8^2)\), where values are measured in units. Find the cutoff above the mean that has the same tail area as the cutoff 42 units, and state the corresponding probabilities.

Check the model. The problem specifies a normal model with mean \(\mu=50\) units and standard deviation \(\sigma=8\) units. Symmetry is a property of this model; using it does not require estimating parameters from a sample.

The lower cutoff is 42, which is \(50-42=8\) units below the mean. Reflect it across 50:

U=2\mu-L=2(50)-42=58\text{ units}.

Thus, the two tail probabilities are equal: \(P(X\leq42)=P(X\geq58)\). A normal-model probability calculator gives either tail as approximately 0.1587. The probability between 42 and 58 is therefore \(1-0.1587-0.1587=0.6826\).

The endpoints are 8 units from the mean on opposite sides, and the two omitted tail areas match. This conclusion depends on the specified normal model, not on the values being integer-valued or on a particular sample.

Equal-Tail Cutoffs for a Chosen Central Probability

Suppose the desired central probability is \(1-\alpha\), where \(0<\alpha<1\). An equal-tail interval assigns half of the excluded probability to each side: the lower tail has probability \(\alpha/2\), and the upper tail also has probability \(\alpha/2\). This makes the central interval symmetric about \(\mu\). If an inverse normal calculator gives one endpoint, symmetry determines the other.

Result: Let \(q_\gamma\) denote the \(\gamma\)-quantile of a continuous normal model, defined by \(P(X\leq q_\gamma)=\gamma\). Symmetry gives \(q_\gamma+q_{1-\gamma}=2\mu\). Therefore, the equal-tail interval with central probability \(1-\alpha\) has endpoints \(q_{\alpha/2}\) and \(q_{1-\alpha/2}\), with probability \(\alpha/2\) below the interval and \(\alpha/2\) above it.

The result can be justified directly. The lower endpoint has probability \(\alpha/2\) below it by the definition of a quantile. Its reflection across \(\mu\) is the same distance above the mean. By the tail-area symmetry, the probability above that reflected endpoint is also \(\alpha/2\). Since the total area is 1, the probability between the endpoints is \(1-\alpha/2-\alpha/2=1-\alpha\).

Worked Example: A Central 90% Interval

Suppose a hypothetical measurement \(X\) follows \(N(50,8^2)\). Find the equal-tail interval containing 90% of the model probability.

Check the model and allocation. The specified normal model is continuous and symmetric about 50 units. A central probability of 0.90 leaves \(1-0.90=0.10\) outside the interval, split equally into tails of \(0.10/2=0.05\).

An inverse normal calculator for this model gives the lower 5th percentile as \(L\approx36.8412\) units. Reflect that cutoff about the mean to obtain the upper endpoint:

U=2\mu-L =2(50)-36.8412 =63.1588\text{ units}.

The endpoints are each about 13.1588 units from 50: \(50-36.8412=13.1588\), and \(63.1588-50=13.1588\). By symmetry, approximately 0.05 of the probability lies below 36.8412 and 0.05 lies above 63.1588. Thus the probability between the cutoffs is \(1-0.05-0.05=0.90\), or 90%.

The rounded endpoints are approximations, so a calculator using more digits may show a tiny rounding difference. The unrounded lower quantile is approximately 36.84117098, giving a reflected upper endpoint of approximately 63.15882902.

Finding a Central Interval from a Tail Probability

Equal-tail questions can be stated in either direction. A question may give the desired central probability and ask for endpoints, or it may give a tail probability and ask for the remaining central probability. The calculation is the same: identify the probability assigned to each tail, find or reflect the corresponding cutoff, and subtract the two tail areas from 1 to obtain the central area.

Worked Example: A Central 98% Interval

Suppose a hypothetical laboratory measurement \(X\) follows \(N(70,5^2)\), with values in the measurement's units. Find the equal-tail interval containing 98% of the model probability.

Check the model and tail allocation. The stated normal model is symmetric about 70 units and has standard deviation 5 units. The excluded probability is \(1-0.98=0.02\), so each tail must contain \(0.02/2=0.01\).

An inverse normal calculator gives the lower 1st percentile as approximately 58.3683 units. This is \(70-58.3683=11.6317\) units below the mean. Reflecting it gives the upper endpoint:

U=2\mu-L =2(70)-58.3683 =81.6317\text{ units}.

The upper endpoint is \(81.6317-70=11.6317\) units above the mean, as required. The lower and upper tails each have probability approximately 0.01, so the probability between the endpoints is \(1-0.01-0.01=0.98\).

The inverse-calculator output and reflected endpoint are rounded to four decimal places. The probability statement is consequently approximate at that displayed precision; with unrounded quantiles, each tail has probability 0.01 under the model.

Checks, Limits, and a Common Mistake

Before reporting equal-tail cutoffs, check three features. First, the variable must be modeled by a normal distribution for the symmetry result to apply. Second, the endpoints must be equidistant from the mean, which can be checked with \(L+U=2\mu\). Third, the tail probabilities must add to the excluded probability: for central probability \(1-\alpha\), they should be \(\alpha/2\) each.

The equal-tail interval is determined by symmetry and the chosen probability, but it is not the only possible interval containing that probability. A non-symmetric interval could place different amounts of probability in its tails. The phrase equal-tail specifies that the two omitted areas match; it does not simply mean that the interval contains a given central proportion.

Mistake: Reflecting a cutoff around zero, regardless of the model's mean. Correction: Reflect it around \(\mu\), so the reflected point is \(2\mu-x\).
Mistake: Assigning all of the excluded probability to one tail when equal tails are requested. Correction: Divide the excluded probability by 2 before finding the lower quantile.
Mistake: Treating symmetry as guaranteed for any distribution. Correction: The equal-area argument here relies on a normal model; a general distribution need not have matching opposite tails.
Mistake: Reporting rounded endpoints and claiming exact tail probabilities from the rounded values. Correction: Say the results are approximate, or retain more digits when the requested precision matters.

Key takeaway: In a normal model, opposite points equally far from \(\mu\) have equal tail areas. For an equal-tail interval with central probability \(1-\alpha\), put probability \(\alpha/2\) in each tail and reflect either endpoint across the mean to obtain the other.

Check Your Understanding

Use a normal model where specified, and explain how symmetry supports each answer.

A normal model has mean 30. A cutoff lies 4 units below the mean. What is its reflected cutoff, and what relationship holds between the two tail probabilities?
A central equal-tail interval is intended to contain 96% of a normal model's probability. What probability belongs in each tail?
A normal model has mean 100, and a lower cutoff is 88. Find the symmetric upper cutoff. How can you check that your answer is symmetric about the mean?
Why does a 90% central equal-tail interval leave 5% in each tail rather than 10% in each tail?
Does the symmetry rule guarantee matching opposite tail areas for every possible distribution? Explain.

What You'll Learn