Finding Values Between Symmetric Boundaries

Two Cutoffs Around the Center

In Finding a Value from a Percentile with invNorm, you used a cumulative area to find one cutoff. Some questions ask for two cutoffs that enclose a central portion of a normal model: for example, the lower and upper values containing the middle 80% of measurements. Others ask for the first and third quartiles, the boundaries of the middle 50%.

For these questions, the important idea is symmetry. A normal model is centered at its mean, and equal distances on opposite sides of the mean have matching areas. So a central interval is found by dividing the area outside it equally between the two tails, then using invNorm to find the lower and upper cutoffs.

Finding the Middle 80%

If the middle 80% lies between two cutoffs, the remaining 20% must lie outside them. Because the interval is symmetric, half of that 20% is in the lower tail and half is in the upper tail: 10% in each. The lower cutoff is therefore at the 10th percentile, and the upper cutoff is at the 90th percentile.

Here, \(L\) is the lower cutoff and \(U\) is the upper cutoff. The two area inputs add to 1, reflecting their symmetry around the mean. The values returned by invNorm are in the original units of the random variable.

You can also think in standard-deviation units. The standard-normal cutoffs for the 10th and 90th percentiles are about \(-1.282\) and \(1.282\). Thus, the middle 80% of a normal model extends about 1.282 standard deviations below the mean and 1.282 standard deviations above it. For a particular model, invNorm applies those cutoffs to its mean and standard deviation.

An equivalent check is to confirm that the two calculated cutoffs are equally far from the mean. If the model is \(N(\mu,\sigma)\), then \(L\) should be below \(\mu\), \(U\) should be above \(\mu\), and the distances \(\mu-L\) and \(U-\mu\) should match apart from rounding.

Finding the Quartiles

The first quartile \(Q_1\) has 25% of the model’s area to its left. The third quartile \(Q_3\) has 75% to its left, which means 25% to its right. These percentiles leave equal areas in the two tails, so the interval from \(Q_1\) to \(Q_3\) is centered at the mean in a normal model. It contains the middle 50%.

For a standard normal distribution, the quartile z-scores are approximately \(-0.6745\) and \(0.6745\). The quartiles of any normal model therefore lie about 0.6745 standard deviations below and above its mean. The difference \(Q_3-Q_1\), called the interquartile range or IQR, measures the width of the middle 50% in the variable’s original units.

To use these results, name the random variable and its model, decide which central interval is requested, and enter the appropriate left-tail areas into invNorm. The calculator input order is area, mean, standard deviation, as in the previous tutorial. A sketch with equal tails marked can help you verify the areas before calculating.

A Procedure for Symmetric Cutoffs

The normal model must be appropriate for the variable, as assumed or established in the problem. These calculations describe proportions under that model; they do not prove that real-world values follow a normal distribution. Keep the mean, standard deviation, and reported cutoffs in the same units.

Worked Example: The Middle 80% of Battery Life

Worked Example: The Quartiles of Assessment Scores

Worked Example: The Middle 80% of Delivery Times

Common Mistakes and AP Exam Tips

The main challenge is translating “middle” into the correct tail areas. The middle percentage is not the area input for either individual cutoff. First find what is outside the interval, split that area equally between the tails, and then express each boundary as a left-tail area for invNorm.

• Using 0.80 for one boundary. The middle 80% is between two values; it does not mean either boundary is the 80th percentile. Use 0.10 for the lower cutoff and 0.90 for the upper cutoff.
• Forgetting to divide the outside area in half. For the middle 80%, 20% is outside the interval, but each tail has only 10%. For quartiles, each outside tail has 25%.
• Reversing the lower and upper areas. A smaller left-tail area produces a lower cutoff. Thus 0.10 gives the lower middle-80% cutoff, while 0.90 gives the upper cutoff.
• Using the wrong calculator order. The inputs are area, mean, and standard deviation. A command with the model parameters in a different order does not give the intended cutoff.
• Reporting only z-scores. The z-scores \(\pm1.282\) and \(\pm0.6745\) describe standardized distances. If the question asks for measurement cutoffs, use the model’s mean and standard deviation to report values in the original units.
• Giving an incomplete interpretation. State both cutoff values with units and identify the proportion of modeled values between them. For the middle 80%, also identify the 10% in each tail when useful.

A full-credit response names the random variable and normal model, shows how the requested central area determines both tail areas, gives both invNorm calculations, and interprets the results in context. As a final reasonableness check, confirm that the lower cutoff is below the mean, the upper cutoff is above it, and their midpoint is approximately the mean.

Key Takeaway

Symmetric boundaries of a normal model come from equal tail areas. For the middle 80%, find the 10th and 90th percentiles. For the quartiles, find the 25th and 75th percentiles; these bound the middle 50%, and their difference is the IQR.