The Empirical Rule for Mound-Shaped Data

Using Standard Deviations to Describe a Mound-Shaped Distribution

In Z-Scores: Standardizing a Value, you learned that a z-score measures how many standard deviations a value is above or below the mean. When a distribution is approximately bell-shaped, that distance also helps describe where observations tend to fall. The empirical rule gives useful approximate percentages for this kind of distribution.

The rule is useful for questions such as “About what percent of values fall between these two measurements?” or “Roughly how many observations are within two standard deviations of the mean?” It does not require the individual data values, but it does require a distribution whose shape is reasonably close to a symmetric, single-peaked mound. As in Describing Shape: Symmetric, Skewed, Uniform and Unimodal, Bimodal, and Multimodal Distributions, use the overall pattern—not one uneven bar—to judge the shape.

The Percentages Around the Mean

The three headline percentages describe regions centered at the mean. If a distribution has mean \(\mu\) and standard deviation \(\sigma\), the regions extend from \(\mu-\sigma\) to \(\mu+\sigma\), from \(\mu-2\sigma\) to \(\mu+2\sigma\), and from \(\mu-3\sigma\) to \(\mu+3\sigma\). For sample data, the corresponding summaries are \(\bar{x}\) and \(s\).

Because a bell-shaped distribution is approximately symmetric, each percentage is divided approximately equally between the two sides of the mean. About 34% lies between the mean and one standard deviation above it, and about 34% lies between the mean and one standard deviation below it. The 68% total is therefore \(34\%+34\%\).

The regions farther from the mean can also be found by subtraction. About 95% is within two standard deviations, while about 68% is within one. The difference, \(95\%-68\%=27\%\), lies between one and two standard deviations from the mean on both sides. Split evenly, that is about 13.5% in each band. Similarly, about \(99.7\%-95\%=4.7\%\) lies between two and three standard deviations from the mean, or about 2.35% in each band.

The remaining \(100\%-99.7\%=0.3\%\) lies more than three standard deviations from the mean. For a roughly symmetric mound, about 0.15% lies in each tail. These smaller pieces are useful when a question asks for a percentage in only part of the distribution.

A Reliable Method for Percent-Between Questions

A “percent between” question may ask for a region that is not one of the three centered intervals. The key is to locate both endpoints relative to the mean in standard-deviation units, mark the relevant regions, and add the pieces between the endpoints. When a requested region crosses the mean, include the appropriate portion on each side. When it does not cross the mean, do not accidentally include the other side of the distribution.

A useful sketch can prevent sign and region errors. Put the mean in the middle, mark one, two, and three standard deviations on both sides, and label each band with its approximate percent. The sketch does not need a vertical scale; it is a map for deciding which percentages to add.

Worked Examples: Applying the Empirical Rule

When the Rule Is Appropriate—and What It Does Not Say

The empirical rule is a rule of thumb for a distribution that is approximately bell-shaped: roughly symmetric, with one main mound and no severe departure from that pattern. A histogram is useful for judging whether this description fits. The shape matters because the percentages rely on the balanced pattern of a mound; the same percentages should not be assumed for a strongly skewed, bimodal, or irregular distribution.

The rule does not say that every bell-shaped data set has exactly 68%, 95%, and 99.7% in the stated intervals. Those are approximate benchmarks. Actual sample percentages can differ, especially in smaller data sets, and a few observations may fall beyond three standard deviations. Use words such as “about” or “approximately” when reporting the result.

The empirical rule describes proportions of observations in regions around the mean. It does not identify which particular observations are in those regions, and it does not provide an exact value for a percentile or probability beyond the approximations it states. If the distribution is not reasonably mound-shaped, use the graph or other information available rather than forcing the empirical rule onto it.

Common Mistakes and AP Exam Tips

• Applying the rule to any distribution. A mean and standard deviation alone do not justify the 68-95-99.7 percentages. A full-credit response first notes that the distribution is approximately bell-shaped or explains why the rule is appropriate.
• Using 68%, 95%, or 99.7% for the wrong interval. Those percentages describe regions centered at the mean. For example, 95% is within two standard deviations on both sides—not between one and two standard deviations on one side.
• Forgetting to split a two-sided band. About 27% lies between one and two standard deviations on both sides combined, so one side has about \(27\%/2=13.5\%\), assuming approximate symmetry.
• Adding regions outside the requested interval. Mark the endpoints first. For a percent-between question, add only the bands that actually lie between them.
• Reporting an estimate as exact. Say “about 81.5%” or “approximately 32 seedlings,” not that exactly that proportion or count must occur.
• Giving a count when asked for a percent, or vice versa. To convert an estimated percentage to a count, change the percent to a proportion and multiply by the number of observations. Include the count’s context and acknowledge rounding.

A strong AP response identifies why the rule is suitable, shows which bands are being combined, and gives the estimate in context. For example: “Because the distribution is approximately bell-shaped, the interval from one standard deviation below the mean to two above it contains about \(34\%+34\%+13.5\%=81.5\%\) of observations.” This makes the reasoning visible and keeps the answer appropriately approximate.