In an approximately normal distribution, roughly what share of observations falls within one standard deviation of the mean?
Correct Answer
B) About 68 percent of the observations
For a normal distribution, about 68.27% falls between the mean minus one SD and the mean plus one SD. The corresponding two-SD and three-SD shares are about 95.45% and 99.73%. Half an SD covers about 38.29%, not 50%. These normal-distribution percentages should not be assumed for an arbitrary sale-price distribution.
Why This Is the Correct Answer
Option B is correct because for a normal distribution about 68% (68.27%) of observations fall between the mean minus one standard deviation and the mean plus one standard deviation. This is the first step of the familiar 68-95-99.7 pattern.
Why the Other Options Are Wrong
Option A: About 50 percent of the observations
Option A, about 50 percent, does not match one standard deviation; it corresponds to the middle half of the data, which in a normal distribution is a narrower band than one standard deviation on each side. About 68% falls within one standard deviation.
Option C: About 95 percent of the observations
Option C, about 95 percent, is the share within two standard deviations of the mean (about 95.45%), not one. Within one standard deviation the share is about 68%.
Option D: About 99 percent of the observations
Option D, about 99 percent, is close to the share within three standard deviations (about 99.73%), not one. Within one standard deviation the share is about 68%.
68, 95, 99.7
Count the standard deviations one, two, three and pair them with 68, 95 and 99.7 percent.
How to use: Use this sequence whenever a question asks what share of a normal distribution falls within a given number of standard deviations.
Exam Tip
Match the number of standard deviations to the right percentage, and remember the rule applies to approximately normal data.
Common Mistakes to Avoid
- -Confusing the one- and two-standard-deviation percentages
- -Assuming half the data fall within one standard deviation
- -Applying normal percentages to clearly skewed price data
Concept Deep Dive
Analysis
This question tests the empirical rule for normal distributions. About 68% of observations fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. The distractors correspond to other intervals.
Background Knowledge
The normal distribution is symmetrical and bell-shaped, with fixed proportions of observations falling within each standard-deviation band around the mean. These percentages hold for normal data and should not be assumed for sale-price distributions that are skewed or otherwise non-normal.
Real-World Application
An appraiser finds that adjusted sale prices in a uniform tract are roughly normal with a mean of $400,000 and a standard deviation of $12,000. About two-thirds of the adjusted prices fall between $388,000 and $412,000.
More Statistics Questions
Sale prices are $270,000, $290,000, $300,000, $310,000, $330,000. Treat the listed observations as the complete population and divide by n when computing the standard deviation. Report to the nearest dollar.
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