Choose the statement that correctly describes a normal distribution. A. The approximate percent of values lying within two standard deviations of the mean is 47.5% B. Approximately 68% of the values are greater than the mean value. C. Approximately 68% of the values lie within one standard deviation of the mean. D. The approximate percent of values lying within three standard deviations of the mean is 49.85%
The correct answer and explanation is:
Correct Answer: C. Approximately 68% of the values lie within one standard deviation of the mean.
Explanation (300 words):
A normal distribution, also known as a Gaussian distribution, is a symmetrical, bell-shaped curve that describes how data is distributed around the mean (average). In a normal distribution, the mean, median, and mode are all equal and lie at the center of the curve. The spread of the distribution is measured by the standard deviation (σ), which tells us how much the values deviate from the mean.
The Empirical Rule, also known as the 68-95-99.7 Rule, is used to describe the percentage of data that falls within certain standard deviations from the mean in a normal distribution:
- Approximately 68% of the data values lie within one standard deviation of the mean (from μ – σ to μ + σ). This supports Option C, making it the correct choice.
- Approximately 95% of the data lie within two standard deviations of the mean (from μ – 2σ to μ + 2σ), not 47.5% as stated in Option A.
- Approximately 99.7% of the data fall within three standard deviations (from μ – 3σ to μ + 3σ), not 49.85% as stated in Option D.
- Because the distribution is symmetrical, about 50% of the values lie above the mean and 50% lie below. So, Option B is incorrect because only about 50%, not 68%, are greater than the mean.
Understanding the properties of the normal distribution is essential in fields such as statistics, psychology, and economics because it provides a basis for making predictions and understanding variability in data. The 68% rule is especially helpful in quickly estimating probabilities and analyzing the reliability of measured data.