What is the 68-95-99.7 rule for normal distributions? Explain how it can be used to answer questions about frequencies of data values in a normal distribution Choose the correct answer below
A. The rule states that about 68%, 95%, and 99 7% of the data points in a normal distribution tle within 0.1 and 2 standard deviations of the mean, respectively,
B. The rule states that about 1, 2 and 3 data points lia in 68%, 95% and 99 7% of the data points, respectively, in a normal distribution
C. The rule states that about 68% 95% and 99 7% of the data points in a normal distribution lie within 1, 2 and 3 standard deviations of the mean respectively
D. The rule states that about 0.1 and 2 data points lle in 68% 95%, and 99.7% of the data points respectively, in a normal distribution
The correct answer and explanation is :
The correct answer is:
C. The rule states that about 68%, 95%, and 99.7% of the data points in a normal distribution lie within 1, 2, and 3 standard deviations of the mean, respectively.
Explanation:
The 68-95-99.7 rule, also known as the Empirical Rule, describes how data values are distributed in a normal distribution (bell-shaped curve). It provides a way to estimate probabilities and frequencies without performing complex calculations.
- 68% of the data lies within one standard deviation (σ) of the mean (µ). This means that most of the values are relatively close to the mean.
- 95% of the data lies within two standard deviations (2σ) of the mean. This covers a broader range and includes most typical observations.
- 99.7% of the data lies within three standard deviations (3σ) of the mean, meaning almost all data points are within this range.
How It Is Used:
- Estimating Probabilities: If a variable follows a normal distribution, you can quickly determine the probability of a value falling within a certain range. For example, if test scores are normally distributed, about 95% of students score within two standard deviations of the mean.
- Identifying Outliers: Values beyond three standard deviations from the mean are rare (only 0.3% of data), making them potential outliers.
- Making Predictions: Businesses and scientists use this rule to predict outcomes, such as analyzing manufacturing defects, IQ scores, or financial market fluctuations.