Final answer:
According to the empirical rule, approximately 95 percent of observations from a bell-shaped normal distribution are within 2 standard deviations of the mean. This represents the 68-95-99.7 rule. It's essential for interpreting the variability and probability of outcomes in statistical analysis.
Step-by-step explanation:
According to the empirical rule, if the data form a "bell-shaped" normal distribution, approximately 95 percent of the observations will be contained within 2 standard deviations around the mean. The rule states that for a symmetric bell-shaped distribution, about 68 percent of the data is within one standard deviation of the mean, roughly 95 percent is within two standard deviations, and more than 99 percent is within three standard deviations of the mean. This pattern of distribution is often referred to as the 68-95-99.7 rule.
In practical terms, for a given set of data with a bell-shaped distribution and known mean (μ) and standard deviation (σ), you can predict that approximately 95 percent of data values will fall in the range (μ - 2σ) to (μ + 2σ). This is a crucial concept in statistics, especially when assessing the probabilities of certain outcomes within a dataset. It helps statisticians to understand and interpret the variability of data in relation to the mean.
This rule is especially relevant in fields like quality control, finance, and any research involving normally distributed data. When analyzing sample data, knowing that the majority of values are close to the mean provides valuable insights into the data's nature and potential outliers.