The Empirical Rule and Chebyshev's Inequality both describe how data are spread around the mean, but they apply in different situations. The Empirical Rule is used for distributions that are approximately normal, or bell-shaped. It gives quick percentages for how much data lies within 1, 2, and 3 standard deviations of the mean.
Chebyshev's Inequality is more general because it works for any distribution with a finite mean and standard deviation.
Key Facts
- Empirical Rule: about 68% of normal data lie within μ ± 1σ.
- Empirical Rule: about 95% of normal data lie within μ ± 2σ.
- Empirical Rule: about 99.7% of normal data lie within μ ± 3σ.
- Chebyshev's Inequality: at least 1 - 1/k^2 of data lie within k standard deviations of the mean, for k > 1.
- For k = 2, Chebyshev guarantees at least 1 - 1/4 = 75% of data within μ ± 2σ.
- For k = 3, Chebyshev guarantees at least 1 - 1/9 = 88.9% of data within μ ± 3σ.
Vocabulary
- Mean
- The mean is the average value of a data set and is often written as μ for a population.
- Standard deviation
- Standard deviation measures the typical distance of data values from the mean and is often written as σ for a population.
- Normal distribution
- A normal distribution is a symmetric, bell-shaped distribution centered at its mean.
- Empirical Rule
- The Empirical Rule states that about 68%, 95%, and 99.7% of normal data fall within 1, 2, and 3 standard deviations of the mean.
- Chebyshev's Inequality
- Chebyshev's Inequality gives a minimum percentage of data within k standard deviations of the mean for any distribution with finite mean and standard deviation.
Common Mistakes to Avoid
- Using the Empirical Rule for a skewed distribution. The 68-95-99.7 percentages are reliable only when the distribution is approximately normal.
- Forgetting that Chebyshev gives a minimum, not an exact percentage. It guarantees at least a certain amount of data, but the actual percentage may be higher.
- Using Chebyshev with k = 1. Chebyshev's formula 1 - 1/k^2 requires k > 1, so k = 1 gives no useful guarantee.
- Confusing standard deviation intervals with raw data intervals. An interval like μ ± 2σ must be converted using the actual mean and standard deviation before comparing it to data values.
Practice Questions
- 1 A normal distribution has mean 50 and standard deviation 6. Using the Empirical Rule, what interval contains about 95% of the data?
- 2 A data set has mean 80 and standard deviation 10, but its shape is unknown. Using Chebyshev's Inequality, what minimum percent of data lies between 60 and 100?
- 3 A teacher has a data set that is strongly right-skewed. Should the teacher use the Empirical Rule or Chebyshev's Inequality to make a guaranteed statement about spread, and why?