Chebyshev's Theorem provides a bound that, for any k > 1, at least a certain percent of data lie within k standard deviations of the mean. Which expression gives this bound?

Chebyshev's Theorem provides a bound that, for any k > 1, at least a certain percent of data lie within k standard deviations of the mean. Which expression gives this bound?

• A. At least (1 - 1/k^2) × 100 percent of the data fall within k standard deviations.
• B. Exactly 68% of data within one standard deviation.
• C. All data lie within k standard deviations.
• D. More than half the data lie within k standard deviations.

Answer: (A)

Chebyshev's inequality gives a universal bound: for any distribution with mean μ and standard deviation σ, the proportion of observations within k standard deviations of the mean is at least 1 − 1/k^2 for any k > 1. Translating that to percent, at least (1 − 1/k^2) × 100% lie within k standard deviations. This matches the statement you’re selecting. For example, with k = 2, at least 75% of data fall within 2 standard deviations. The other options aren’t general guarantees: exactly 68% within one standard deviation is true for a normal distribution, not all distributions; all data within k standard deviations is false in general; and “more than half” within k standard deviations isn’t guaranteed for all k > 1 since the bound can be less than 50% when k is just above 1.

Chebyshev's inequality gives a universal bound: for any distribution with mean μ and standard deviation σ, the proportion of observations within k standard deviations of the mean is at least 1 − 1/k^2 for any k > 1. Translating that to percent, at least (1 − 1/k^2) × 100% lie within k standard deviations. This matches the statement you’re selecting. For example, with k = 2, at least 75% of data fall within 2 standard deviations. The other options aren’t general guarantees: exactly 68% within one standard deviation is true for a normal distribution, not all distributions; all data within k standard deviations is false in general; and “more than half” within k standard deviations isn’t guaranteed for all k > 1 since the bound can be less than 50% when k is just above 1.