Chebyshev’s inequality says that at least 1 -1/K2 of data from a sample must fall within K standard deviations from the mean, where K is any positive real number greater than one. This means that we don’t need to know the shape of the distribution of our data. With only the mean and standard deviation, we can determine the amount of data a certain number of standard deviations from the mean.
The following are some problems to practice using the inequality. A version of the worksheet with solutions is located here.
- A class of second graders has mean height of 5 feet with standard deviation of one inch. At least what percent of the class must be between 4’10”and 5’2”?
- Computers from a particular company are found to last on average for three years without any hardware malfunction, with standard deviation of two months. At least what percent of the computers last between 31 months and 41 months?
- Bacteria in a culture live for an average time of three hours with standard deviation of 10 minute. At least what fraction of the bacteria live between two and four hours?
- What is the smallest number of standard deviations from the mean that we must go if we want to ensure that we have at least 50% of the data of a distribution?
- Bus route #25 takes a mean time of 50 minutes with standard deviation of 1 minute. A promotional poster for this bus system states that “95% of the time bus route #25 lasts from ____ to _____ minutes.” What numbers would you fill in the blanks with?