Chebyshev’s Theorem and the Empirical Rule

Chebyshev’s Theorem The proportion of values from a data set that will fall within k standard deviations of the mean will be at least 1 – 1/k ², where k is a number greater than 1. (Does not need to be an integer) Can be applied to any distribution, regardless of it’s shape. Can also be used to find the minimum percentage of data values that will fall between any two given values.

Types of Distribution Positively skewed: Majority of the data values fall to the left of the mean. Negatively skewed: Majority of the data values fall to the right of the mean. Symmetrical: Data values are evenly distributed.

Chebyshev’s Theorem Example 1 A sample of the hourly wages of employees who work in large city restaurants has a mean of $7.35 with a standard deviation of $0.08. Use his theorem to find the range in which at least 75% of the data falls.

CT Example 2 The average number of trials it took a sample of rats to traverse a maze was 18. The standard deviation was 5. Use his theorem to find the range in which at least 88.89% of the data will fall.

CT Finding % of DataValues Steps A distribution has a mean of 80 and a standard deviation of 10. Find the minimum percentage of the data that will fall between 60 and 100. Then 65 and 95. Step 1: Subtract the mean from the larger value of each group. Step 2: Divide the answer by the standard deviation. (This will get you k ) Step 3: Use k in his theorem to get %

CT Finding % of Data Values Example 2 In a distribution of 300 values the mean is 62 and the standard deviation is 7. Find the % of data that falls between 30 and 70.

Empirical Rule Approximately 68% of the data values fall within 1 standard deviation of the mean. Approximately 95% of the data values fall within 2 standard deviations of the mean. Approximately 99.7% of the data values fall within 3 standard deviations of the mean.