Section 2.5 notes continued Statistics Section 2.5 notes continued

Range Rule of Thumb – s ≈ highest value – lowest value 4 estimate of standard deviation

minimum “usual” value = mean – 2(std dev) maximum “usual” value = mean + 2(std dev) Anything below the minimum or above the maximum is unusual.

Empirical Rule: Also called the 68-95-99.7 Rule Used for data with a bell-shaped distribution About 68% of the data is within one std dev of the mean About 95% of the data is within 2 std devs of the mean About 99.7% of the data falls within 3 std devs of the mean

(applies to bell-shaped distributions) The Empirical Rule (applies to bell-shaped distributions) FIGURE 2-15 page 79 of text x

(applies to bell-shaped distributions) The Empirical Rule (applies to bell-shaped distributions) FIGURE 2-15 68% within 1 standard deviation Some student have difficulty understand the idea of ‘within one standard deviation of the mean’. Emphasize that this means the interval from one standard deviation below the mean to one standard deviation above the mean. 34% 34% x - s x x + s

(applies to bell-shaped distributions) The Empirical Rule (applies to bell-shaped distributions) FIGURE 2-15 95% within 2 standard deviations 68% within 1 standard deviation 34% 34% 13.5% 13.5% x - 2s x - s x x + s x + 2s

The Empirical Rule x - 3s x - 2s x - s x x + s x + 2s x + 3s (applies to bell-shaped distributions) FIGURE 2-15 99.7% of data are within 3 standard deviations of the mean 95% within 2 standard deviations 68% within 1 standard deviation These percentages will be verified by the concepts learned in Chapter 5. Emphasize the Empirical Rule is appropriate for data that is in a BELL-SHAPED distribution. 34% 34% 2.4% 2.4% 0.1% 0.1% 13.5% 13.5% x - 3s x - 2s x - s x x + s x + 2s x + 3s

Chebyshev’s Theorem: Chebyshev’s Theorem can be applied to any data set. The results are more approximate than the Empirical Rule. At least 75% of all data values lie within 2 std devs of the mean. At least 89% of all values lie within 3 std devs of the mean.

Measures of Variation Summary For typical data sets, it is unusual for a score to differ from the mean by more than 2 standard deviations. This idea will be revisited throughout the study of Elementary Statistics.