By Ashlee Driscoll The Empirical Rule What is it? When do you use it? How do you use it? Solution: Tips to help get to the solution Things to remember: The Empirical Rule is most often used in statistics for forecasting final outcomes. After a standard deviation is calculated, and before exact data can be collected, this rule can be used as a rough estimate as to the outcome of the impending data. This probability can be used in the meantime as gathering appropriate data may be time consuming, or even impossible to obtain. The scores for all high school seniors taking the verbal section of the Scholastic Aptitude Test (SAT) in a particular year had a mean of 490 and a standard deviation of 100. The distribution of SAT scores is bell-shaped.

A. What percentage of seniors scored between 390 and 590 on this SAT test?

B. One student scored 795 on this test. How did this student do compared to the rest of the scores?

C. A rather exclusive university only admits students who were among the highest 16% of the scores on this test. What score would a student need on this test to be qualified for admittance to this university? The data being described are the verbal SAT scores for all seniors taking the test one year. Since this is describing a population, we will denote the mean and standard deviation as m = 490 and s = 100, respectively. A bell shaped curve summarizing the percentages given by the empirical rule is below Since about 16% of the students scored above 590 on this SAT test, to be qualified for admittance to this university, a student would need to score 590 or above on this test. Before applying the empirical rule it is a good idea to identify the data being described, and the value of the mean and standard deviation.

You should also sketch a graph summarizing the information provided by the empirical rule. If helpful, more than one graph may be needed to help find the desired solution. •68% of values fall within ±1 standard deviation of the mean

•95% fall within ± 2 standard deviations of the mean

•99% fall within ± 3 standard deviations of the mean A statistical rule stating that for a normal distribution, almost all data will fall within three standard deviations of the mean. Broken down, the empirical rule shows that 68% will fall within the first standard deviation, 95% within the first two standard deviations, and 99.7% will fall within the first three standard deviations of the mean.

Also referred to as the Three Sigma Rule, or the 68-95-99.7 Rule. C. Since about 68% of the scores are between 390 and 590, this leaves 32% of the scores outside this interval. Since a bell-shaped curve is symmetric, one-half of the scores, or 16%, are on each end of the distribution. The figure below shows these percentages. B. Since about 99.7% of the scores are between 190 and 790, a score of 795 is excellent. This is one of the highest scores on this test. A. From the figure above, about 68% of seniors scored between 390 and 590 on this SAT test.

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# The Empirical Rule

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