Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

Document related concepts

Transcript

2.6 - Interpreting the standard deviation Often it makes sense to describe the distance from the mean in terms of the standard deviation in order to interpret it. And we can talk about the percentage of the data that falls within one standard of the mean, two standard deviations of the mean etc. For a symmetric bell shaped distribution, we have the following empirical rule: 1) Approximately 68% of the observations lie within one standard deviations of the mean. 2) Approximately 95% of the observations lie within two standard deviations of the mean. 3) Approximately all the observations lie within three standard deviations of the mean. For a general distribution, however we have the following: Chebyshev's Rule : Given a number k ≥ 1, at least (1 - 1/k2) of the observations lie within k standard deviations of the mean. k 1 2 3 1-1/k2 Exercises: 2.86, 2.87, 2.91, 2.95, 2.97, 2.102 2.7 Measures of Relative Standing Here we are interested in describing the relative location of a particular meaasurement within a data set. One such measure is percentile ranking. Suppose you scored 80 on an exam - but the instructor tells you that you scored the 90th percentile. What does that mean? Definition : The pth percentile of a data set is the number (when the data are ordered) such that p% of the measurements fall below it and (100-p)% of the measurements fall above it. This is something that can be calculated exactly for a large data set only else it has to be approximated. What percentile is the median? Recall the job data: 3.1, 3.1, 3.5, 4.14, 4.32, 4.47, 4.5, 4.68, 4.8, 4.9 Another measure of relative standing is the zscore. The sample z score for a sample is The population z score is xx s x How do we interpret this?? e.g. There are two sections of STA 2122 – In section I for the first exam , the mean is 62 and s = 5 while in Section II , the mean is 62, but s = 10. Sue is in Section I and scores 72. Amy is in section II and scores 72 also. Who did better relative to their classmates? Exercises: 2.103, 2.108, 2.110, 2.117