chap7 - 7 Expectation Averages Variability 7.1 Summarizing...

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7. Expectation, Averages, Variability 7.1 Summarizing Data on Random Variables When we return midterm tests, someone almost always asks what the average was. While we could list out all marks to give a picture of how students performed, this would be tedious. It would also give more detail than could be immediately digested. If we summarize the results by telling a class the average mark, students immediately get a sense of how well the class performed. For this reason, “summary statistics” are often more helpful than giving full details of every outcome. To illustrate some of the ideas involved, suppose we were to observe cars crossing a toll bridge, and record the number, X , of people in each car. Suppose in a small study 10 data on 25 cars were collected. We could list out all 25 numbers observed, but a more helpful way of presenting the data would be in terms of the frequency distribution below, which gives the number of times (the “frequency”) each value of X occurred. X Frequency Count Frequency 1 | | | | | 6 2 | | | | | | | 8 3 | | | | 5 4 | | | 3 5 | | 2 6 | 1 We could also draw a frequency histogram of these frequencies: Frequency distributions or histograms are good summaries of data because they show the variability in the observed outcomes very clearly. Sometimes, however, we might prefer a single-number sum- 10 "Study without desire spoils the memory, and it retains nothing that it takes in." Leonardo da Vinci 93
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