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7. Expectation, Averages, Variability7.1Summarizing Data on Random VariablesWhen we return midterm tests, someone almost always asks what the average was. While we couldlist out all marks to give a picture of how students performed, this would be tedious. It would alsogive more detail than could be immediately digested. If we summarize the results by telling a classthe 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, andrecord the number,X, of people in each car. Suppose in a small study10data on 25 cars were collected.We could list out all 25 numbers observed, but a more helpful way of presenting the data would be interms of thefrequency distributionbelow, which gives the number of times (the “frequency”) eachvalue ofXoccurred.XFrequency CountFrequency1| | | | |62| | | | | | |83| | | |54| | |35| |26|1We could also draw afrequencyhistogram of these frequencies:Frequency distributions or histograms are good summaries of data because they show the variabilityin 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 Vinci93
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