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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94 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 Figure 7.1: Frequency Histogram mary. The most common such summary is the average, or arithmetic mean of the outcomes. The mean of n outcomes x 1 ,...,x n for a random variable X is n P i =1 x i /n , and is denoted by ¯ x . The arithmetic mean for the example above can be calculated as (6 × 1) + (8 × 2) + (5 × 3) + (3 × 4) + (2 × 5) + (1 × 6) 25 = 65 25 =2 . 60 That is, there was an average of 2.6 persons per car. A set of observed outcomes x 1 n for a random variable X is termed a sample in probability and statistics. To reflect the fact that this is the average for a particular sample, we refer to it as the sample mean . Unless somebody deliberately “cooked” the study, we would not expect to get precisely the same sample mean if we repeated it another time. Note also that ¯ x is not in general an integer, even though X is. Two other common summary statistics are the median and mode. Definition 12 The median of a sample is a value such that half the results are below it and half above it, when the results are arranged in numerical order. If these 25 results were written in order, the 13 th outcome would be a 2. So the median is 2. By convention, we go half way between the middle two values if there are an even number of observations. Definition 13 The mode of the sample is the value which occurs most often. In this case the mode is 2.
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chap7 - 7 Expectation Averages Variability 7.1 Summarizing...

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