s21purvesnotation

s21purvesnotation - Some Statistical Notation 1...

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Page 1 1. Introduction. Statistical procedures are often given in mathematical notation. Sec- tions 2 to 8 below introduce a notation based on high school algebra, and sections 9 to 20 show you how to use the algebra implicit in the notation. 2. Notation for lists of numbers. Data comes in many forms. One of the simplest is a list of numbers: 6 3 1 8 2 Usually there is a label at the top of the list–e.g: “Income (thousands of dollars)” , “Time (hours)”, “Number of Visits”, etc. For statistical formulas, it is traditional to use letters of the alphabet as labels: x 6 3 1 8 2 y 7 4 9 5 The letters make it easy to talk about lists; for example: The largest number on the list x is 8. The smallest number on y is 5. The sum of y is 25. Informal language is not always clear. The list x has five numbers on it; but what about the list below? z 6 7 6 6 7 Does z have five numbers on it—or two? To avoid this possibility for confusion, the word “entry” is often used. Then z is said to have five Some Statistical Notation.
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Page 2 entries: the first one is 6, the second is 7, and so on. 3. Putting lists together. The two lists: u 2 8 5 14 v 6 1 7 can be combined into one list by writing the entries of v below the entries of u: w 2 8 5 14 6 1 7 Example 1 in the next section involves a list built up in this way. There is no widely accepted term for the operation of putting lists together like this, even though it is what happens when data from separate sources is compiled into one source. Programmers might refer to w as the result of appending v to u, but the term will not be used here. 4. The sum and average of a list of numbers. The sum of a list of numbers is the result of adding up all the entries on the list, and the average is the result of dividing that sum by the number of entries on the list. A straightforward notation for the sum and average of a list is: sum(x), av(x) For average, statisticians often use the more symbolic notation: x which is read “x-bar”. In that notation, the definition of average
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Page 3 becomes: x = number of entries of x sum(x) Or, more briefly: x = n sum(x) where n is the number of entries of x. Multiplying both sides by n leads to the following: sum(x) = nx The sum of a list is the number of entries times the average. Simple as it is, this little equation is useful. Example 1. An income study involves 300 men and 200 women; the average income of the men is $38,000 and the average income of the women is $33,000. Find the average income of the 500 people. (Try to do this before reading further.) Hint and answer. You are asked to find the average of a list of 500 incomes. The first step is to get its sum. Think of the list as the result of putting togeth- er two lists: the list of 300 incomes of the men in the study, the list of 200 incomes of the women in the study So, if you can get the sum of each of these lists, you can get the sum of the 500 incomes. To get the two sums, use the above equation twice.
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