LISTS AND SUBSCRIPTED VARIABLES

Throughout this book, I make frequent use of tables. Tables are lists of numbers that relate variables in different situations. This isn’t as bad as it first sounds. I’m sure you’ve all seen this many times—everything from income tax tables that the Internal Revenue Service provides to automobile value depreciation tables.

Table 1.1 is a hypothetical automobile value depreciation table. Don’t worry about what kind of car it is—I just made up the numbers for the sake of this example.

Looking from left to right, you see two columns: the age of the car and the car’s wholesale price. Looking from top to bottom you see six rows. The top row contains the headings, or descriptions, of what the numbers beneath mean. Then there are

Table 1.1 Hypothetical Automobile Value Depreciation Table

Age of car (years)

Wholesale price

($)

0

32,000

1

26,500

2

21,300

3

18,000

4

15,500

5

13,250

Table 1.2 Hypothetical Automobile Depreciation Table with Air-Conditioning Option

Age of car (years)

Wholesale price ($)

Extra for air- conditioning

0

32,000

1,200

1

26,500

1,050

2

21,300

850

3

18,000

650

4

15,500

550

5

13,250

450

five rows of numbers. The numbers on each row “belong together.” For example, when the car is 2 years old, the wholesale price is $21,300.

An important point about the headings is that whenever appropriate, the units should be listed. In Table 1.1, the age of the car is expressed in years. If I didn’t say so, how would you know I didn’t mean months, or decades? The value of the car is expressed in dollars. To be very precise, maybe I should have said U. S. dollars (if that’s what I meant). Someone in Great Britain could easily assume that the prices are in pounds if I didn’t clearly state otherwise.

Very often a table will have many columns. Table 1.2 is a repeat of Table 1.1, but with a third column added: How much more the car is worth if it has air­conditioning. Notice that I was a little sloppy here. I didn’t say that the extra amount was in dollars. In this case, however, a little sloppiness is harmless. Once you know that we’re dealing in dollars, you can be pretty sure that things will be consistent.

Again, the numbers in a given row belong together: A 3-year-old car is worth $18,000, and it is worth $650 more if it has air-conditioning.

Tables 1.1 and 1.2 tell you some dollar amounts based on the age of the car. It’s therefore typical for the age of the car to appear in the leftmost column. I could have put the car’s age in the middle column (of Table 1.2) or in the right column. Even though doing this wouldn’t introduce any real errors, it makes things harder to read.

Whenever convenient, columns are organized from left to right in order of decreasing importance. That is, I could have made the air-conditioning increment the second column and the car value the third column (always count columns from the left), but again it’s clearer if I put the more important number to the left of the less important number.

Some tables have many, many rows. The Life Tables presented in Chapter 10, the chapter about life insurance, have 102 rows—representing ages from 0 to 100, plus the heading row. The second column in the Life Tables is a number represented by the variable q, the third by the variable -, and so on. Don’- worry about what these letters mean now; this is a topic in Chapter 10.

In Table 1.3, I’ve extracted a piece of the Life Table shown in Table 10.1, As you can see, for every age there are six associated pieces of information. Suppose I wanted to compare the values of q for two different ages, or to make some

Table 1.3 Part of the 2004 U. S. Life Table for All Men

Age

q

l

d

L

T

e

0

0.007475

100,000

747

99,344

7,517,501

75.2

1

0.000508

99,253

50

99,227

7,418,157

74.7

2

0.000326

99,202

32

99,186

7,318,929

73.8

3

0.000250

99,170

25

99,157

7,219,744

72.8

4

0.000208

99,145

21

99,135

7,120,586

71.8

5

0.000191

99,124

19

99,115

7,021,451

70.8

6

0.000182

99,105

18

99,096

6,922,336

69.8

7

0.000171

99,087

17

99,079

6,823,240

68.9

8

0.000152

99,070

15

99,063

6,724,161

67.9

9

0.000125

99,055

12

99,049

6,625,098

66.9

10

0.000105

99,043

10

99,038

6,526,049

65.9

11

0.000111

99,033

и

99,027

6,427,011

64.9

12

0.000162

99,022

16

99,014

6,327,984

63.9

13

0.000274

99,006

27

98,992

6,228,970

62.9

14

0.000431

98,978

43

98,957

6,129,978

61.9

15

0.000608

98,936

60

98,906

6,031,021

61.0

16

0.000777

98,876

77

98,837

5,932,116

60.0

17

0.000935

98,799

92

98,753

5,833,278

59.0

18

0.001064

98,706

105

98,654

5,734,526

58.1

19

0.001166

98,601

115

98,544

5,635,872

57.2

20

0.001266

98,486

125

98,424

5,537,328

56.2

21

0.001360

98,362

134

98,295

5,438,904

55.3

22

0.001419

98,228

139

98,158

5,340,609

54.4

generalizations of some sort. As I go through my discussion, I find that it’s very cumbersome repeating terms like “the value of q for age 10” over and over again.

I can develop a much more concise, easy to read, notation by taking advantage of the fact that the left-hand column is a list of nonrepeating numbers that increase monotonically. By this I mean that 1 is below 0, 2 is below 1, 3 is below 2, and so on, so that it’s easy to understand what row I’m looking at just by referring to the age (the left-hand column). Then I use a subscript (a little number placed low down on the right) tied to any variable that I want to discuss to tell you what I’m looking at. This is hard to describe but easy to show with examples:

q3 refers to the value of q for age 3: q3 = 0.000250.

q12 refers to the value of q for age 12: q12 = 0.000162.

d15 refers to the value of d for age 15: d15 = 60.

Now I can easily discuss the table using this subscript notation. In Table 1.3, qio is the smallest of all the values of q, l22 is about 2% smaller than l0, and so on. Asking why I’d want to be saying these things depends on the topic and the table

under discussion. It’s like asking why I’d ever want to multiply two numbers together.

Leave a reply

You may use these HTML tags and attributes: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <s> <strike> <strong>