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numbers drawn). Depending on how you choose to look at these results, you can again make the statistics appear to tell very different stories.

Table 2-3: Number of Times Each Number Was Drawn (Kansas Pick 3 Lottery, through 3/15/97)
Number Drawn
Number of Times Drawn
----
0
485
----
1
468
----
2
513
----
3
491
----
4
484
----
5
480
----
6
487
----
7
482
----
8
475
----
9
474

    The way lotteries typically display results like those in Table 2-3 is shown in Figure 2-1 . Notice that in this chart, it seems that the number 1 doesn't get drawn nearly as often (only 468 times) as number 2 does (513 times). The difference in the height of these two bars appears to be very large, exaggerating the difference in the number of times these two numbers were drawn. However, to put this in perspective, the actual difference here is 513 − 468 = 45, out of a total of 4,839 numbers drawn. In terms of the total number of individual numbers drawn, the difference between the number of times the number 1 and the number 2 are drawn is 45 ÷ 4,839 = 0.009, or only nine-tenths of one percent.
    Figure 2-1: Bar chart showing number of times each number was drawn.
    What makes this chart exaggerate the differences? Two issues come to the surface here, both affecting the appearance of the chart. First, notice that the vertical axis shows the number of times (or frequency) that each number is drawn, and it goes up by 5s. So a difference of 5 out of a total of 4,839 numbers drawn appears as if it actually means something. This is a common trick used to exaggerate results — stretching the scale so that differences appear larger than they really are. Second, the chart starts counting not at zero, but at 465, so it really is only showing the top part of each bar, where the differences are. This also exaggerates the results.
    Table 2-4 shows a more realistic summary for each of the numbers drawn in the Pick 3 Lottery, by showing the percentage of times each number was drawn.

Table 2-4: Percentage of Times Each Number Was Drawn
Number Drawn
Number of Times Drawn
Percentage of Times Drawn
----
0
485
10.0% = 485 ÷ 4,839
----
1
468
9.7% = 468 ÷ 4,839
----
2
513
10.6% = 513 ÷ 4,839
----
3
491
10.1% = 491 ÷ 4,839
----
4
484
10.0% = 484 ÷ 4,839
----
5
480
9.9% = 480 ÷ 4,839
----
6
487
10.0% = 487 ÷ 4,839
----
7
482
10.0% = 482 ÷ 4,839
----
8
475
9.8% = 475 ÷ 4,839
----
9
474
9.8% = 474 ÷ 4,839

    Figure 2-2 is a bar chart showing the percentage of times each number was drawn, rather than the number of times each number was drawn. Note that this chart also uses a more realistic scale than the one in Figure 2-1 , and that it also starts at zero, both of which make the differences appear as they really are — not much different at all. Boring, huh?
    Figure 2-2: Bar chart showing percentage of times each number was drawn.
    Now why would a lottery do this? Maybe it wants you to believe you're getting some inside information, and thinking that the number 1 doesn't get drawn very much will make you want to buy a lottery ticket and choose 1, because it's "due" to happen (which is not true, by the way; see Chapter 7 for more on this). Or, you may want to choose the number 2, because it has been drawn a lot, and it's "on a roll" (again, no dice). However you look at it, the lottery folks want you to think that some "magic" is involved in the numbers, and you can't blame them; that's their business.
HEADS UP 
Misleading graphs occur all the time in the media! Reporters and others can stretch the scale out (make the tick marks represent increments of small amounts) and/or start at a number other than zero, to make differences appear larger than they really are. Or the scale can also be squeezed down (make the tick marks represent increments of large amounts) to give the appearance of "no change." These are examples of misleading representations of the truth. (See Chapter 4 for more information on spotting misleading graphs.)
REMEMBER 
Looking at the scale of a