Statistics for Dummies (8 page)
Depending on how you spin the numbers, the results could be made to show opposite trends: that crime went up or down between 1987 and 1993. But now that you know the difference between the number of crimes and the crime rate, you know that some statistics should not simply be reported as the total number of events, but instead should be reported as rates (that is, the number of events divided by the number in the entire group).
| REMEMBER | Question the type of statistic that was used before you try to make sense of the results. Is it a fair and appropriate measurement? Is it an accurate way to portray the real story behind the data, or is there a better way? |
| Tip | The scale tells you a lotto! Charts and graphs are good ways of showing clearly and quickly the point that you want to make, as long as the drawings are done correctly and fairly. And just to be clear, what's the difference between a chart and a graph? Not much: Statisticians use these terms quite interchangeably when talking about visual displays of statistical information. But a good rule is that if the picture shows bars or pies or other figures or shapes, you're looking at a chart. Otherwise, you are looking at a graph, which plots numbers as they change over time, or as they appear as pairs on the |
Unfortunately, many times, the charts and graphs accompanying everyday statistics aren't done correctly and/or fairly, and you need to be on the look-out for problems. One of the most important elements to watch for is the way that the chart or graph is scaled. The
scale
of a graph is the quantity used to represent each tick mark on the axis of the graph. Do the tick marks increase by 10s, 20s, 100s, 1,000s, or what? The scale can make a big difference in terms of the way the graph or chart looks.
For example, the Kansas Lottery routinely shows its recent results from the Pick 3 Lottery. One of the statistics reported is the number of times each number (0 through 9) is drawn among the three winning numbers.
Table 2-3
shows a chart of the number of times each number was drawn through March 15, 1997 (during 1,613 total Pick 3 games, for a total of 4,839 numbers drawn). Depending on how you choose to look at these results, you can again make the statistics appear to tell very different stories.
 |
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.
|
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 |
| REMEMBER | Looking at the scale of a graph or chart can really help you keep the results in perspective. |
Check the source of the information; the best results are often published by a journal that's known by the experts in the field. For example, in the world of medical science, the
Journal of the American Medical Association
(JAMA), the
New England Journal of Medicine, The Lancet
, and the
British Medical Journal
are all among the reputable journals in which doctors publish results and read about new findings.
| Tip | When examining the results of any study, consider the source and question all of the studies that were conducted, not just those whose results were published in journals or appeared in advertisements. A conflict of interest on the part of researchers can lead to incorrect information. |
