Statistics for Dummies (24 page)
Suppose that hypothetical physical therapy student Rhodie took a standardized test for certification to become a physical therapist, and her results indicated that she got a score of 235. All you know is that the scores for this test had a normal distribution. Is Rhodie's score a good one, a bad one, or is it just a middle-of-the-road result? You can't answer this question without a measure of where Rhodie stands among the other people who took the test. In other words, you need to determine the relative standing of Rhodie's test score on the distribution of all the scores.
You can determine the relative standing for Rhodie's test score in a number of ways — some better than others. First you can look at the score in terms of the total points possible, which for this test turns out to be 300. This doesn't compare her score to anyone else's, it just compares her score to the possible maximum. So you really don't know where Rhodie's score stands relative to the other scores. Next, you can try to compare her result to the average. Suppose the average was 250. This does provide a bit more information. At this point you know Rhodie's score of 235 is below average; in fact, her score is 15 points below average (because 235
−
250 =
−
15). But what does a difference of 15 points mean in this situation?
As you see in the "Straghtening Out the Bell Curve" section earlier in this chapter, which discusses the light bulb lifetimes from the two companies (refer to
Figure 8-4
), to get an understanding of relative standing for any value on a distribution, you have to know what the standard deviation is. Suppose the standard deviation of the distribution of scores on Rhodie's test was 5 (as in
Figure 8-6
).
Figure 8-6:Scores having a normal distribution with mean of 250 and standard deviation of 5.
A distribution with a standard deviation of only 5 means that the scores were fairly close together, and 15 points below average is really quite a bit, relatively speaking. In this case, Rhodie's score is interpreted as being pretty far below average, because the 15 point difference is 3 standard deviations below the mean (because each standard deviation is worth 5, and –15 ÷ 5 =
−
3). Only a tiny fraction of the other test takers scored lower than she did. (You know that 99.7% of the test takers scored between 235 and 265 by the empirical rule, and the total percentage for all possible scores is 100. This means that the percentage who scored outside of the 235 to 265 range is 100
−
99.7 = 0.3%. You want the percentage falling below Rhodie's score of 235, which is half of 0.3%. This means that only 0.15%, or 0.0015 of the test takers scored lower than Rhodie in this case.)
Now suppose that the standard deviation is a different value, say 15, but the mean of the test scores is the same (250).
Figure 8-7
shows what this distribution looks like.
Figure 8-7:Scores having a normal distribution with mean of 250 and standard deviation of 15.
A standard deviation of 15 means the scores are much more variable (or spread out) than they were in the previous situation. In this case, Rhodie's score being 15 points below average isn't so bad, because these 15 points represent only 1 standard deviation below the mean (because
−
15 ÷ 15 =
−
1). In this example, 68% of the scores are between 235 and 265 by the empirical rule, and half of the remaining 32% (those in the lower tail) scored lower than Rhodie. So in this scenario, 16% of the test takers scored lower than Rhodie. Her relative standing still isn't great, but this second scenario improves her relative standing compared to the first scenario. Notice that her score didn't change from one scenario to the other; what changed was the interpretation of her score due to the difference in the standard deviations.
| REMEMBER | The relative standing of any score on a distribution depends greatly on the standard deviation. Distances in original units don't mean much without it. |
| HEADS UP | Many times, in the media, the standard deviation is never to be found. Never interpret any statistical result by comparing it only to the mean without knowing what the standard deviation is. Numbers may be farther from the mean than they appear. |
To find, report, and interpret the relative standing of any value on a normal distribution (such as Rhodie's score), you need to convert the score to what statisticians call a standard score. A
standard score
is a standardized version of the original score; it represents the number of standard deviations above or below the mean. The formula for calculating a standard score is the following:
Standard score = (original score
−
mean) ÷ standard deviation. Or, usingshorthand notation,
To convert an original score to a standard score:
Find the mean and the standard deviation of the population that you're working with.
For example, you can convert Rhodie's exam score of 235 to a standard score under each of the two scenarios discussed in the preceding section. In the first scenario, the mean is 250 and the standard deviation is 5, so Step 1 is done.
Take the original score, and subtract the mean.
In Rhodie's first scenario, you find the actual distance from the mean by taking 235
−
250 =
−
15 (which means her score is fifteen points below the mean).Divide your result by the standard deviation.
In Rhodie's case, the distance is
−
15. Converting this distance in terms of number of standard deviations means taking
−
15 ÷ 5 =
−
3, which is Rhodie's standard score. In the first scenario (standard deviation = 5), Rhodie's score of 235 is 3 standard deviations below the mean.In the second scenario (standard deviation = 15), Rhodie's standard score is (235
−
250) ÷ 15 =
−
15 ÷ 15 =
−
1. So her score is 1 standard deviation below the mean in the second scenario.
| HEADS UP | To avoid errors in converting to standard scores, be sure to do Steps 2 and 3 in the order given. |
The following properties may prove helpful when interpreting standard scores:
Almost all standard scores (99.7% of them) fall between the values of
−
3 and +3, because of the empirical rule.A negative standard score means the original score was below the mean.
A positive standard score means the original score was above the mean.
A standard score of 0 means the original score was the mean itself.
Scores that come from a normal distribution, when standardized, have a special normal distribution with mean 0 and standard deviation 1. This distribution is called the
standard normal distribution
(see
Figure 8-8
).
Figure 8-8:
The standard normal distribution.
| REMEMBER | Standard scores have a universal interpretation, which is what makes them so great. If someone gives you a standard score, you can interpret it right away. For example, a standard score of +2 says that this score is 2 standard deviations above the mean. To interpret a standard score, you don't need to know what the original score was or what the mean and standard deviation were. The standard score gives you the relative standing of that value, which, in most cases, is what matters most. |
| HEADS UP | Converting to standard scores doesn't change the relative standing of any of the values on the distribution; it simply changes the units. (This is analogous to changing the units of temperature when converting from the Fahrenheit scale to the Celsius scale. The temperature outside doesn't change, but the units used to measure that temperature do.) Subtracting the mean from the |
One common use for standard scores is to compare scores from different distributions that would otherwise not be comparable. For example, suppose Bill applies to two different colleges (call them Data University and Ohio Stat), and he has to take a math placement test for each of the colleges. The tests are totally different (they even had a different number of questions) and when Bob gets his scores back, he wants to be able to compare his scores and determine which college gives him a better relative standing on its math placement test.
Data University tells Bob that his score is 60 and that the distribution of all scores is normal with a mean of 50 and standard deviation of 5. Ohio Stat tells Bob that his score is 90 and that the scores have a normal distribution with mean 80 and standard deviation 10. On which test does Bob perform better? You can't compare the 50 to the 90 outright, because they're on totally different scales. And you can't say that he performed the same on each just because he was 10 points above the average on each one. Here, the standard deviation is an important factor. The way to make a fair and equitable comparison of these scores is to convert each of them to standard scores, allowing them to both be on the same scale (where most of the values lie between
−
3 and +3 with units of 1).
Again, Bob's score on Data University's placement exam is 60 with a mean of 50 and standard deviation of 5 for the population of all test scores. His standard score, therefore, is (60
−
50) ÷ 5 = 10 ÷ 5 = +2, meaning his relative standing on Data University's placement exam is 2 standard deviations above the mean. His score on Ohio Stat's placement test is 90, where the mean is 80 and the standard deviation is 10. His standard score, therefore, is (90
−
80) ÷ 10 = 10 ÷ 10 = +1, meaning Bob's relative standing on Ohio Stat's math placement exam is 1 standard deviation above the mean. This score is not as high, relatively speaking, as his Data University math placement exam score. Bob, therefore, performs better on Data University's math placement exam.
| HEADS UP | Don't compare results from different distributions without first converting everything to standard scores. Standard scores allow for fair and equitable comparisons on the same scale. |
