Standard Errors
The second major issue arising from the game of statistical inference is that every measurement has some associated error that takes two forms: systematic error and random error. An example will clarify this. Suppose, for some obscure reason, we want to determine the height of 14-year-old boys in the United States. That’s our population, but no agency will fund us to measure all 14-year-old boys in the country, so we settle for some samples of boys. For example, we randomly sample from 10 schools that were, in turn, randomly sampled throughout New York. Let’s look at how our calculated mean height may differ from the true value, namely, that number obtained by measuring every boy in the United States and then averaging the results.
The idea of systematic error is fairly easy. If the ruler used is an inch short or if the schools have many northern Europeans, then regardless of how large a sample is or how many times we do the study, our results would always differ from the true value in a systematic way.
The notion of random error is a bit trickier but is fundamental to statistical inference. As an educated guess, the mean height of 14-year-olds is probably approximately 5 feet 8 inches, with a standard deviation (SD) of approximately 4 inches. This means that every time the experiment is performed, the results will differ depending on the specific kids who are sampled. This variation in the calculated mean, caused by individual measurements scattered around the true mean value, is called “random error.” Actually, statistics are lousy at dealing with systematic error but awfully good at determining the effect of random error.
To illustrate, if we grabbed a few kids off the street, the chances are good that some of them might be very short or very tall, so our calculated mean may be far from the truth. Conversely, if we used thousands of kids, as long as they were sampled at random, their mean height value should fall close to the true population mean. As it turns out, the mean values determined from repeated samples of a particular size are distributed near the true mean in a bell-shaped curve with an SD equal to the original SD divided by the square root of the sample size. This new SD, describing the distribution of mean values, is called the standard error (SE) of the mean and is found as follows:
Figure 1344 –
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Fig.1118 illustrates how the SE is related to the SD and sample size.
Putting it another way, because every measurement is subject to some degree of error, every sample mean we calculate will be somewhat different. Most of the time, the sample means will cluster closely to the population mean, but every so often, we’ll end up with a screwball result that differs from the truth just by chance. So if we drew a sample, did something to it, and measured the effect, we’d have a problem. If the mean differs from the population mean, is it because our intervention really had an effect, or is it because this is one of those rare times when we drew some oddballs? We can never be sure, but statistics tell a lot about how often we can expect the group to differ by chance alone (more on this later).
Figure 1118 – Figure 3-1: Original distribution and distribution of means related to sample size.
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Content on this page was last changed on March 19, 2009.
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| 5343. | Norman GR, Streiner DL. PDQ Statistics . 3rd ed. Hamilton, Ontario: BC Decker Inc.; 2003. |
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