Sample Size

Statistical Analysis

Why do we have a **margin of error** in statistics? If statistics are meant to be accurate, why are they sometimes accompanied by an estimate of doubt? Well, if we are simply describing something we know everything about, we don’t really need to include an estimate of error. For example, *descriptive statistics* are often used to summarize observations in some way; so if we have tabulated all there is to know (from every member) of the population we are studying, then we don’t need a margin of error. Our statistic is not a guess. But if we need to *infer* something about a larger population—and have only a sample to work with—our statistic will be a guess, and that guess will contain some degree of potential error. Weisstein (2013) defines *error* as the difference between an actual quantity and our estimate of it. In *inferential statistics* a margin of error is how much this difference probably is, either side of the correct figure.

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To appreciate this better, let’s introduce two terms: the *point estimate*, and the *interval estimate*. A point estimate is a single guess about the actual figure. Say for instance, from a population of 10,000 we surveyed 100 people whether they like yogurt, and the mean answer was that 45% said they like yogurt. We don’t really know what all 10,000 people like, but perhaps it turns out to be 48% that like yogurt. Our 45% is a point estimate (and we’re a bit off). Now say we use an interval estimate instead, using our same 45% guess but with a margin of error of plus or minus 3%. Now we are allowing for an interval of 42% to 48%, which happens to include the correct figure. It’s easy to see that an interval estimate allows greater flexibility (Nolan & Heinzen, 2011).

Let’s use a real world example. Say we need to know how many psychologists in North America favor nature over nurture as the explanation for individual behavioral differences (a popular debate and not a bad idea for a survey). We can’t survey everyone, so we use a sample population representative of the whole population and survey that sample. Opinion polls are a common and useful way to gather data. We survey a random sample of 300 psychologists, and we find that *naturists* represent 55% of the sample population, with *nurturists* trailing at 45%. That’s great, but we did not survey every psychologist in North America, so the number we didn’t survey are going to represent our margin of error as some range of possible correct percentages either side of the real percentage. If our margin of error is 6%, naturists may actually be somewhere between 49% and 61% (55 +- 6) and nurturists may be somewhere between 39% and 51% (45 +- 6). As you can see, the winner of the poll can swing either way, depending on our margin of error.

The key is the ratio between our sample size and the full population. If our population is 100,000 and our sample is only 300, we will have a 6% margin of error (at a 95% confidence level). We need to survey at least 1,000 psychologists (preferably 1,500) to reach a 3% margin of error for a race as close as this one. A real plus is that once your full population is 10,000 or more, you can usually sample just 1,000 and still get pretty much the same accuracy as if had you sampled a lot more (American Association for Public Opinion Research, 2007).

**References**

American Association for Public Opinion Research. (2007). *Margin of sampling error*. Available at http://www.aapor.org/

Nolan, S. A., & Heinzen, T. E. (2011). *Essentials of statistics for the behavioral sciences*. New York: Worth Publishers. View

Weisstein, E. W. (2013). *Error*. Available at http://mathworld.wolfram.com/Error.html