**Choosing Frames**

The choice of sampling frames is important, because each frame must be a fair (unbiased) representation of the next larger element set in which it is contained. Let's look at another frames-within-frames example that can be descriptive, even though it does not lend itself to illustration.

Imagine that we want to evaluate some characteristic of real numbers. The population is the set of all real numbers. Sampling frames are a matter of choice. How about the irrational numbers? Or the rational numbers? How about the set of all real numbers that are square roots of whole numbers? Suppose we choose the set of rational numbers as the sampling frame. Within this set, we might further specify subframes. How about the set of integers? Or the set of rational numbers whose quotients have denominators that are natural numbers between and including 1 and 100? How about the set of even integers? Or do you prefer the set of odd integers? Finally, within this set, we choose a sample. How about the set of integers divisible by 100? Or the set of odd integers that are 1 greater than every integer divisible by 100?

Throughout this process, we must keep one thing in mind: All the sampling frames we choose, and the final sample as well, must be an unbiased representation of the population for the purposes of our experiment. Depending on what this purpose happens to be, the whittling-down process we choose might be satisfactory, or it might be questionable, or it might put us way off the track.

In any real-life experiment, the sample should not be too large or too small. If the sample is too large, it becomes difficult to collect all the data because the process takes too many human-hours, or requires too much travel, or costs too much. If the sample is too small, it will not be a fair representation of the population for the purposes of the experiment. As the sample gets smaller, the risk of its being a poor representation increases.

**Source Data and Sampling Frames Practice Problems**

**Practice 1**

Suppose you want to describe the concept of a "number" to pre-college school students. In the process of narrowing down sets of numbers described above into sampling frames in an attempt to make the idea of a number clear to a child, name a few possible assets, and a few limitations.

**Solution 1**

Think back to when you were in first grade. You knew what a whole number is. The concept of whole number might make a good sampling frame when talking about the characteristics of a number to a six-year-old. But by the time you were in third grade, you knew about fractions, and therefore about rational numbers. So the set of whole numbers would not have been a large enough sampling frame to satisfy you at age eight. But try talking about irrational numbers to a third grader! You won't get far! A 12th-grader would (we hope) know all about the real numbers and various subcategories of numbers within it. Restricting the sampling frame to the rational numbers would leave a 12th-grader unsatisfied. Beyond the real numbers are the realms of the complex numbers, vectors, quaternions, tensors, and transfinite numbers.

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