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Unformatted text preview: Prof. Green Stat 102 The Statistical Underpinnings of Experimentation Randomization and Sampling Variability R.A. Fisher saw that not only did random assignment make for unbiased inference, it also allows one to make useful statements about the degree to which two groups may be expected to differ due to chance assignment alone. For example, suppose N people are randomly assigned to treatment and control groups (N t in the treatment group and N c in the control group). Suppose that some outcome measure in this experiment has a standard deviation of σ , and suppose for the sake of illustration that this standard deviation is known to the researcher beforehand. Fisher pointed out that the difference between the average score in the treatment group and the average score in the control group has an expectation of zero. That is, if the experiment were replicated an infinite number of times, the average difference would be zero. This point is in some sense obvious – of course randomly assigned groups have the same expectation. Nevertheless, the insight represents one of the great turning points in the intellectual history of science. Random assignment represents a procedure for generating unbiased comparisons . Just as importantly, Fisher pointed out that the standard deviation of this difference across an infinite number of replications may be expressed: Standard Error = T C T C N N N N 1 1 2 2 + = + σ σ σ The term “standard error” and the formula above are important to understand. In fact, in some ways, this is the critical idea in any introductory statistical class. Even if the treatment has no effect, we should nonetheless expect to see differences between the treatment and control groups unless the sample sizes are very large or the experimental groups are homogeneous. A sampling distribution describes the pattern of results that one obtains through some statistical procedure when an experiment is performed repeatedly. Estimates vary from one experiment to the next because the disturbances that generate any particular set of observations vary. The more dispersion one observes from experiment to experiment, the more “sampling variability” there is in the results. The “standard error” is the standard deviation of the sampling distribution . The standard error is a convenient way to measure our uncertainty about an estimate. If the standard error were zero, we would know the exact truth based on a single experiment, and any subsequent experiment would reproduce the same answer. If the standard error were infinite, no single experiment could tell us what to expect from any subsequent experiment....
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This note was uploaded on 04/07/2008 for the course STAT 102 taught by Professor Jonathanreuning-schererdonaldgreen during the Fall '05 term at Yale.
- Fall '05