# Section 6.1

### Section 6.1: Error and Confidence Intervals

In social scientific research utilizing samples, population parameters are often estimated using sample statistics.  For example, the mean of a sample is often used to approximate the mean of the population.  When we do this, we go into the process knowing that our sample mean will be different from the population mean (this idea applies to any statistic that we can compute based on sample data; we’re just focusing on a single statistic—the mean— right now to keep things from getting confusing).  The process is useful because most of the time, the sample mean will be very close to the population mean.  There are times, however, that we’ll be way off the mark.  The rules of mathematics and probability allow the researcher to estimate how big the discrepancy (difference) between the sample mean and the population mean is likely to be.    One such estimation is known as the standard error of the mean.

To understand how standard error works, it is important to understand what a sampling distribution is.  Suppose that we have a standardized test with a population mean of 100.00 and a population standard deviation of 15.00.  As with most research situations, we have no idea what these population values are in reality.  We will need to estimate them using sample data.  As we previously discussed, there will always be some discrepancy between any given population parameter and its corresponding sample statistic.  Say for example that we took a sample from our standardized test with a mean of 100.00.  It is very unlikely that we would get a sample with a mean of exactly 100.00.  Most samples, however, would result in a mean very near the actual mean, with extreme deviations being very rare.

If we obtain all possible samples of a particular sample size (n) from a given population, and then compute a statistic (mean, standard deviation, proportion, etc.) for each sample, the probability distribution of that statistic is called a sampling distribution.  This idea is extremely useful when we examine it in light of the central limit theorem.  The central limit theorem states that the sampling distribution of any statistic will be approximately normal if the sample size is large enough.  (As a rough rule of thumb, many researchers say that a sample size of 30 is large enough).  Remember that the means only vary from each other because of random chance due to random sampling error.  These rules do not apply to systematically biased samples.

In our hypothetical example, the mean of the sampling distribution would be 100.00, and other sample mean values would cluster around the population mean in a normal distribution.  The standard deviation of a sampling distribution of means is known as the standard error of the mean.  Of course, we do not actually know the standard deviation of the sampling distribution, so we use the standard deviation of the sample as an estimate of it.

The standard error of the mean (SEM) is an estimate of the amount of error in estimating the mean of a population based on sample data.  Recall that there will always be chance error (i.e., sampling error) that crops up in samples drawn from populations by random selection.  The standard error of the means tells us how large that error is likely to be.

When we report the value of an estimate for a population, it is often called a point estimate.  It is so called because it represents a single point estimated to be the mean of the population based on the sample data.  Some critics argue that reporting point estimates can be misleading.  After all, we know that it is unlikely that the population mean is exactly equal to the one we estimated using the sample data.  It would be better, they argue, to provide a range within which we can be fairly sure the population mean really falls.

The standard error of the mean allows us to construct a confidence interval.  A confidence interval is simply a range within which we are fairly sure (to a degree specified by the researcher) the population mean will fall.  When we report such an interval, we are said to be reporting an interval estimate rather than a single point estimate.  When a researcher reports a 95% confidence interval (CI) for a mean, we can have 95% confidence that the true mean lies within that interval.  The true mean is the mean of the population if we could somehow determine it (which we can’t, or we wouldn’t be doing sample research in the first place).  Obviously, if we had a complete set of data for the population, we would not be using samples.  Therefore, we have to estimate how good an estimate the sample mean is of the population mean.

### Computing the Standard Error of the Mean

 Standard Error of the Mean (SEM) is a simple computation involving the standard deviation of the scores and the sample size.  The formula is as follows:

Confidence intervals for estimated means are very easy to construct once you have computed the standard error of the mean.  Recall the earlier section where we examined areas under the normal curve in terms of standard deviation units.  The 68% Rule, the 95% Rule, and the 99% Rule apply to sampling distributions just as they did with frequency distributions.

### Computing Confidence Intervals for the Mean

 Use the following formulas to compute the lower limit and upper limit for the 68%, 95%, and 99% Confidence Intervals: Recall that the constants 1.96 and 2.58 are the same ones we learned about previously.

You may have wondered why researchers do not simply compute a 100% confidence interval so that they are sure the mean falls within it.  The problem with that goes back to the very nature of sample research:  It is probabilistic.  Random sampling produces some degree of error.  Unfortunately, we can only estimate the magnitude of that error.  The only way to be 100% confident that the mean lies within the confidence interval would be to construct a confidence interval within which all possible means fall.  This would merely tell us the minimum and maximum possible score and would be essentially useless.

### Key Terms

Standard Error of the Mean, Point Estimate, Confidence Interval, Interval Estimate, True Mean

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