# THE STANDARD ERROR AND CONFIDENCE INTERVALS

In the hypothetical example on page 131 we took a sample of 100 merchants in a Malaysian town and found that the mean income was RM12,600, standard error 400. We know from figure 6.1 that 68.26% of all samples of size 100 from this population will produce an estimate that is between 1 standard error above and 1 standard error below the mean— that is, between RM12,200 and RM13,000. The 68.26% confidence interval, then, is $400.

We also know from figure 6.1 that 95.44% of all samples of size 100 will produce an estimate of 2 standard errors, or between RM13,400 and RM11,800. The 95.44% confidence interval, then, is RM800. If we do the sums for the example, we see that the 95% confidence limits are:

and the 99% confidence limits are:

Our “confidence” in these 95% or 99% estimates comes from the power of a random sample and the fact that—by the central limit theorem—*sampling distributions are known to be normal irrespective of the distribution of the variable whose mean we are estimating.*

**FIGURE 6.6.**

**Visualizing the central limit theorem: The distribution of sample means approximates a normal distribution. Means for 10 (left) and 20 (right) samples of 5 from the 50 GDP values in table 6.1.**

**What Confidence Limits ****Are**** and What They ****Aren't**

If you say that the 95% confidence limits for the estimated mean income are RM11,816-13,384, this does *not* mean that there is a 95% chance that the true mean, ^, lies somewhere in that range. The true mean may or may not lie within that range and we have no way to tell. What we can say, however, is that:

- 1. If we take a very large number of suitably large random samples from the population (we’ll get to what ‘‘suitably large’’ means in a minute); and
- 2. If we calculate the mean, x, and the standard error,
*SE,*for each sample; and - 3. If we then calculate the confidence intervals for each sample mean, based on ± 1.96
*SE*; then - 4. 95% of these confidence intervals will contain the true mean, ^.