CHAPTER 10 Having Confidence in Your Results 131

If you want to have a more precise estimate of your population parameter from

your sample statistic, it’s best if the SEs are small and the CIs narrow. One impor-

tant property of both CIs and SEs is that how big they are varies inversely with the

square root of the sample size. For example, if you were to blow up your sample

size — let’s pretend to quadruple it — it would cut the size of the SE and the width

of the CI in half! This square root law is one of the most widely applicable rules in

all of statistics, and is the reason why you often hear researchers trying to find

ways to increase the sample size in their studies. In practice, a reasonable sample

size is reached based on budget and historical studies, because including the whole

population is usually not possible (or necessary).

Understanding and interpreting

confidence levels

The probability that the CI encompasses the true value of the population parame-

ter is called the confidence level of the CI. You can calculate a CI for any confidence

level, but the most commonly seen value is 95 percent. Whenever you report a CI,

you must state the confidence level. As an example, let’s restate our CI from the

analysis of mean blood glucose levels in a sample of adult diabetics to express that

we used the 95 percent confidence level: 95 percent CI = 114 – 126 mg/dL.

In general, higher confidence levels correspond to wider confidence intervals (so

you can have greater confidence that the interval encompasses the true value),

and lower confidence level intervals are narrower. As an example, a 90 percent CI

for the same data is a smaller range (115–125 mg/dL) and the 99 percent CI is a

larger range (112–128 mg/dL).

Although a 99 percent CI may be attractive, it can be hard to achieve in practice

because an exponentially larger sample is needed (as described earlier in this sec-

tion). Also, the wide range it provides can be relatively unhelpful. While dropping

to a 90 percent CI would reduce the range and sample size needed, having only

90 percent confidence that the true value is in the range is also not very helpful.

This may be why there seems to be an industry standard to use the 95 percent

confidence level when calculating and reporting CIs.

The confidence level is sometimes abbreviated CL, just like the confidence limit,

which can be confusing. Fortunately, the distinction is usually clear from the con-

text in which CL appears. When it’s not clear, we spell out what CL stands for.