95%

0.05

1.96

98%

0.02

2.33

99%

0.01

2.58

For the most commonly used confidence level, 95 percent, k is 1.96, or approximately 2. This

leads to the very simple approximation that 95 percent upper confidence limit is about two SEs

above the value, and the lower confidence limit is about two SEs below the value.

The confidence interval around a mean

Suppose that you enroll a sample of 25 adult diabetics (N = 25) as participants in a study, and find that

they have an average fasting blood glucose level of 130 mg/dL with a standard deviation (SD) of ±40

mg/dL. What is the 95 percent confidence interval around that 130 mg/dL estimated mean?

To calculate the confidence limits around a mean using the formulas in the preceding section, you first

calculate the SE, which in this case is the standard error of the mean (SEM). The formula for the SEM

is

, where SD is the SD of the sample values, and N is the number of values included

in the calculation. For the fasting blood glucose study sample, where your SD was 40 mg/dL and your

sample size was 25, the SEM is

, which is equal to 40/5, or 8 mg/dL.

Using k = 1.96 for a 95 percent confidence level (from Table 10-1), the sample mean of 130 mg/dL,

and the SD you just calculated of 8 mg/dL, you can compute the lower and upper confidence limits

around the mean using these formulas:

On the basis of your calculations, you would report your result this way: mean glucose = 130 mg/dL

(95 percent CI = 114 – 116 mg/dL).

Please note that you should not report numbers to more decimal places than their precision

warrants. In this example, the digits after the decimal point are practically meaningless, so the

numbers are rounded off.

A version of the formula in the preceding section is designed to be utilized with smaller

samples, and uses k values derived from a table of critical values of the Student t distribution. To

calculate CIs this way, you need to know the number of degrees of freedom (df). For a mean

value, the df is always equal to N – 1, so in our case, df = 25 – 1 = 24. Using a Student t table

(see Chapter 24), you can find that the Student-based k value for a 95 percent confidence level

and 24 degrees of freedom is equal to 2.06, which is a little bit larger than the normal-based k

value of 1.96. Using this k value instead of 1.96, you can calculate the 95 percent confidence