Degrees of Freedom Five pulse rates randomly selected from Data Set 1 “Body Data” in Appendix B have a mean of 78.0 beats per minute. Four of the pulse rates are 82, 78, 56, and 84.

a. Find the missing value.

b. We need to create a list of n values that have a specific known mean. We are free to select any values we desire for some of the n values. How many of the n values can be freely assigned before the remaining values are determined? (The result is referred to as the number of degreesof freedom.)

The mean of five pulse rates is 78.0 beats per minute.

The four known observations are 82, 78, 56, and 84.

a.

Let c be the missing pulse rate observation.

The formula for the mean of n observations is stated as follows.

$\overline{x}=\frac{\sum x}{n}$ for n observation with values x.

Substitute the pulse rates observation in the formula as shown.

$$\begin{gathered} 78=\frac{82+78+56+84+c}{5} \\ 78=\frac{300+c}{5} \\ 390=300+c \\ c=90 \end{gathered}$$

Thus, the missing observation is 90 beats per minute.

b.

It is required to list n values for a known mean measure.

The degree of freedom is the number of values that are free to be selected from n.

For the mean value of $\overline{x}$ for n counts of observations, the sum of all observations is defined as $\sum _{i=1}^{n}x$, such that the formula holds true.

$\overline{x}=\frac{{\sum }_{i=1}^{n}x}{n}$is the known sum of all observations. It can be true for many possible combinations of observations.

To determine a feasible combination, you can select n – 1 observations freely (randomly), but the last observation will always be$\sum _{i=1}^n{x}_i-\sum _{i=1}^{n-1}{x}_i$ .

As a result, there are only n – 1 observations free to be chosen.

Thus, the degree of freedom is n – 1.