Chapter 8, Problem 124CP

Why called “degrees of freedom”? You know the sample mean $\bar{x}$ of n observations. Once you know (n – 1) of the observations, show that you can find the remaining one. In other words, for a given value of $\bar{x}$, the values of (n – 1) observations determine the remaining one. In summarizing scores on a quantitative variable, having (n – 1) degrees of freedom means that only that many observations are independent. (If you have trouble with this, try to show it for n = 2, for instance showing that if you know that $\bar{x}=80$ and you know that one observation is 90, then you can figure out the other observation. The df value also refers to the divisor in $s^2={\displaystyle \sum {\left(x-\bar{x}\right)}^2}/\left(n-1\right)$.)