Given the sample data set consisting of a single value $8$, what is the variance of the sample?

• A. $0$
• B. $8$
• C. $64$
• D. The variance is undefined for a sample of size $1$.

Answer: (D)

Verified step by step guidance

1. Recall the formula for the sample variance, which is given by: \[ S^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 \] where \(n\) is the sample size, \(x_i\) are the sample values, and \(\bar{x}\) is the sample mean.
2. Calculate the sample size \(n\). In this problem, there is only one data point, so \(n = 1\).
3. Calculate the sample mean \(\bar{x}\). Since there is only one value, \(\bar{x} = 8\).
4. Notice that the denominator in the variance formula is \(n - 1\). For \(n = 1\), this becomes \$1 - 1 = 0$, which means division by zero would occur.
5. Since division by zero is undefined, the sample variance cannot be calculated for a sample size of 1. Therefore, the variance is undefined for this sample.