If a distribution has zero variance, which of the following statements is true about the data values? $\mathrm{Var}\left(X\right)=0$

• A. All data values are equal.
• B. The data values are spread out over a wide range.
• C. The mean is $0$.
• D. There are both positive and negative values.

Answer: (A)

Verified step by step guidance

1. Recall the definition of variance: Variance measures how much the data values deviate from the mean. It is calculated as \(\sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2\), where \(x_i\) are the data values and \(\mu\) is the mean.
2. Understand that if the variance is zero, it means that the sum of squared deviations from the mean is zero. Since squared terms are always non-negative, the only way for their sum to be zero is if each individual term is zero.
3. This implies that for every data value \(x_i\), the difference from the mean \(x_i - \mu\) must be zero, meaning \(x_i = \mu\) for all \(i\).
4. Therefore, all data values must be exactly equal to the mean, indicating no spread or variability in the data.
5. Conclude that the correct statement is: 'All data values are equal.' The other options contradict the idea of zero variance.