To determine whether the statement "The variance (if it exists) is always the square of the standard deviation" is true or false, let's break it down step by step with a detailed explanation:
1. Definition of Variance:
- Variance is a measure of how much the values in a data set differ from the mean (average) of the data set.
- Mathematically, for a population, the variance ([tex]\(\sigma^2\)[/tex]) is defined as the average of the squared differences from the mean:
[tex]\[
\sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2
\][/tex]
where [tex]\(x_i\)[/tex] are the data points, [tex]\(\mu\)[/tex] is the mean of the data points, and [tex]\(N\)[/tex] is the number of data points.
2. Definition of Standard Deviation:
- Standard deviation is a measure of the amount of variation or dispersion in a set of values.
- It is the square root of the variance:
[tex]\[
\sigma = \sqrt{\sigma^2}
\][/tex]
3. Relationship between Variance and Standard Deviation:
- Since the standard deviation ([tex]\(\sigma\)[/tex]) is defined as the square root of the variance ([tex]\(\sigma^2\)[/tex]), squaring the standard deviation will give you back the variance:
[tex]\[
\sigma^2 = (\sqrt{\sigma^2})^2
\][/tex]
- Therefore, by definition, the variance is indeed always the square of the standard deviation.
Given this detailed analysis and explanation, we can conclude that the statement "The variance (if it exists) is always the square of the standard deviation" is:
True
