- Understand the variance formula: \(V = \sigma^2\).
- Substitute the standard deviation: \(V = (2.5)^2 = 6.25\).
- Round the result to the nearest tenth: \(6.3\).
- Final answer: \(\boxed{6.3}\).
Explanation
- Understand the relationship between variance and standard deviation
To find the variance of a data set, we need to understand the relationship between variance and standard deviation. The variance, denoted by \(V\), is defined as the square of the standard deviation, denoted by \(\sigma\). Mathematically, this relationship is expressed as:
\[V = \sigma^2\]
Given in the problem, the standard deviation is \(\sigma = 2.5\).
- Calculate the variance
Now, we substitute the given value of the standard deviation into the variance formula:
\[V = 2.5^2\]
Performing the multiplication:
\[2.5 \times 2.5 = 6.25\]
The variance of the data set is \(6.25\).
- Round and select the correct option
The question asks to round the variance to the nearest tenth. Looking at our calculated value of \(6.25\), the digit in the tenths place is 2, and the digit in the hundredths place is 5. According to the standard rounding rules, since the digit in the hundredths place is 5 or greater, we round up the tenths digit. Therefore, \(6.25\) rounds to \(6.3\). Comparing this to the given options, we select the correct one.
Examples
In experimental physics, knowing the variance helps researchers understand the spread of their measurements. For example, if students measure the acceleration due to gravity, the standard deviation tells them how consistent their results are, while the variance provides a measure of total error dispersion, helping them evaluate the reliability of their lab experiments. If they obtain \(\sigma = 2.5 \text{ m/s}^2\), then the variance is \(\sigma^2 = 6.25 \text{ m}^2/\text{s}^4\).