To determine which formula should be used to calculate the standard deviation for the areas of houses recorded by the contractor, let's consider the nature of the data recorded. The contractor is likely using a sample of houses from the entire neighborhood rather than the whole population of houses.
When dealing with a sample, we use the sample standard deviation formula. The sample standard deviation accounts for the fact that the data represents only a subset of the total population, which involves using [tex]\( n - 1 \)[/tex] (where [tex]\( n \)[/tex] is the sample size) in the denominator. This adjustment is called Bessel's correction and helps to provide an unbiased estimate of the population standard deviation.
The formulas provided are:
1. [tex]\( s^2 = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1} \)[/tex] — This is the formula for the sample variance.
2. [tex]\( s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} \)[/tex] — This is the formula for the sample standard deviation.
3. [tex]\( \sigma^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_n - \mu)^2}{n} \)[/tex] — This is the formula for the population variance.
4. [tex]\( \sigma = \sqrt{\frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_n - \mu)^2}{n}} \)[/tex] — This is the formula for the population standard deviation.
Among these formulas:
- [tex]\( s \)[/tex] represents the sample standard deviation.
- [tex]\( \sigma \)[/tex] represents the population standard deviation.
- [tex]\( \bar{x} \)[/tex] is the sample mean.
- [tex]\( \mu \)[/tex] is the population mean.
Since the contractor is using a sample of houses to estimate data about the neighborhood:
- The correct formula involves [tex]\( s \)[/tex], not [tex]\( \sigma \)[/tex], because we are calculating the sample standard deviation.
- We also use [tex]\( n - 1 \)[/tex] in the denominator, which further narrows it down to a formula involving the sample calculation that reflects the sample standard deviation.
Therefore, the correct formula to use is:
[tex]\[ s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} \][/tex]
This is the second formula in the list. Thus, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
