To calculate the standard deviation of the areas of several houses in a neighborhood, we need to use the formula that measures the dispersion of a sample.
1. First, let's understand the different elements in the problem:
- [tex]\( x_i \)[/tex] represents the individual data points (areas of the houses).
- [tex]\( \bar{x} \)[/tex] represents the sample mean (average area of the houses).
- [tex]\( n \)[/tex] is the number of data points (houses measured).
- [tex]\( s \)[/tex] is the sample standard deviation.
- [tex]\( \sigma \)[/tex] is the population standard deviation.
- [tex]\( \mu \)[/tex] is the population mean.
2. The correct formula to calculate the sample standard deviation takes into account n-1 (degrees of freedom) and is expressed as:
[tex]\[ s = \sqrt{\frac{\left(x_1-\bar{x}\right)^2+\left(x_2-\bar{x}\right)^2+\ldots+\left(x_n-\bar{x}\right)^2}{n-1}} \][/tex]
This formula calculates the square root of the average of the squared deviations of each data point from the sample mean.
3. The alternative options provided involve different computations:
- [tex]\( s^2 = \frac{\left(x_1-\bar{x}\right)^2+\left(x_2-\bar{x}\right)^2+\ldots+\left(x_n-\bar{x}\right)^2}{n-1} \)[/tex]: This is the formula for the sample variance, not the standard deviation.
- [tex]\( \sigma^2 = \frac{\left(x_1-\mu\right)^2+\left(x_2-\mu\right)^2+\ldots+\left(x_n-\mu\right)^2}{n} \)[/tex]: This is the population variance, using the population mean [tex]\( \mu \)[/tex] and the total number of data points [tex]\( n \)[/tex].
- [tex]\( \sigma = \sqrt{\frac{\left(x_1-\mu\right)^2+\left(x_2-\mu\right)^2+\ldots+\left(x_n-\mu\right)^2}{n}} \)[/tex]: This is the formula for the population standard deviation, which uses the population mean [tex]\( \mu \)[/tex] and [tex]\( n \)[/tex] as the divisor.
4. Given that we are dealing with a sample of house areas to understand the neighborhood, we need the sample standard deviation formula.
Therefore, the correct formula to use in this case is:
[tex]\[ s = \sqrt{\frac{\left(x_1-\bar{x}\right)^2+\left(x_2-\bar{x}\right)^2+\ldots+\left(x_n-\bar{x}\right)^2}{n-1}} \][/tex]
