To determine the numerator in the calculation of variance and standard deviation, we must follow these steps:
1. Find the Mean of the Sample:
The areas of the houses in the sample are:
[tex]\[ 2400, 1750, 1900, 2500, 2250, 2100 \][/tex]
To find the mean area, sum all the values and divide by the number of houses:
[tex]\[
\text{Mean} = \frac{2400 + 1750 + 1900 + 2500 + 2250 + 2100}{6} = \frac{12900}{6} = 2150
\][/tex]
2. Calculate the Squared Deviations from the Mean for Each Area:
Subtract the mean from each area, then square the result:
- For [tex]\(2400\)[/tex]:
[tex]\[
(2400 - 2150)^2 = 250^2 = 62500
\][/tex]
- For [tex]\(1750\)[/tex]:
[tex]\[
(1750 - 2150)^2 = (-400)^2 = 160000
\][/tex]
- For [tex]\(1900\)[/tex]:
[tex]\[
(1900 - 2150)^2 = (-250)^2 = 62500
\][/tex]
- For [tex]\(2500\)[/tex]:
[tex]\[
(2500 - 2150)^2 = 350^2 = 122500
\][/tex]
- For [tex]\(2250\)[/tex]:
[tex]\[
(2250 - 2150)^2 = 100^2 = 10000
\][/tex]
- For [tex]\(2100\)[/tex]:
[tex]\[
(2100 - 2150)^2 = (-50)^2 = 2500
\][/tex]
3. Sum Up the Squared Deviations:
Add all the squared deviations to get the numerator for the variance calculation:
[tex]\[
62500 + 160000 + 62500 + 122500 + 10000 + 2500 = 420000
\][/tex]
Therefore, the option that represents the numerator in the calculation of variance and standard deviation is:
[tex]\[
\boxed{(250)^2+(-400)^2+(-250)^2+(350)^2+(100)^2+(-50)^2=420,000}
\][/tex]
