To solve the problem, let's go through it step-by-step.
### Step-by-Step Solution
1. List of Areas:
We have the areas in square feet: [tex]\( 2400, 1750, 1900, 2500, 2250, 2100 \)[/tex].
2. Calculate the Mean:
First, calculate the mean of these areas.
[tex]\[
\text{Mean} = \frac{2400 + 1750 + 1900 + 2500 + 2250 + 2100}{6} = \frac{12900}{6} = 2150 \, \text{square feet}
\][/tex]
3. Calculating Squared Deviations from the Mean:
Next, calculate how much each value deviates from the mean, square these deviations, and sum them up.
- Deviation for 2400: [tex]\( (2400 - 2150) = 250 \)[/tex]
[tex]\[
(250)^2 = 62500
\][/tex]
- Deviation for 1750: [tex]\( (1750 - 2150) = -400 \)[/tex]
[tex]\[
(-400)^2 = 160000
\][/tex]
- Deviation for 1900: [tex]\( (1900 - 2150) = -250 \)[/tex]
[tex]\[
(-250)^2 = 62500
\][/tex]
- Deviation for 2500: [tex]\( (2500 - 2150) = 350 \)[/tex]
[tex]\[
(350)^2 = 122500
\][/tex]
- Deviation for 2250: [tex]\( (2250 - 2150) = 100 \)[/tex]
[tex]\[
(100)^2 = 10000
\][/tex]
- Deviation for 2100: [tex]\( (2100 - 2150) = -50 \)[/tex]
[tex]\[
(-50)^2 = 2500
\][/tex]
Adding up these squared deviations:
[tex]\[
62500 + 160000 + 62500 + 122500 + 10000 + 2500 = 420000
\][/tex]
Therefore, the correct numerator in the calculation of variance and standard deviation is [tex]\( 420000 \)[/tex].
4. Calculating Variance:
Variance is calculated by dividing the sum of squared deviations by the number of observations.
[tex]\[
\text{Variance} = \frac{420000}{6} = 70000 \, \text{square feet}^2
\][/tex]
5. Calculating Standard Deviation:
The standard deviation is the square root of the variance.
[tex]\[
\text{Standard Deviation} = \sqrt{70000} \approx 264.575
\][/tex]
Rounding to the nearest whole number, the standard deviation is [tex]\( 265 \)[/tex].
### Summary
- The numerator in the calculation of variance and standard deviation is [tex]\( 420000 \)[/tex].
- The variance is [tex]\( 70000 \, \text{square feet}^2 \)[/tex].
- The standard deviation, rounded to the nearest whole number, is [tex]\( 265 \)[/tex].
