Question

calculate standard deviation
this is a multi - part item.
a contractor records the areas, in square feet, of a small sample of houses in a neighborhood to determine data about the neighborhood. they are:
2,400; 1,750; 1,900; 2,500; 2,250; 2,100
which of the following represents the numerator in the calculation of variance and standard deviation?
$(250)^2 + (-400)^2 + (-250)^2 + (350)^2 + (100)^2 + (-50)^2 = 420,0$
$(650)^2 + (-150)^2 + (-600)^2 + (250)^2 + (150)^2 + (-300)^2 = 980,$
$(225)^2 + (-425)^2 + (-275)^2 + (325)^2 + (75)^2 + (-75)^2 = 423,75

Explanation:

Step1: Calculate the mean

First, find the mean (\(\bar{x}\)) of the data set: \(2400, 1750, 1900, 2500, 2250, 2100\).
The sum of the data is \(2400 + 1750 + 1900 + 2500 + 2250 + 2100 = 12900\).
The mean \(\bar{x}=\frac{12900}{6} = 2150\).

Step2: Find deviations from the mean

For each data point, calculate \((x_i - \bar{x})\):

• \(2400 - 2150 = 250\)
• \(1750 - 2150 = -400\)
• \(1900 - 2150 = -250\)
• \(2500 - 2150 = 350\)
• \(2250 - 2150 = 100\)
• \(2100 - 2150 = -50\)

Step3: Square the deviations

Square each of the deviations:

• \((250)^2\)
• \((-400)^2\)
• \((-250)^2\)
• \((350)^2\)
• \((100)^2\)
• \((-50)^2\)

Step4: Sum the squared deviations

Sum these squared deviations: \((250)^2 + (-400)^2 + (-250)^2 + (350)^2 + (100)^2 + (-50)^2\). Calculating each term:

• \(250^2 = 62500\)
• \((-400)^2 = 160000\)
• \((-250)^2 = 62500\)
• \(350^2 = 122500\)
• \(100^2 = 10000\)
• \((-50)^2 = 2500\)

Summing them: \(62500 + 160000 + 62500 + 122500 + 10000 + 2500 = 420000\) (assuming the first option's "420,0" is a typo for 420000).

Answer:

The first option \((250)^2 + (-400)^2 + (-250)^2 + (350)^2 + (100)^2 + (-50)^2 = 420000\) (or as presented in the option with the corrected sum) represents the numerator (sum of squared deviations) for variance and standard deviation calculation.