Question

Find the population variance for the following data points. 10 15 16 16 16 19 25What is x bar: _____What is the sum of the square deviations: _____What is the population variance: _____Round all to the nearest tenth.

Answer 1

Let's recall the formulas for getting the different values:

`x\text{ bar = }\bar{x}\text{ = }(\Sigma x)/(n)`
`\text{Sum of the square deviations = }\Sigma(\text{x - }\bar{\text{x}})^2`
`\text{Population variance = }\frac{\Sigma(\text{x - }\bar{\text{x}})^2}{n}`

A.) Let's now get the x bar (mean):

`x\text{ bar = }\bar{x}\text{ = }(\Sigma x)/(n)`
`\bar{x}\text{ = }\frac{10\text{ + 15 + 16 + 16 + 16 + 19 + 25}}{7}\text{ = }(117)/(7)=\text{ 16.7142857143 }\approx\text{ 16.7}`

B.) Let's get the sum of the square deviations:

`\text{Sum of the square deviations = }\Sigma(\text{x - }\bar{\text{x}})^2`
`\text{ = (10-16.7)}^2+(15-16.7)^2+(16-16.7)^2(3)+(19-16.7)^2+(25-16.7)^2`
`=44.9\text{ + 2.9 + (0.5)(3) + 5.3 + 68.9}`
`=44.9\text{ + 2.9 + 1.5 + 5.3 + 68.9}`
`\Sigma(\text{x - }\bar{\text{x}})^2\text{ = 123.5}`

C.) Let's get the population variance:

`\text{Population variance = }\frac{\Sigma(\text{x - }\bar{\text{x}})^2}{n}\text{ ; but }\Sigma(\text{x - }\bar{\text{x}})^2\text{ = 123.40 and n = 7}`
`\frac{\Sigma(\text{x - }\bar{\text{x}})^2}{n}\text{ = }(123.5)/(7)`
`\frac{\Sigma(\text{x - }\bar{\text{x}})^2}{n}\text{ = 17.64285714286 }\approx\text{ 17.60}`