Question

Calculate the range, population variance, and population standard deviation for the following data set. If necessary, round to one more decimal place than the largest number of decimal places given in the data. 2,3,4,5,6,7,8,9,10,11

Answer 1

Answer:

The range of the data is 9.0.

The population variance is 8.25.

The population standard deviation is 2.87.

Step-by-step explanation:

The data set is: S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

(A)

The range of a data set is the difference between the maximum and minimum value of the data set.

Compute range as follows:

$Range=Max.-Min.=11-2=9$

Thus, the range of the data is 9.0.

(B)

The formula to compute the population variance is:

$Var=\frac{1}{n} \sum (x_{i}-\bar x)^{2}$

Compute the mean of the data set:

$\bar x=\frac{1}{n} \sum x_{i}\\=\frac{1}{10}(2+3+4+5+6+7+8+9+10+11)\\=6.5$

Compute the variance as follows:

$Var=\frac{1}{n} \sum (x_{i}-\bar x)^{2}\\=\frac{1}{10}[(2-6.5)^{2}+(3-6.5)^{2} +(4-6.5)^{2}+...(11-6.5)^{2}]\\=8.25$

Thus, the population variance is 8.25.

(C)

The population standard deviation is:

$SD=\sqrt{Var} = \sqrt{\frac{1}{n} \sum (x_{i}-\bar x)^{2}}$

Compute the population standard deviation of the data set:

$SD=\sqrt{Var} = \sqrt{8.25}=2.87$

Thus, the population standard deviation is 2.87.