Chapter 11.1, Problem 23E

(a)

To determine

To find the Standard Deviation of Golfer A.

Answer to Problem 23E

Standard Deviation = $4.63$

Explanation of Solution

Given information: $83,84,91,90,98,88,95,89,87,95$

Formula Used: Standard Deviation = $\sqrt{\frac{\sum {\left(\left|x_i-\mu \right|\right)}^2}{N}}$ and Mean $\mu =\frac{Sumofallobservations}{Totalobservations}$

Calculation:

  $83,84,91,90,98,88,95,89,87,95$

Now, in Ascending Order

  $83,84,87,88,89,90,91,95,95,98$

• Mean

Mean = $\frac{Sumofallobservations}{Totalobservations}$

So,

Mean = $\frac{83+84+87+88+89+90+91+95+95+98}{10}$

Now,

Mean = $\frac{900}{10}$

Hence,

Mean $\mu =90$

• Standard Deviation

  $x_i=83,84,87,88,89,90,91,95,95,98$

and

Mean $\mu =90$

So,

  $\left|x_i-\mu \right|=7,6,3,2,1,0,1,5,5,8$

Now,

  ${\left(\left|x_i-\mu \right|\right)}^2=49,36,9,4,1,0,1,25,25,64$

So,

Standard Deviation = $\sqrt{\frac{\sum {\left(\left|x_i-\mu \right|\right)}^2}{N}}$

Now,

Standard Deviation = $\sqrt{\frac{49+36+9+4+1+0+1+25+25+64}{10}}$

So,

Standard Deviation = $\sqrt{\frac{214}{10}}$

Now,

Standard Deviation = $\sqrt{21.4}$

Hence,

Standard Deviation = $4.63$

(b)

To determine

To find the Standard Deviation of Golfer B.

Answer to Problem 23E

Standard Deviation = $2.53$

Explanation of Solution

Given information: $89,93,92,88,89,87,95,94,91,92$

Formula Used: Standard Deviation = $\sqrt{\frac{\sum {\left(\left|x_i-\mu \right|\right)}^2}{N}}$ and Mean $\mu =\frac{Sumofallobservations}{Totalobservations}$

Calculation:

  $89,93,92,88,89,87,95,94,91,92$

Now, in Ascending Order

  $87,88,89,89,91,92,92,93,94,95$

• Mean

Mean = $\frac{Sumofallobservations}{Totalobservations}$

So,

Mean = $\frac{87+88+89+89+91+92+92+93+94+95}{10}$

Now,

Mean = $\frac{910}{10}$

Hence,

Mean $\mu =91$

• Standard Deviation

  $x_i=87,88,89,89,91,92,92,93,94,95$

and

Mean $\mu =91$

So,

  $\left|x_i-\mu \right|=4,3,2,2,0,1,1,2,3,4$

Now,

  ${\left(\left|x_i-\mu \right|\right)}^2=16,9,4,4,0,1,1,4,9,16$

So,

Standard Deviation = $\sqrt{\frac{\sum {\left(\left|x_i-\mu \right|\right)}^2}{N}}$

Now,

Standard Deviation = $\sqrt{\frac{16+9+4+4+0+1+1+4+9+16}{10}}$

So,

Standard Deviation = $\sqrt{\frac{64}{10}}$

Now,

Standard Deviation = $\sqrt{6.4}$

Hence,

Standard Deviation = $2.53$

(c)

To determine

Compare the Standard Deviation of Golfer A and B.

Answer to Problem 23E

Standard Deviation of Golfer A > Standard Deviation of Golfer B

Explanation of Solution

Given information: Standard Deviation of Golfer A = $4.63$ and Standard Deviation of Golfer B = $2.53$

Formula Used: Compatibility

Calculation:

Standard Deviation of Golfer A = $4.63$ and Standard Deviation of Golfer B = $2.53$

Hence, compare

Standard Deviation of Golfer A > Standard Deviation of Golfer B