Chapter 11.1, Problem 37E

(a)

To determine

To find the mean, median, mode, range and standard deviation of data set.

Answer to Problem 37E

Mean = $4$

Median = $3$

Mode = $1$

Range = $14$

Standard Deviation = $4.15$

Explanation of Solution

Given information: $3,5,1,5,1,1,2,3,15$

Formula Used:Mean $\mu =\frac{Sumofallobservations}{Totalobservations}$ , Median = ${\left(\frac{n+1}{2}\right)}^{th}term$ , Mode = Most repeated value, Range = Highest − Lowest and Standard Deviation = $\sqrt{\frac{\sum {\left(\left|x_i-\mu \right|\right)}^2}{N}}$

Calculation:

  $3,5,1,5,1,1,2,3,15$

Now, in Ascending Order

  $1,1,1,2,3,3,5,5,15$

• Mean

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

So,

Mean = $\frac{1+1+1+2+3+3+5+5+15}{9}$

Now,

Mean = $\frac{36}{9}$

Hence,

Mean $\mu =4$

• Median

Median = ${\left(\frac{n+1}{2}\right)}^{th}term$

So,

Median = ${\left(\frac{9+1}{2}\right)}^{th}term$

Now,

Median = ${\left(5\right)}^{th}term$

As, ${\left(5\right)}^{th}term=3$

Hence,

Median = $3$

• Mode

Mode = Most repeated value.

So,

1 is the most repeated value.

Mode = $1$

• Range

Range = Highest − Lowest

So,

Range = $15-1$

Hence,

Range = $14$

• Standard Deviation

  $x_i=1,1,1,2,3,3,5,5,15$

and

Mean $\mu =4$

So,

  $\left|x_i-\mu \right|=3,3,3,2,1,1,1,1,11$

Now,

  ${\left(\left|x_i-\mu \right|\right)}^2=9,9,9,4,1,1,1,1,121$

So,

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

Now,

Standard Deviation = $\sqrt{\frac{9+9+9+4+1+1+1+1+121}{9}}$

So,

Standard Deviation = $\sqrt{\frac{155}{9}}$

Now,

Standard Deviation = $\sqrt{17.2}$

Hence,

Standard Deviation = $4.15$

(b)

To determine

To find the mean, median, mode, range and standard deviation of data set when height 7 feet join

Answer to Problem 37E

Mean = $4.3$

Median = $3$

Mode = $1$

Range = $14$

Standard Deviation = $4.05$

Explanation of Solution

Given information: $3,5,1,5,1,1,2,3,15,7$

Formula Used:Mean $\mu =\frac{Sumofallobservations}{Totalobservations}$ , Median = $\frac{{\left(\frac{n}{2}\right)}^{th}+{\left(\frac{n}{2}+1\right)}^{th}}{2}term$ , Mode = Most repeated value, Range = Highest − Lowest and Standard Deviation = $\sqrt{\frac{\sum {\left(\left|x_i-\mu \right|\right)}^2}{N}}$

Calculation:

  $3,5,1,5,1,1,2,3,15,7$

Now, in Ascending Order

  $1,1,1,2,3,3,5,5,7,15$

• Mean

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

So,

Mean = $\frac{1+1+1+2+3+3+5+5+7+15}{10}$

Now,

Mean = $\frac{43}{10}$

Hence,

Mean $\mu =4.3$

• Median

Median for even number = $\frac{{\left(\frac{n}{2}\right)}^{th}+{\left(\frac{n}{2}+1\right)}^{th}}{2}term$

So,

Median = $\frac{{\left(\frac{10}{2}\right)}^{th}+{\left(\frac{10}{2}+1\right)}^{th}}{2}term$

Now,

Median = $\frac{{\left(5\right)}^{th}+{\left(6\right)}^{th}}{2}term$

As, ${\left(5\right)}^{th}term=3$ and ${\left(6\right)}^{th}term=3$

So,

Median = $\frac{3+3}{2}$

Hence,

Median = $3$

• Mode

Mode = Most repeated value.

So,

1 is the most repeated value.

Mode = $1$

• Range

Range = Highest − Lowest

So,

Range = $15-1$

Hence,

Range = $14$

• Standard Deviation

  $x_i=1,1,1,2,3,3,5,5,7,15$

and

Mean $\mu =4.3$

So,

  $\left|x_i-\mu \right|=3.3,3.3,3.3,2.3,1.3,1.3,0.7,0.7,2.7,10.7$

Now,

  ${\left(\left|x_i-\mu \right|\right)}^2=10.89,10.89,10.89,5.29,1.69,1.69,0.49,0.49,7.29,114.49$

So,

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

Now,

Standard Deviation = $\sqrt{\frac{10.89+10.89+10.89+5.29+1.69+1.69+0.49+0.49+7.29+114.49}{10}}$

So,

Standard Deviation = $\sqrt{\frac{164.1}{10}}$

Now,

Standard Deviation = $\sqrt{16.41}$

Hence,

Standard Deviation = $4.05$