Exercise 4 Page 775

Consider the given data set.

42, 39, 45, 44, 37, 42, 38, 37, 41, 49, 42, 36

We want to find the mean, median, mode, range, and standard deviation of data set obtained by multiplying the given constant,

k = 0.5,

by each value. If every value in the data set is multiplied by the constant

0.5,

the mean, median, mode, range, and standard deviation of the new data set can be found by multiplying each original statistic by

0.5 .

To begin, let's find the statistics of the original data set.

Mean

The mean of a data set

\overline{x}

is calculated by finding the sum of all of the values in the set and then dividing by the number of values in the set. In this case, there are

12

values in the set.

Mean = \frac{Sum of values}{Number of values}

SubstituteValues

Substitute values

\overline{x} = \frac{4 2 + 3 9 + 4 5 + 4 4 + 3 7 + 4 2 + 3 8 + 3 7 + 4 1 + 4 9 + 4 2 + 3 6}{1 2}

AddTerms

Add terms

\overline{x} = \frac{4 9 2}{1 2}

CalcQuot

Calculate quotient

\overline{x} = 41

The mean of the set is

41 .

Median

When the data are arranged in numerical order, the median is the middle value — or the mean of the two middle values. Let's arrange the given values and find the median.

36, 37, 37, 38, 39, 41 ∣ 42, 42, 42, 44, 45, 49

Since there are

12

values, there is no one middle value. Therefore, the median is the mean of the two middle values.

Median : \frac{4 1 + 4 2}{2} = 41.5

Mode

The mode is the value or values that appear most often in a set of data. Let's find the mode of the given values.

42, 39, 45, 44, 37, 42, 38, 37, 41, 49, 42, 36

The value that appears most often is

42

, so this is our mode.

Range

The range is the difference between the least and greatest values in a set of data.

42, 39, 45, 44, 37, 42, 38, 37, 41, 49, 42, 36

For this set, the greatest value is

49

and the least value is

36 .

Range : 49 - 36 = 13

Standard Deviation

The standard deviation of a set of data is the average amount by which each individual value deviates or differs from the mean.

\underline{Standard Deviation} \frac{( x _{1} - x ) ^{2} + ( x _{2} - x ) ^{2} + \dots + ( x _{n} - x ) ^{2}}{n}

In this formula,

x_{1}, \dots, x_{n}

are the values of the set of data,

\overline{x}

is the mean, and

n

is the number of values. We have

12

values and the mean is

\overline{x} = 41 .

Let's apply the formula to each value in the set.

$x$	$x - \overline{x}$	$(x - \overline{x})^{2}$
$42$	$42 - 41 = 1$	$1^{2} = 1$
$39$	$39 - 41 = - 2$	$(- 2)^{2} = 4$
$45$	$45 - 41 = 4$	$4^{2} = 16$
$44$	$44 - 41 = 3$	$3^{2} = 9$
$37$	$37 - 41 = - 4$	$(- 4)^{2} = 16$
$42$	$42 - 41 = 1$	$1^{2} = 1$
$38$	$38 - 41 = - 3$	$(- 3)^{2} = 9$
$37$	$37 - 41 = - 4$	$(- 4)^{2} = 16$
$41$	$41 - 41 = 0$	$0^{2} = 0$
$49$	$49 - 41 = 8$	$8^{2} = 64$
$42$	$42 - 41 = 1$	$1^{2} = 1$
$36$	$36 - 41 = - 5$	$(- 5)^{2} = 25$
Sum of Values		$= 162$

Finally, since

n = 12,

we need to divide by

12

and then calculate the square root.

Standard Deviation : \frac{1 6 2}{1 2} \approx 3.67

Multiplying by a Constant

Finally, we can find new values of the statistics by multiplying each value by $0.5 .$

Statistic	Original Value	Required Change	New Value
Mean	$41$	$41 \cdot 0.5$	$20.5$
Median	$41.5$	$41.5 \cdot 0.5$	$20.75$
Mode	$42$	$42 \cdot 0.5$	$21$
Range	$13$	$13 \cdot 0.5$	$6.5$
Standard Deviation	$3.67$	$3.67 \cdot 0.5$	$1.84$