Exercise 4 Page 775

Consider the given data set. 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. 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. Since there are $12$ values, there is no one middle value. Therefore, the median is the mean of the two middle values.

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. 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. For this set, the greatest value is $49$ and the least value is $36.$

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. 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.

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

Finally, since $n=12,$ we need to divide by $12$ and then calculate the square root.

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	$3.67$	$3.67\cdot 0.5$	$1.84$