Exercise 1 Page 775

Consider the given data set. We want to find the mean, median, mode, range, and standard deviation of data set obtained by adding the given constant, $k=\text{-}7,$ to each value. If every value in the data set is increased by the constant $\text{-}7,$ then the statistics of the new data set will behave in a consistent, predictable way.

• The new mean, median, and mode can be found by adding $\text{-}7$ to the mean, median, and mode of the original data set.
• The range and standard deviation will not change.

Notice that only the measures of center are increased by the constant and the measures of spread will not change. This is because the distances between the individual values do not change. 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 $10.5.$

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 $8$, 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 $15$ and the least value is $7.$

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 $\overline{x}=10.5.$ Let's apply the formula to each value in the set.

$x$	$x-\overline{x}$	$(x-\overline{x}{)}^2$
$10$	$10-10.5=\text{-}0.5$	$(\text{-}0.5{)}^2=0.25$
$13$	$13-10.5=2.5$	$2.5^2=6.25$
$9$	$9-10.5=\text{-}1.5$	$(\text{-}1.5{)}^2=2.25$
$8$	$8-10.5=\text{-}2.5$	$(\text{-}2.5{)}^2=6.25$
$15$	$15-10.5=4.5$	$4.5^2=20.25$
$8$	$8-10.5=\text{-}2.5$	$(\text{-}2.5{)}^2=6.25$
$13$	$13-10.5=2.5$	$2.5^2=6.25$
$12$	$12-10.5=1.5$	$1.5^2=2.25$
$7$	$7-10.5=\text{-}3.5$	$(\text{-}3.5{)}^2=12.25$
$8$	$8-10.5=\text{-}2.5$	$(\text{-}2.5{)}^2=6.25$
$11$	$11-10.5=0.5$	$0.5^2=0.25$
$12$	$12-10.5=1.5$	$1.5^2=2.25$
Sum of Values		$=71$

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

Adding a Constant

Finally, we can find new values of the statistics by adding $\text{-}7$ to the mean, median, and mode.

Statistic	Original Value	Required Change	New Value
Mean	$10.5$	$10.5+(\text{-}7)$	$3.5$
Median	$10.5$	$10.5+(\text{-}7)$	$3.5$
Mode	$8$	$8+(\text{-}7)$	$1$
Range	$8$	-	$8$
Standard Deviation	$2.43$	-	$2.43$