a Start by finding the outlier. Then calculate the mean, median, and mode for the given values with and without the outlier.
B
b What could be the reasons for getting paid more than the others?
A
aOutlier: $72 000
Mean: The outlier increases the mean by about $4161. Median: The outlier increases the median by $1500.
Mode: The outlier does not affect the mode.
B
bSample Answer: It represents an employee's salary who has more responsibilities.
Practice makes perfect
a We are given a table that represents the annual salaries of employees of an auto repair service.
Annual Salaries
$ 32 000
$ 42 000
$ 41 000
$ 38 000
$ 38 000
$ 45 000
$ 72 000
$ 35 000
We are asked to find the outlier of the given data set. An outlier is a data value that is much greater than or much less than the other values in a data set. In this case, $72 000 is much greater than the other salaries. Therefore, it is the outlier. To find out how this value affects the measures of center, let's analyze each measure one at a time.
Mean
The mean of a numerical data set is the sum of the data divided by the number of data values. Let's calculate the mean without removing the outlier. Since we have 8 data values the mean will be the sum of the data values divided by 8.
Now we need to remove the outlier from the given values.
32 000,41 000,38 000,72 000,
42 000,38 000,45 000,35 000
⇓
32 000,41 000,38 000,42 000,
38 000,45 000,35 000
Proceeding in the same way, we can calculate the mean for the values without the outlier. In this case, we have 7 data values.
We can see that the mean of the values with the outlier is greater than the mean of the values without the outlier. Let's find the difference of these means.
$42 875-$38 714=$4161
Therefore, the outlier increases the mean by about $4161.
Median
The median of a numerical data set is the middle number when the values are written in numerical order. If the data set has an even number of values, the mean of the two middle values will be the median. Let's first find the median without removing the outlier. To do so, let's write the given values in numerical order.
32 000, 35 000, 38 000, 38 000,
41 000, 42 000, 45 000, 72 000
Since we have an even number of data values, we have to calculate the mean of the two middle values.
Median&=38 000+ 42 000/2
&⇓
Median&=$39 500
Now, let's take a look at the given values without the outlier.
32 000, 35 000, 38 000, 38 000,
41 000, 42 000, 45 000
In this case, the median is $38 000. Comparing the medians, we can see that the median with the outlier is greater. Let's calculate the difference of these medians.
$39 500-$38 000=$1500
Therefore, the outlier increases the median by $1500.
Mode
The mode of a data set is the value or values that occur most often. Again, we will first find the mode for the data without removing the outlier. Let's take a look at the given values.
32 000, 35 000, 38 000, 38 000,
41 000, 42 000, 45 000, 72 000
Note that the mode is $38 000. Now, let's see at our data without the outlier.
32 000, 35 000, 38 000, 38 000,
41 000, 42 000, 45 000
We can see that the mode is the same. Therefore, the outlier does not affect the mode.
b One possible explanation for the outlier could be that it represents an employee's salary who has more responsibilities. Please note that there are many possible explanations. Here we are only considering one possibility.
