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 usually happens when you attach images or files into an email?
A
aOutlier: 46 kilobytes
Mean: The outlier increases the mean by about 4.3 kilobytes. Median: The outlier increases the mean by 0.5 kilobytes.
Mode: The outlier does not affect the mode.
B
bSample Answer: It represents an email with an attached image.
Practice makes perfect
a The following data represents the sizes of emails (in kilobytes) in our inboxs.
2, 3, 5, 2, 1, 46, 3, 7, 2, 1
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, 46 is much greater than the other sizes. 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 10 data values, the mean will be the sum of the data values divided by 10.
Now, we need to remove the outlier from the given values.
2, 3, 5, 2, 1, 46, 3, 7, 2, 1
⇓
2, 3, 5, 2, 1, 3, 7, 2, 1
Proceeding in the same way, we can calculate the mean for the values without the outlier. In this case, we have 9 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.
7.2-2.9=4.3
Therefore, the outlier increases the mean by about 4.3 kilobytes.
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.
1,1,2,2, 2, 3,3,5,7,46
Since we have an even number of data values, we have to calculate the mean of the two middle values.
Median&=2+ 3/2
&⇓
Median&=2.5
Now, let's take a look at the given values without the outlier.
1,1,2,2, 2,3,3,5,7
In this case, the median is 2 kilobytes. Comparing the medians, we can see that the median with the outlier is greater. Let's calculate the difference of these medians.
2.5-2=0.5
Therefore, the outlier increases the median by 0.5 kilobytes.
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.
1,1, 2, 2, 2,3,3,5,7,46
Note that the mode is 2 kilobytes. Now, let's see at our data without the outlier.
1,1, 2, 2, 2,3,3,5,7
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 email that has an image. Please note that there are many possible explanations. Here we are only considering one possibility.
