The mean of a data set is the sum of the data divided by the number of values. Therefore, to find the mean we first need to calculate how many students we have that are 18, 19, 20, 21, and 37. To do so let's use the percentages.
Years
Number of Students ( %)
Number of Students
18
35 %
70
19
30 %
60
20
14 %
28
21
20 %
40
37
1 %
2
The mean will be the sum of the product of each age and the number of students that are that age, divided by the total number of students.
The median of a numerical data set is the middle number when the values are written in numerical order. If there is an even number of values, the mean of the two middle values is the median. Let's order the given values to find the median.
18, 18,...,18,_(70)19,19,...,19,_(60)
20, 20, ..., 2,_(28)21,21,..., 21,_(40values)37,37
In this case we have an even number of values, 200. Therefore, the mean of the 100^\text{th} and 101^\text{th} ages will be the median. Observing the ordered data, we can see that both of these values are 19.
Median: 19+19/2=19 years
Mode
The mode of a data set is the value or values that occur most often. Again, let's observe the given values.
18, 18,...,18,_(70)19,19,...,19,_(60)
20, 20, ..., 2,_(28)21,21,..., 21,_(40values)37,37
We can see that the age of 18 is the most repeated value. Therefore, the mode is 18 years.
b Observing the given data, we can see that 37 is the value that occurs least often. Additionally, this values are much greater than the other values. Therefore, these two values are the outliers. Let's find the measures of center after removing the outliers.
Mean
Let's first remove the outliers from the original data
18, 18,...,18,_(70)19,19,...,19,_(60)
20, 20, ..., 20,_(28)21,21,..., 21,_(40values)37,37
⇓
18, 18,...,18,_(70)19,19,...,19,_(60)
20, 20, ..., 20,_(28)21,21,..., 21,_(40values)
This time we have 198 data values.
Therefore, the mean without the outliers is about 19.19 years. The mean with the outliers, 19.37 years, is greater. Let's calculate the difference of these means.
19.37-19.19=0.18 years
Therefore, the outliers increase the mean by about 0.18 years.
Median
Let's take a look at our data to find the median.
18, 18,...,18,_(70)19,19,...,19,_(60)
20, 20, ..., 20,_(28)21,21,..., 21_(40values)
In this case the mean of the 99^\text{th} and 100^\text{th} ages will be the median. Both values are 19.
Median: 19+19/2=19years
In this case, the outliers do not affect the original median, since is still 19 years.
Mode
Again, let's observe the data to find the mode.
18, 18,...,18,_(70)19,19,...,19,_(60)
20, 20, ..., 20,_(28)21,21,..., 21_(40values)
We can see that the mode is still 18 years. Therefore, the outliers do not affect the mode.
c All 200 students take the same Psychology II class exactly 1 year later. We are asked to draw a new circle showing the distribution of the ages. In this case, the distribution is the same, except that each age is increased by 1.
We will find the mean, median and mode of this distribution. To do so, let's first recall that when a real number k is added to each value in a numerical data set, the measures of center can be found by adding k to the original measures of center. Therefore, we can add 1 to each measure we found previously.
Mean: &19.37+ 1=20.37
Median: &19+ 1=20
Mode: &18+ 1=19
