a Calculate the mean, median, and mode for the data values after the student is hired. How does each measure of center vary from the previous one?
B
b Which of the measures of center splits the data evenly?
A
aMean: The additional value decreases the mean by about $0.13.
Median: The additional value decreases the median by $0.05. Mode: The additional value changes the mode to $8.25 and $8.45.
Explanation: See solution.
B
b The median best represents the data. See solution.
Practice makes perfect
a The following table represents the students' hourly wages that an amusement park hires for the summer.
Student's Hourly Wages
$16.50
$8.25
$8.75
$8.45
$8.65
$8.25
$9.10
$9.25
The given data has a mean of $9.65, a median of $8.7, and a mode of $8.25. We are asked how these measures will be affected if the park hires another student at an hourly wage of $8.45. Let's analyze each measure of center 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. After hiring the student, we will have 9 data values. Therefore we will divide the sum of the data by 9.
The mean would be about $9.52. Comparing the original mean, $9.65, to the new mean, we can see that the original mean is greater. Let's find the difference of these means.
$9.65- $9.52=$0.13
Therefore, after the student is hired the mean decreases by about $0.13.
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. Similarly, we first need to find the median after the student is hired. Let's write the data in numerical order.
8.25, 8.25, 8.45, 8.45, 8.65,
8.75, 9.10, 9.25, 16.50
We can see that the median is now $8.65. Comparing it with the original median, $8.70, we can see that the original median is greater. Let's find the difference of these medians.
$8.70- $8.65=$0.05
Therefore, after the student is hired the median decreases by $0.05.
Mode
The mode of a data set is the value or values that occur most often. Again, we first need to find the mode for the data with the new wage added. Let's take a look at the given values.
8.25, 8.25, 8.45, 8.45, 8.65,
8.75, 9.10, 9.25, 16.50
This time we have two values that occur most often. Therefore, instead of only having the mode $8.25, it will now be $8.25 and $8.45.
b To find which measure of center best describes the data, we will compare each measure with the given values. Let's first recall the data.
8.25, 8.25, 8.45, 8.45, 8.65,
8.75, 9.10, 9.25, 16.50
First, the mean $9.52 is greater than most of the data. Second, the modes $8.25 and $8.45 do not divide the data into equal groups. Finally, since the median $8.65 splits the data evenly it best represents the data.
