Question 631·Easy·Math

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Number of High School Students Who Completed Summer Internships

High school	2008	2009	2010	2011	2012
Foothill	87	80	75	76	70
Valley	44	54	65	76	82
Total	131	134	140	152	152

The table above shows the number of students from two different high schools who completed summer internships in each of five years. No student attended both schools.

Which of the following statements are true about the number of students who completed summer internships for the 5 years shown?

I. The mean number from Foothill High School is greater than the mean number from Valley High School.

II. The median number from Foothill High School is greater than the median number from Valley High School.

A table that asks about both the mean and the median is asking for two different centers of each school's yearly counts. Enter each school's five counts as a Desmos list and compare the means. Then compare the middle values after sorting. A mean uses every count; a median uses the middle after sorting. One year's count cannot stand in for either five-year center.

Hints

Hint 1
The mean is the total split evenly among the values. Both schools have five yearly counts, so enter each school's row as a Desmos list and compare the means of those lists.
Hint 2
The median is the middle value after sorting. The years are in order, but Foothill's counts aren't. Which count is third when you put each school's five counts in order from least to greatest?

Step-by-step

Compare two Desmos lists
Step 1
Find Foothill's mean
Use Foothill's row, not the Total row. Type $a=[87,80,75,76,70]$ in Desmos, then $\operatorname{mean}(a)$ . The mean splits the sum of those five counts evenly across the five years. Desmos shows $77.6$ students per year.
Step 2
Compare Valley's mean
Type Valley's five counts as $b=[44,54,65,76,82]$ , then type $\operatorname{mean}(b)$ . Desmos shows $64.2$ . Since $77.6>64.2$ , Foothill's mean is greater, so statement I is true.
Step 3
Find Foothill's middle count
Type $\operatorname{median}(a)$ . Desmos shows $76$ : Foothill's counts in order are $70,75,76,80,87$ , so $76$ is third. The median is the middle after sorting the counts. Don't pick $75$ because 2010 is the middle year.
Step 4
Compare Valley's middle count
Type $\operatorname{median}(b)$ . Desmos shows $65$ , the third count in Valley's ordered list $44,54,65,76,82$ . Foothill's median is greater because $76>65$ , so statement II is true too. Both statements hold for the five years shown. Choice C.

Question 631·Easy·Math

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