Henri surveys his 20 classmates about the number of times they go swimming per week during the summer.

He finds that 6 of his classmates go less than 2 times a week, while 9 of his classmates go more than 10 times a week. The other 5 classmates go between 4-6 times a week.

Would the mean be greater than or less than the median?

the answer is greater than the median

Answer: even with a contrived example yielding minimal mean for a given median necessarily between 4 and 6, the mean is always greater than the median.

Step-by-step explanation:

Total number of students: 20

c (1 through 6) < 2

4 < = c (7 through 11) < = 6

10 < c (12 through 20)

Median is sum of students 10 and 11 divided by 2.

Median is < = 6

Minimal mean = (6*0+5*4+9*11) / 20 = 5.95

Maximal median for that mean = (4+4) / 2=4, smaller.

Maximal median for any mean = (6+6) / 2=6

Minimal mean for that median = (6*0+3*4+2*6+9*11) / 20 = 6.15

General case: given

median = (c (10) + c (11)) / 2

mean = (6*0+3*4+2*median+9*11) / 20

= (12+99+2*median) / 20

= (12+99) / 20 + 2*median/20

= 111/20 + median/10

Let x = median.

mean > median when

111/20+x/10 > x

111/20 > 9x/10

111*10 / (9*20) > x

6.1667 > x

But the median has to be < = 6, so the mean is necessarily greater than the median.