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

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.