mean, median and mode - which is bigger?

for a symmetrical distribution, mean = median = mode
(one simple way of checking for a normal distribution but you need more information)

for a (simple, not overly lumpy) negatively skewed distribution
[mean - mode] is negative (since mode is bigger then mean)
the median is greater than the mean

(averages are the x-values)














for a positively skewed distribution
the mean is greater than the median

for a small data set of five numbers
the data needs to be kind of
negatively skewed for the median to be bigger than the mean
e.g. 1 , 1 , 3 , 4 , 5
e.g. 2 , 2 , 8 , 9 , 10

how much bigger can the median be when compared with the mean (and also be bigger than the mode)?

for: a , a , b , b + 1 , b + 2
show that b needs to be bigger than a + 1.5 for the median to be bigger than the mean