I hope you're having a good day, I'm very new to statistics, and I'm having trouble with this question: "The average age of 100 people is 30 years. Prove by contradiction that the median of the ages cannot be larger than 60". (Note: Not sure what they mean by "average", but I suppose it's the arithmetic mean). Thank you. Thank you guys for the response, this is just some practice, I'm self studying, I've tried to come up with a proof, and this is what I could come up with for now, though I'm not proud of it, and it feels rather weak: "Assume that the Median is strictly larger than 60. That means that there are 50 people with ages between 0 and 59 and another 50 with ages 61 and up (60 cannot be an element of the sample considering the number of samples is even)
If we take the smallest possible values, age 0 (for people less than 60) and age 61 (for people over 60), and find the mean (so we assume everyone has the same age corresponding to their position in relation to the Median)(
). In fact it appears that the sum of the numerator has to be always 3000 in order for the mean to be exactly 30, and that cannot happen for a median larger than 60 because the smaller values have to step into the negatives to satisfy that, which is impossible for out data set (unless you count in unborn people), therefore, with our parameters, the mean cannot be equal to 30 because the sum would always be larger than 3000, which is a contradiction, meaning the first statement is false, thus making the Median less or equal to 60.
Please feel free to correct or critique this proof, and lend me your feed back."