Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If you look at Wolfram Alpha

alt text

or this Wikipedia page List of countries by median age

alt text

Clearly median seems to be the statistic of choice when it comes to ages.

I am not able to explain to myself why arithmetic mean would be a worse statistic. Why is it so?

share|improve this question
    
Just a thought: Perhaps it is so, because it easier to estimate the median as compared to the mean from a given sample? –  Aryabhata Sep 10 '10 at 20:18
    
@Moron: Why do you think it is easier to estimate the median? –  Lazer Sep 10 '10 at 20:23
    
@Lazer: It was just a thought. What I was thinking: say I gave you a random sample of 1000 people. Now if you calculate the median and mean of those, I would expect the median to be much more accurate than the mean. –  Aryabhata Sep 10 '10 at 20:27
2  
Better for what? –  Mariano Suárez-Alvarez Sep 10 '10 at 20:41
2  
This question has been cross-posted at stats.stackexchange.com/q/2547/159 which is a more appropriate site for it (IMO). –  Rob Hyndman Sep 11 '10 at 2:19
show 1 more comment

2 Answers

up vote 7 down vote accepted

Median is what many people actually have in mind when they say "mean." It's easier to interpret the median: half the population is above this age and half are below. Mean is a little more subtle.

People look for symmetry and sometimes impose symmetry when it isn't there. The age distribution in a population is far from symmetric, so the mean could be misleading. Age distributions are something like a pyramid. Lots of children, not many elderly. (Or at least that's how it is in a sort of steady state. In the US, for example, the post-WWII baby boom generation has distorted the distribution as they age.)

With an asymmetrical distribution, it may be better to report the median because it is a symmetrical statistic in the sense that it splits the population in half. Said another way, the median is symmetrical even if the distribution isn't.

Update: I got my logic backward when I first answered and said the mean would be lower than the median. I meant the opposite.

share|improve this answer
2  
Why do you say the mean should be lower than the median? While not always true, I would typically expect the mean to be higher if the distribution is positively skewed as age is. –  Larry Wang Sep 10 '10 at 21:58
3  
You have a population of five people, four are age 5 and the other age 80. Median = 5 and mean = 20, so here we have median<mean and more young people than old people. –  Derek Jennings Sep 11 '10 at 7:27
    
Thank you for responding so gently! You're absolutely right. I updated my response. –  John D. Cook Sep 11 '10 at 14:03
add comment

The best statistic to summarize a distribution depends upon the distribution and what you want to use it for. For distributions that are nicely bell-shaped, the mean, median, and mode are close together and it doesn't matter. For skew distributions the mean is out on the skew side from the median, but it still represents the expected value of the average of a large number of samples. The median is closer to more of the individuals than the mean. For stranger distributions no one number can provide a useful summary.

share|improve this answer
1  
To use more terminology: the median is a "robust" estimator of "central tendency": the mean is easily thrown off by even a few outliers. But yes, for badly behaved distributions, merely reporting central tendency alone is not enough. In any event: remember that there are countries with more tots than seniors, and then there are countries where most of the people are over the hill (due to e.g. sociological pressures for not having kids). –  J. M. Sep 10 '10 at 23:21
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.