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?
Originally posted here because I did not know this site existed.
Statistics does not provide a good answer to this question, in my opinion. A mean can be relevant in mortality studies for example, but ages are not as easy to measure as you might think. Older people, illiterate people, and people in some third-world countries tend to round their ages to a multiple of 5 or 10, for instance.
The median is more resistant to such errors than the mean. Moreover, median ages are typically 20 – 40, but people can live to 100 and more (an increasing and noticeable proportion of the population of modern countries now lives beyond 100). People of such age have 1.5 to 4 times the influence on the mean than they do on the median compared to very young people. Thus, the median is a bit more up-to-date statistic concerning a country's age distribution and is a little more independent of mortality rates and life expectancy than the mean is.
Finally, the median gives us a slightly better picture of what the age distribution itself looks like: when you see a median of 35, for example, you know that half the population is older than 35 and you can infer some things about birth rates, ages of parents, and so on; but if the mean is 35, you can't say as much, because that 35 could be influenced by a large population bulge at age 70, for example, or perhaps a population gap in some age range due to an old war or epidemic.
Thus, for demographic, not statistical, reasons, a median appears more worthy of the role of an omnibus value for summarizing the ages of relatively large populations of people.
John gave you a good answer on the sister site.
One aspect he didn't mention explicitly is robustness: median as a measure of central location does better than the mean as it has a higher breakdown point (of 50%) whereas the mean has a very low one of 0 (see wikipedia for details).
Intuitively, it means that individual bad observations do not skew the median whereas they do for the mean.
Here's my answer first posted on math.stackexchange:
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, the post-WWII baby boom generation has distorted this distribution as they age. Some people have called this "squaring the pyramid" because the boomers have made the top of the pyramid wider than it has been in the past.)
With an asymmetrical distribution, it may be better to report the median because it is a symmetrical statistic. The median is symmetrical even if the sampling distribution isn't.
Why is an axe better than a hatchet?
That's similar to your question. They just mean and do different things. If one is talking about medians then the story they are trying to convey, the model they are trying to apply to the data, is different than one with means.
For a concrete example, consider the mean ages for the Congo (DRC) and Japan. One is devastated by civil war, the other is well developed with an ageing population. The mean isn't terribly interesting for an apples to apples comparison. On the other hand, the median can be informative as a measure of central tendency since by definition we have half above, half below. The wikipedia article on Population Pyramid might be enlightening (see the sections on youth bulge, ageing populations).
Public Health Data repositories in the United States are moving toward an AGE in years format of five year increments due to the impact of the HIPAA regulations regarding the intentional blinding and masking of data for personal privacy reasons.
Given this challenge to what was had been in the past (prior to HIPAA) a fairly scale level of measure data element based upon the difference between date of birth and date of death, we may need to reconsider AGE as a scale variable that can be parametrically described at all in public health data sets, in favor of models that describe AGE in a non-parametric fashion, as an ordinal level of measure. I know this may seem "over the top" to many factions within the biomedical informatics community, but this idea may have some merit in terms of "interpretation" as described in the comments above.
What about all of the analytical power that is available to the non-parametric approaches? Yes, it is true that every one of us almost universally will attempt to apply GLM (general linear model) techniques to a variable that presents itself to us in distributions that behave the way AGE does.
At the same time the shape of that distribution and how that shape is being determined by multiple dimension interaction effects upon multi-dimensional centroids and sub-group centroids present in the distribution, must be taken into consideration. What to do with these very complex data sets?
When a data element fails to meet the "assumptions of the model", we progressively scan across (I said across, not down; we should be equal opportunity employers of method, each tool comes from the factory with form follows function rules) the list of other possible models to find the ones that "do not fail" the assumptions tests.
In the present format in public health data sets, we really do need (as a data visualization community) to come up with a more standard model for handling AGE in five year increments (5YI). My vote for data visualization of AGE (given the new 5YI format) is to use histograms and box and whisker plots. Yes, this means the median. (No pun intended!)
Sometimes a picture really is worth a thousand words, and an abstract is a summary of a thousand words. The box and whisker plot shows the "shape" of the distribution as a meaningful symbolic representation of the histogram at nearly an iconic level of resolution. Comparing the distributions of five year age increments by showing "side by side" box and whisker plots where one can instantly visually compare patterns of 75th to 50th (median) to lower 25th ntiles, would make an elegant "universal standard" for comparing AGE across the world. For those of us that continue to enjoy the thrill of data representation through the textual mechanics of tabular display, the "stem and leaf" diagram may also be of service when employed as an animated visual graphics element in a "sparkline" approach that portrays variation of the shapes of distributions over time.
AGE has come of age. It needs to be explored further with the more powerful computational algorithms that are now available.
I don't think there is a good descriptive reason for choosing median over mean for age distributions. There is one of practicality when comparing reported data.
Many countries report their population in 5-year age intervals with the top band open-ended. This causes some difficulties calculating the mean from the intervals, especially for the youngest interval (affected by infant mortality rates), the top "interval" (what is the mean of an 80+ "interval"?), and the near top intervals (the mean of each interval is usually lower than the middle).
It is far easier to estimate the median by interpolating within the median interval, often approximating by assuming a flat or trapezium age distribution in that interval (death rates in many countries are relatively low around the median age, making this a more reasonable approximation than it is for the young or old).
To give a useful answer the original question requires we know the question behind the question. In other words, "Why do you want some sort of summary statistic comparing the age distribution of different countries?" The median might be the most useful for some questions. The mean might be the most useful for others. And there are probably questions where "percent above (or below) some particular age" would be the most useful statistic.
You're getting good answers here, but let me just add my 2 cents. I work in pharmacometrics, which deals in things like blood volume, elimination rate, base level of drug effect, maximum drug effect, and parameters like that.
We make a distinction between variables that can take on any value plus or minus, versus values that can only be positive. An example of a variable that can take on any value, plus or minus, would be drug effect, which could be positive, zero, or negative. An example of a variable that can only realistically be positive is blood volume or drug elimination rate.
We model these things with distributions that are typically either normal or lognormal, normal for the any-valued ones, and lognormal for the only-positive ones. A lognormal number is the number E taken to the power of a normally distributed number, and that is why it can only be positive.
For a normally distributed variable, the median, mean, and mode are the same number, so it doesn't matter which you use. However, for a lognormally distributed variable, the mean is larger than both the median and the mode, so it is not really very useful. In fact, the median is where the underlying normal has its mean, so it is a much more attractive measure.
Since age (presumably) can never be negative, a lognormal distribution is probably a better description of it than normal, so median (E to the mean of the underlying normal) is more useful.
I've been taught that median should be used with range and mean with standard deviation. When we talk about age, I think range is a more relevant way to express the spread, and easier to understand for most. For example in a study population, the mean age was 53 (SD 5.4) or the median age was 48 (range 23-77). For that reason, I would prefer to use median rather than mean. But I would be very interested to here what a statistician or stats pro would say about using mean with range? I see this quite a bit in scientific papers.
John's answer on math.stackexchange can be viewed as the following:
When you have a skewed distribution the median may be a better summary statistic than the mean.
Note that when he says that there are more infants than adults he essentially is suggesting that the age distribution is a skewed distribution.
I hope the mean age would be influenced by the outliers in your data set while this is not the case for a median age. Let us take an example of a data set vaccinated patients: 1,2,3,4,4,5,6,6,6,78 years the mean would be:11.5 and median age of these patients is 4.5. this mean age has been affected by the outlier 78. median is the best while dealing with data sets of the skewed distribution.
Certainly in the case of demographic analysis, I would think that both the mean and median would be valuable, especially in combination with each other, if you are looking for outliers or areas of growth that may be mislabeled by the median alone. In communities with a large retirement community or in an area with a birth rate explosion, the median alone may not give you the whole picture, and that is where the mean, in comparison, can be very useful.
SHORT ANSWER: Median age is not simply better than mean age; however, you may have noticed that more people use it. So a better question might be: "Why would more demographers use median age than mean age?"
A statistic, as a vocabulary term, has its origins in the state (nominally a legal entity) attempting to understand its human population. So think about the people in those governments and how much information they want or need, as well as how much time they have to devote to understanding the precise mathematical meaning of scientific words.
The easiest way to sum up a lot of data, without using a picture, is to report a single number; this is known as an estimator for the parameter in question (in this cage, the time elapsed since birth of a human being, precise to the level of years). As E.T. Jaynes showed in his book Probability Theory: The Logic of Science, one could choose to construct an estimator based on a utilitarian loss function summarizing the consequences of making a mistake based on using a single number instead of an entire data set when making decisions based on that information.
In Jaynes' book, he shows via mathematical proof that the mode, or maximum likelihood estimator, is the estimator minimizing loss shaped like a Dirac delta function. The mean minimizes quadratic loss functions, such that the further one gets from the estimate, the quantity of loss (undesirable consequence) goes up very quickly once you pass the unit scale.
The median, on the other hand, minimizes a loss function shaped like an inverted triangle, such that it is only five times less desirable to be off by one unit of precision, rather than 25 times (as in the case when using the mean). In fact, the unit of precision makes no difference whatsoever, because there is no curvature in such a triangular pointy loss function.
With this theoretical foundation, one could literally draw loss functions that are not symmetric at all and form an infinite number of new estimators custom-suited to the needs of their consumers/users. Another alternative to dealing with the cultural expectation of a single number is to educate those same users/consumers of information that a measure of central tendency can provide more information when paired with other parameters of a distribution, such as variance, skew, and kurtosis (might want to start with just variance and skew to ease them into it.
The variance is just one example of a dispersion measure; another Jaynes suggests (in other writings) is to form a Bayesian posterior distribution and calculate the width of the shortest credible interval with value 0.5 (or confidence interval/standard deviation etc. if you do not buy into Bayesian theory--please, let's not get sidetracked). A more intuitive method which is possibly easier for more people to grasp would be the inter-quartile range, especially when reported with the median as its corresponding measure of central tendency.
I am not sure if there is a non-parametric form of skew or kurtosis, but if they do exist they will almost certainly be easier to understand than these parametric analogs. I have a hunch that a major, if not dominant, part of the reason median age crops up more often than mean age is because it simply appeals more to people with less time or desire to go into theoretical details about things like sigma-algebras, Lebesgue measure theory, etc. which are all technically necessary to understand the more common foundations of probabilistic reasoning.
