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This question already has an answer here:

Wondering if any general guidelines whether using Mean or Median is better to represent the statistics of underlying data? I think using Median is always better, especially better when standard deviation is large for a data set.

Your advice is appreciated.

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marked as duplicate by Nick Cox, Greenparker, whuber Aug 12 '16 at 18:25

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    $\begingroup$ Neither the mean nor the median are always better than the other. The median is a more robust estimator (median had breakdown point of 50% while the mean has breakdown of 0%)which is important if you have outliers, or data that's not continuous. However there are also (many) cases you would want to use the mean. A simple example of when the mean is better is if I'm interested in how far I drove today, and I know how many hours I drove, then I would multiply my average speed by hours, the median would not be as helpful. Not to mention if you're analyzing Normally distributed data, use the mean $\endgroup$ – RNM Aug 12 '16 at 17:23
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    $\begingroup$ The breakdown point can be thought of (intuitively) as the proportion of your data that can be incorrect before it significantly impacts your estimator. If I have 100 data points, and one of them was corrupted and entered as infinity then my mean would be infinity. Thus the breakdown point of the arithmetic mean is 0. The median would have to have 50% of its data corrupted before the estimator was affected, thus it has a breakdown point of 50%. $\endgroup$ – RNM Aug 12 '16 at 18:01
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    $\begingroup$ @LinMa as to your other question you would need to quantify "a lot" but I suppose that wouldn't be a bad conjecture. If they mean and median differ significantly there has to be at least one observation pulling the mean away from the median. $\endgroup$ – RNM Aug 12 '16 at 18:07
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    $\begingroup$ Also see stats.stackexchange.com/questions/2547 $\endgroup$ – whuber Aug 12 '16 at 18:25
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    $\begingroup$ I see examples in the answer Matthew Gunn posted in this thread! Here's another: suppose you had a set of quantities of a long-lasting drug administered in widely varying doses to a medical patient and your purpose was to estimate how much of that drug they had consumed on average. The median could be misleading whereas the mean is directly related to what you want to know. Here's another: you would like to estimate a typical salary earned by a writer during their career. Most years she earned nothing, but in a few years earned millions from best-sellers. The median (zero) would be misleading. $\endgroup$ – whuber Aug 15 '16 at 14:27
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To give a counter-example, you almost certainly want the mean and not the median when calculating returns in finance.

Examples where the median is horribly misleading compared to the mean:

  • If you're looking at a set of bond returns, the median will effectively ignore those observations where the bond defaults and you lose half or more of your money.
  • If you're looking at venture capital returns, it's in some sense the reverse. The median company in VC or angel investing is a bust, and the median will effectively ignore the big winners like Google. The return for Ron Conway's first angel fund came largely from one company, Google.

Sometimes insensitivity to outliers is NOT what you want!

Good luck explaining to investors, "I know our fund is down 40 percent this year because nearly half our bonds went bust with no recovery, but our median bond is returning one percent!"

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  • $\begingroup$ Thanks Matt for the advice and vote up. I am not finance expert and I trust all you mentioned. Actually I want to draw correlation between kids age and weight, to see the trend/correlation, and how correlated. In each age (integer), there are 10-20 kids, I only want to draw one point per age for all kids, I think using median is better? If you could advise for my specific use case, it will be great. $\endgroup$ – Lin Ma Aug 12 '16 at 17:49
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    $\begingroup$ @LinMa In a kind of ambiguous setting where it's unclear what to use, why not calculate both? Use ordinary least squares regression to estimate a conditional expectation function (i.e. linear function of the mean) and use quantile regression to estimate a linear function for quantiles (eg. the median, 80th percentile etc...). $\endgroup$ – Matthew Gunn Aug 12 '16 at 17:56
  • $\begingroup$ Thanks Matt, vote up for your reply. In your funds return example, why using mean is better than median? I think good performance 10% funds might be far from mean, I do not see specific values using mean other than using median to find valuable funds to invest. Maybe my understanding is wrong, please feel free to correct me. If you could elaborate a bit more, it will be great. $\endgroup$ – Lin Ma Aug 14 '16 at 22:14
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    $\begingroup$ @LInMa Imagine you have 5 bonds returns where 4 return 2% and 1 defaults with no recovery. The returns are {-100%, 2%, 2%, 2%, 2%}. The median is 2%. The mean is -18.4%. $\endgroup$ – Matthew Gunn Aug 16 '16 at 12:26
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    $\begingroup$ @LinMa No. Imagine asset A has an equal probability of returning {-100%, 4.1%, or 4.2%}. Imagine asset B returns 4% with certainty. The median return of Asset A is 4.1%. The median return of Asset B is 4%. Which one do you want? $\endgroup$ – Matthew Gunn Aug 17 '16 at 15:07

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