# Median or mode for measurements with erroneous outliers

Background: I am working with real measurements that likely contain two sources of error, (1) measurements that were performed incorrectly, and (2) natural variability of the measured quantity and measurement sensors since different units of the same instrument were used to make the measurements. The real distribution is not necessarily normal, though I expect it to have a single peak, and I can't discard outliers because I can't consistently identify them. (I don't have duplicated values so I would estimate the mode as the peak point of the distribution of measured values.)

I want to find the typical value of the measured quantity. In the past I've had good results using the median, but a colleague asked why I would not use the mode instead. Is the mode more suitable than the median for noisy, possibly skewed data?

• Do you have duplicated values?
– Dave
Jan 13, 2020 at 16:17
• How do you determine the "distribution peak," then? Usually it requires some kind of density estimate that is bandwidth-dependent, as illustrated at stats.stackexchange.com/a/428083/919.
– whuber
Jan 13, 2020 at 18:32
• The answer depends on (a) the actual data-generation process; (b) what you mean by "sensible function;" (c) how you fit it; and (d) the sample size. That's why we're probing for details: there's no generic solution or universally correct answer.
– whuber
Jan 13, 2020 at 18:43
• Scenario 1) Let the natural variability and sensor error create a fairly symmetric, single modal distribution with a small variance -- like a Normal distribution covering a contextually tight range. At this point the mode=median. Then let the incorrect measurement process randomly take 10% of the measurements and shift them dramatically to the right -- shifting all of them to the right of the median. Now the mode is in the same spot but the median has shifted a little to the right. Under this data generating mechanism, you can make a case for the mode as a "typical" value. Jan 13, 2020 at 19:54
• However, observations that are only a bit erroneous and do not show up as outliers can affect the mode more than the median. Jan 13, 2020 at 23:30