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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?

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  • $\begingroup$ Do you have duplicated values? $\endgroup$
    – Dave
    Jan 13, 2020 at 16:17
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    $\begingroup$ 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. $\endgroup$
    – whuber
    Jan 13, 2020 at 18:32
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    $\begingroup$ 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. $\endgroup$
    – whuber
    Jan 13, 2020 at 18:43
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    $\begingroup$ 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. $\endgroup$ Jan 13, 2020 at 19:54
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    $\begingroup$ However, observations that are only a bit erroneous and do not show up as outliers can affect the mode more than the median. $\endgroup$ Jan 13, 2020 at 23:30

1 Answer 1

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It may help to consider two hypothetical scenarios.

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.

Scenario 2) Let the natural variability and the sensor error create a right skewed distribution with a single mode and a large variance. At this point the mode does not equal the median. Then let the incorrect measurement process randomly take 10% of the measurements and shift them dramatically to the right. Again the mode may be stable and the median may shift to the right, but it is questionable whether either measure represents a "typical" value for this distribution. A single number summary may be too simplistic for this setting.

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  • $\begingroup$ Excellent, these scenarios make clear why my colleague suggested using the mode instead of median. For my particular dataset, where Scenario #1 is likely, the mode is more appropriate than the median. Many thanks to all for this helpful discussion $\endgroup$
    – KAE
    Jan 14, 2020 at 12:54

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