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This question is in regards to the reasoning underpinning the definition of a set with no repeating values as having no mode. I am interested in the analytical philosophy behind this.

A set with no repeating values is defined as having no mode. But a set that has an even number of elements is described as having a median = [(n/2)th term + (n/2)+1 term)]/2 rather than “no median”.

So, in cases where there is no repeating element, for what reason did mathematicians not define such sets as mode = mean?

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  • $\begingroup$ The mode generally isn't very interesting mathematically (and then you probably care about the maximum of a probability distribution, rather than the mode of a (multi-)set of observables). If you're looking at a dataset that has no repeated elements, you can just consider the mean to be the mode, but the point you're probably interested is the fact that there aren't any repeated elements. $\endgroup$
    – anomaly
    Commented Feb 25, 2022 at 23:32
  • $\begingroup$ The mode is very interesting in applied mathematics. What I’m interested in is why so many exercise books I’ve looked at define a set with no repeating elements as no mode. $\endgroup$
    – duckegg
    Commented Feb 25, 2022 at 23:34
  • $\begingroup$ The mean does not have to be one of the possible values. The mode really should be (or at least arbitrarily close to possible values) to make any sense. But I have seen various different definitions of mode, including cases of unimodal discrete distributions with multiple modes, so what is possible depends on the precise definition being used. $\endgroup$
    – Henry
    Commented Feb 26, 2022 at 3:09

2 Answers 2

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A mode is, by definition, the value most represented. If no value is represented more than any other, then there is no mode. (Which possible value could it be?)

The case is different for median, which is the value for which half of the data is above, and half of the data is below. Now there always DOES exist such a value (for $n>1$). True, there may be multiple such values, and here it is just a useful and principled convention to choose the average value of all those consistent with the constraint.

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  • $\begingroup$ This is the answer to my question. Apparently my understanding of median was insufficient. I thought that the median was the number under the index of the middle of a set. Thank you. $\endgroup$
    – duckegg
    Commented Feb 25, 2022 at 23:40
  • $\begingroup$ Nice explanation. (+1) $\endgroup$
    – BruceET
    Commented Feb 26, 2022 at 7:22
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Here is a different perspective on finding the 'mode' of a sample from a continuous distribution (where there are theoretically no ties.)

Sometimes one tries to use a large sample from a continuous population distribution to estimate the mode of the population.

Consider the following sample of size $n = 5000$ from a population distributed $\mathsf{Beta}(3, 5),$ which has mode $2/6 = 0.3333$ according to this Wikipedia page. Of course, in a real application we would not know the population mode. [Using R.]

set.seed(225)
x = rbeta(5000, 3, 5)
summary(x)
    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
0.01159 0.25535 0.36605 0.37609 0.48496 0.91409 

Using a histogram. Here is a histogram of the data:

    hdr = "n=5000: From BETA(3,5)"
    hist(x, prob=T, col="skyblue2", main = hdr)

enter image description here

The 'modal interval' of this histogram has the tallest bar with base $(0.30, 0.35),$ so one might guess that the mode of the population is between $0.30$ and $0.35.)$

Some elementary statistics books have formulas for guessing the mode, supposedly more precisely, by considering heights of nearby histogram bars. Bear in mind that various histograms (with different bins) might result in somewhat different answers.

Using a density estimator. Independently of how any particular histogram of the data may be drawn, a kernel density estimator (KDE) attempts to imitate the PDF of the population from the available data.

Below we show the KDE for our data (brown curve), and also the PDF of the population distribution (dotted curve, which would ordinarily not be known in a real application).

lines(density(x), col="brown", lwd=2)
curve(dbeta(x, 3,5), add=T, lwd=2, lty="dotted")

enter image description here

The KDE in R is a sequence of 512 points $(x,y).$ So we can look for the highest point on the KDE plot, which corresponds to about $0.31$ on the horizontal axis.

xx = density(x)$x
yy = density(x)$y
mode = xx[yy==max(yy)];  mode
[1] 0.3080652

So, according to the KDE, the mode of the population is estimated as about $0.31.$

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