Is mode a measure of central tendency?

Question

I am just starting out with my stat basics and that’s where I came across measures of central tendency. In one of the books measure of central tendency is defined as a measure which yield information regarding the central or the middle part of a group of data. As we all know mean, median and mode are measures of central tendency. What I am not clear with is how is mode used to yield information regarding the the central part of a group of data. What if the the group of data has numbers which are occurring most other than in the central part? I am a bit confused here.

Answer 1

Bickel & Lehmann (1975) suggest three criteria that a measure of location $\mu$ ought to meet to be worthy of the characterization, & it seems to me they might as well have talked of central tendency. The criteria are these:

1. $\mu(X) \geq \mu(Y)$ when $X$ is stochastically larger than $Y$, i.e. when $\Pr(X > k) \geq \Pr(Y > k)$ for any $k$ & $\Pr(X > k) > \Pr(Y > k)$ for at least some $k$

2. $\mu(aX+b) = a \mu(X) + b$ for $a>0$

3. $\mu(-X) = -\mu(X)$

(The 2nd & 3rd together ensure that $\mu$ identifies the centre of symmetry of a symmetric distribution.)

It's already clear from @Ben's answer that the mode sometimes only barely meets the first criterion (perhaps oddly, B. & L. don't mention the mode): the scale family of exponential distributions is totally ordered, yet each distribution has the same mode.^† Worse still, it can fail to meet it: let $X$ have a triangular distribution on [0,10] with mode 1, & $Y$ a triangular distribution on [0,4] with mode 2; then $X$ is stochastically larger than $Y$ despite having a lesser mode.

The median (obviously) & the mean (see https://math.stackexchange.com/q/3116135/59351) meet all three criteria, by the way. All the same, the mode seems fairly well entrenched as a measure of "central tendency", whatever exactly that's supposed to be. Perhaps it'd be sensible to follow B. & L. in requiring measures of location to meet all three criteria & then requiring measures of central tendency to meet only the 2nd & 3rd. On the other hand, the geometric, harmonic, &c., means perhaps have a claim to be measures of central tendency in their own right, though not of location; so there's a case for saying measures of central tendency need only meet the 1st criterion. At any rate just making such distinctions is the interesting & useful part.

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† In light of this example it's tempting to propose an additional criterion along the lines of 1.(a) $\mu(X) > \mu(Y)$ when $\Pr(X > k) > \Pr(Y > k)$ for any $k$ such that $0<\Pr(X>k)<1$ or $0<\Pr(Y>k)<1$.

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Bickel & Lehmann (1975), "Descriptive statistics for non-parametric models. II. Location", Ann. Statist., 3, 5, pp. 1045–69

(I discuss other papers in this series dealing with dispersion & spread here, & must again thank Nick Cox for bringing them to my attention.)

Answer 2

The notion that the mode constitutes a measure of "central tendency" is contextual at best, and hinges on the assumption that we are dealing with a distribution with a non-monotonic unimodal density. There are many distributions where the mode is in the tails of the distribution, or even at the extreme edge of its support. For example, the exponential distribution has its mode at zero, which is the extreme left edge of the support; not even close to a measure of central tendency. More generally, one can construct a distribution with any mean and mode one wishes, so it is entirely possible to construct a distribution with a mode that is absurdly far away from the "centre" of the distribution, as measured by the mean.

My view is that your misgivings here are correct --- you should not consider the mode to be a measure of central tendency unless you have strong a priori reasons to believe you are dealing with a non-monotonic unimodal distribution with a "hump" somewhere near the middle of the distribution. (And obviously judging this presupposes that one already has an a priori view of what constitutes the "middle".) It is best just to ignore this aspect of your textbook.

Answer 3

Sometimes the mode is useful as a measure of the 'center' of a distribution. In a right-skewed distribution the mode is often smaller than the median, which is in turn smaller than the mean.

For example, consider a gamma distribution, specifically $\mathsf{Gamma(\text{shape}=\alpha = 5,\,\text{rate}=\theta=2)},$ which has mode $\frac{\alpha-1}{\theta} = 2,$ median $2.335454,$ and mean $\frac{\alpha}{\theta} = 2.5).$ There are convenient formulas in terms of the shape and scale parameters for the mode and mean, but in most cases the median must be computed by numerical integration.

qgamma(.5, 5, 2)
[1] 2.335454

If you have a moderately large sample from a population known to be gamma, but with unknown parameters, then the parameters can be estimated by the method of moments or by the method of maximum likelihood, from which the mean and median can be estimated. (See the Wikipedia link above.)

For a very large sample the population mean is well estimated by the sample mean, and the population median is well estimated by the sample median. The mode can be very roughly estimated by looking at the highest bar in a histogram, or more precisely estimated by finding a kernel density estimator of the sample.

Suppose we have a sample of size $n = 10,000$ from $\mathsf{Gamma}(5,2).$ Then the sample mean 2,506 and median 2.324 are reasonably good estimates of the population mean and median, as shown below.

set.seed(1111)
x = rgamma(10^4, 5, 2)
summary(x)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.1407  1.6806  2.3244  2.5057  3.1290  9.2822 

Below is a histogram of these 10,000 observations along with a plot of the population density curve [solid black] and the a kernel density estimate (KDE) [dotted red].

hist(x, prob=TRUE, col="skyblue2", main="GAMMA(5,2)")
curve(dgamma(x, 5, 2), add=TRUE, lwd=2)
plot(density(x), col="red", lwd=2, lty="dotted")

The highest histogram bar is centered at about 2.25, and there are formulas for trying to refine this to get a slightly better estimate of the population mode (one of them here). (A slightly different choice of histogram bins might improve this estimate.)

The density estimator is made up of about 512 x-points and 512 y-points. The maximum of the KDE, which can be taken as the sample mode 2.082, can be found using $-notation in R as shown below. (I have used the default version of the KDE in R.)

kde = density(x)
kde$x[kde$y == max(kde$y)]
[1] 2.081567