Talk:Central tendency

There are two meanings of central tendency or statistical centrality sourced in the literature. Narrow-sense centrality is synonymous with center, representative, or typical value. It is specified in terms of a location parameter (the reverse is not necessarily true). As such it is also synonymous with average and can thus be realized by any of its variants: mean, median, mode, etc. The wide-sense meaning, arguably closer to the dictionary definition of tendency and centrality, encompasses not only the centrality location but also the degree of centrality, i.e., the statistical dispersion. (I hope this is acceptable under WP:SYNTHESIS.) Fgnievinski (talk) 07:15, 24 April 2013 (UTC)

References for narrow-sense meaning: Cambridge Dictionary of Statistics: "Central tendency - A property of the distribution of a variable usually measured by statistics such as the mean, median and mode."

References for wide-sense meaning: Oxford Dictionary of Statistics: "central tendency (centrality) - The tendency of quantitative data to cluster around some central value. The central value is commonly estimated by the mean, median, or mode, whereas the closeness with which the values surround the central value is commonly quantified using the standard deviation or variance."

Thanks for the references. My interpretation, looking at some other references too, is that the "narrow-sense" is the primary meaning and that the other meaning seems to occur in some encyclopedias but not commonly otherwise. Also "measures of central tendency" is much more frequent than "central tendency" in isolation. I also looked up "Tendance centrale" in the french wikipedia, since the OECD reference mentions the french phrase. My french is rusty, but I think they only acknowledge the narrow meaning. Dingo1729 (talk) 16:42, 24 April 2013 (UTC)

I agree with Dingo. While I don't particularly like the word "average" since for some people it is a synonym for "mean" and for others it means "mean, median, or mode", over the past forty years I have seldom heard of the broad sense encompassing dispersion. I've occasionally heard the term "cluster around" but it has always seemed like an informal freshman oversimplification.

I think it would be a good idea to start the article with something like what I put in a few days ago, or an improved version of it maybe quoted from one of the sources, with citations, and then mention that sometimes the term is used in the broader sense that more nearly matches the common word "tendency", again with citations.

As for the observation that "measure of central tendency" -- in the sense of mean, median, or mode but not in the sense of dispersion -- is much more common than "central tendency" by itself, this is certainly very true. But I think it would be a mistake to rename the article "Measure of central tendency" because we already have the article Location parameter. Duoduoduo (talk) 17:27, 24 April 2013 (UTC)

I'm not happy with the current version of the "wide sense" meaning in the article. The definitions from the sources are all very similar. Here's one:

"The tendency of quantitative data to cluster around some random variable value. The position of the central value is usually determined by one of the measures of location such as the mean , median or mode. The closeness with which values cluster round the central value is measured by one of the measures of dispersion such as the mean deviation or standard deviation."

I think that the only part of this that can be salvaged is the first sentence. I don't read the definition as saying that a central tendency consists of two numbers, a position and a spread. Also, if we're thinking of "central tendency" in this way, I don't believe the actual value of the standard deviation tells us anything. Surely a normal distribution has the same amount of "central tendency" independent of a scale factor? I'm assuming that this is in contrast to, say, a Bernoulli distribution which, so far as I can tell, this definition would say doesn't have a central tendency. If we really want to distinguish between situations like this we'd have to look at something like the coefficient of kurtosis. But that's far into WP:OR.

So I'm changing this part of the leading paragraph again. Dingo1729 (talk) 02:39, 2 May 2013 (UTC)

The quote has three sentences: (1) "The tendency..."; (2) "The position..."; (3) "The closeness..." Rejecting (3) is WP:CHERRYPICKING IMHO. And just stating (1) -- "central tendency means tendency to cluster around center" -- is ^{[circular definition]}. This sourced definition does affirm that a normal distribution with smaller standard deviation is said to exhibit a greater degree of central tendency. As for the Bernoulli distribution, please find a source. Fgnievinski (talk) 06:54, 2 May 2013 (UTC)

No, I don't have anything like a source for this definition and the Bernoulli distribution. We don't have any sources for a practical use of it do we? I'm just trying to understand the definition. At least in previous iterations here it was implied that some distributions have a central tendency and others don't. The Bernoulli distribution is the most extreme example I can think of. Surely we aren't meant to say that a normal distribution has a central tendency if it has standard deviation 1 but not if it has standard ddeviation 20. It seems to me that you were trying to make sense of a bad definition by paraphrasing it and I was trying to cut it down to the minimum that makes sense. Though I agree there are problems with it being circular. But I'd blame the person who wrote it rather than me. I guess I would be OK with including a full exact quote from one of the sources, though it's a bit WP:UNDUE since no-one other than lexicographers seem to use this meaning. Dingo1729 (talk) 14:08, 2 May 2013 (UTC)

I read the three-sentence definition like this: A tendency is something well, tending, toward someplace more or less strongly. A tendency can be measured by where it is tending toward and how strongly it is tending toward there. The implication is that, for example, a low-variance normal has a strong tendency to have values near a particular location, while a high-variance normal has a weak tendency to have values near a particular location. Duoduoduo (talk) 16:55, 2 May 2013 (UTC)

Searching googlescholar, I found examples of "central tendency" being used to mean the location of the central tendency -- i.e., the location measure of the central tendency without using the word "measure": "We assume that the instantaneous riskless rate reverts toward a central tendency which....". "It helps us identify the central tendency of the distribution (called the expected value) and the amount of variability or spread of the distribution (called the variance)." "Many accounts of categorization equate goodness-of-example with central tendency for common taxonomic categories." "...rather than inferring such relationships from models based on conditional central tendency." "...examines the effects of comparative context on central tendency and variability judgements of groups,...." "The Kerala model: its central tendency and the outlier." One paper defines central tendency as the "degree to which units of a distribution tend to cluster around a given point" but then ignores the degree in favor of the location. Basically it's very hard to find any papers that actually define it as "the degree to which" and then actually stick with that definition. Duoduoduo (talk) 17:20, 2 May 2013 (UTC)

Earlier Melcombe was arguing that "Central tendency" is a property of a distribution "the first consideration here is "does this population have a central tendency"". He interpreted the definition to mean that, for example, a bimodal or multimodal distribution doesn't have a central tendency. I think that all this is implied though not stated by the first sentence. On the other hand, Fgnievinski and Duoduoduo seem to be saying that we can't say anything about a single distribution, we can only compare two distributions (which must be on the same scale) and say that one has more of a central tendency than the other. I agree that this is implied though not stated by the third sentence. Overall I come down on Melcombe's side, though I think either meaning is possible. Which is precisely why I think it's a bad definition. So what should we write in the article? I wrote what I did to try to avoid getting any deeper into the interpretation, but I'm hearing disagreement. Is it important or irrelevant that a distribution is bimodal when talking about a central tendency? Dingo1729 (talk) 17:54, 2 May 2013 (UTC)

I don't see the conflict of interpretations. Melcombe also said from his sources that the standard deviation inversely measures the strength of the central tendency (he had a whole subsection on measures of dispersion), which my interpretation agreed with.

In any event, I think the current text is fine in this regard. In the three-sentence definition/description above on this talk page, I think only the first sentence is the definition; the second and third sentences say how to measure the concept defined in the first sentence. Duoduoduo (talk) 18:15, 2 May 2013 (UTC)

I agree with Duoduoduo's paraphrasing about the distinction between weak vs. strong centrality. I also concede to Dingo1729's point that the third sentence only implies this point. Yet I disagree that a direct quotation of the first sentence suffices -- the meaning is not trivial so it has to be spelled out, as the discussion here attests. Melcombe's interpretation, that centrality is an all-or-nothing property, is not supported by the sources. What is sourced is that wide-sense centrality can always be defined because mean and standard deviation can always be calculated, although e.g. a bimodal distribution will exhibit a poor degree of centrality. Fgnievinski (talk) 05:38, 3 May 2013 (UTC)

I think I was wrong to write that Melcombe was saying that it's an all-or-nothing property. Of course there would be intermediate cases.

I've found a couple of sources which I think support the view that what matters is the shape of the distribution rather than the standard deviation. They come from this search . There may be some other sources in there which emphasize standard deviation rather than shape but I haven't found them yet. Dingo1729 (talk) 02:29, 4 May 2013 (UTC)

This would be a third definition. I'll stick with the first two definitions -- narrow and wide. Since these seem to have been agreed upon, I've restored the original edit in which the latter had been introduced. Fgnievinski (talk) 07:12, 6 May 2013 (UTC)

I'm not sure why you say that this has been agreed upon. By my reading the two new sources are consistent with the sourced definitions. Duoduoduo's opinion was that "I think only the first sentence is the definition", which I took to mean that that is what we should use as the definition in the article. Perhaps he could help us here, I don't want to put words in his mouth. The "narrow" and "wide" terminology is just what we're using here, not a sourced use is it? Dingo1729 (talk) 13:44, 6 May 2013 (UTC)

I think I may have mis-represented one of the sources above . Reading it more carefully, it may actually support the use of dispersion rather than shape. But the source may not be useable because the author seems not to know the meaning of kurtosis. Dingo1729 (talk) 17:34, 6 May 2013 (UTC)

This article seems to have moved beyond being a stub; I'm re-rating it as start-class, if no one objects. -Bryanrutherford0 (talk) 20:19, 22 July 2013 (UTC)

An IP has added the quadratic mean (rms) to the list. Is it ever really used as a measure of central tendency? The biggest problem comes when there are negative values, so then it isn't anywhere near the center of the distribution. I'm somewhat inclined to revert, but I thought I'd ask for opinions first. I've mentioned this on the IP's talk page. Dingo1729 (talk) 16:12, 9 February 2014 (UTC)

No one else seems interested so I re-wrote that entry. Dingo1729 (talk) 05:13, 16 February 2014 (UTC)

In the "Solutions to variational problems" section, in paragraph 4, we have this sentence:

For ${\displaystyle p=0}$ the limiting values are ${\displaystyle 0^{0}=0}$ and ${\displaystyle a^{0}=0}$...

By my reading, this should say:

For ${\displaystyle p=0}$ the limiting values are ${\displaystyle 0^{0}=0}$ and ${\displaystyle a^{0}=1}$...

That's because the typical situation where ${\displaystyle a^{0}=1}$ for ${\displaystyle a\neq 0}$ is what makes the formulation of mode in terms of a minimization of a ${\displaystyle 0}$-norm make sense. That whole sentence should probably be rewritten. The overall point should be that ${\displaystyle \left|x_{i}-c\right|^{0}}$ becomes an indicator function on the boolean expression ${\displaystyle (x_{i}-c)\neq 0}$. Thought it's implied, the sentence doesn't make it clear that ${\displaystyle a=\left|x_{i}-c\right|}$. Davesque (talk) 01:16, 27 February 2023 (UTC)

This edit request by an editor with a conflict of interest was declined. Some or all of the changes may be promotional in tone.

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• Reason for the change: add link to convenient online calculator

Pereiratr (talk) 21:03, 8 June 2025 (UTC)

Not done: A majority of the requested changes are currently written in a promotional tone. Please review WP:Neutral point of view and ensure you follow this before submitting any edit requests. Encoded  Talk 💬 23:10, 8 June 2025 (UTC)