# Why are there two spellings of "heteroskedastic" or "heteroscedastic"?

I frequently see both the spellings "heteroskedastic" and "heteroscedastic", and similarly for "homoscedastic" and "homoskedastic". There seems to be no difference in meaning between the "c" and the "k" variants, simply an orthographic difference related to the Greek etymology of the word.

What are the origins of the two distinct spellings?

Is one usage more common than the other, and do they reflect variation between regions or research fields, or nothing more than authorial (or indeed, editorial) preference?

As an aside, other languages have different policies on latinizing Greek roots to English: I note that in French it is, I think, always "hétéroscédasticité" whereas in German it is always "Heteroskedastizität". So I would not be surprised if authors with English as a second language may have a preference for the English spelling corresponding to their mother tongue's. Perhaps the real test is what Greek statisticians call it when writing in English!

• latinizing Greek roots I think that your last paragraph partly answers it. There are many examples when Greek or even earlier (indo-european) k is spelled either as K or, in latinized style, as C. (One example is my own name, Kirill=Cyril). See also this link which specifically links sked/sced with scatter. May 22, 2015 at 9:06
• English Stack Exchange... May 23, 2015 at 3:29
• @Michael Since it is technical vocabulary of interest to practitioners I think it belongs here, particularly any discussion of which fields the usages are more common in. May 23, 2015 at 7:43

Inside this small and vexed question even smaller questions are struggling to get out.

The most detailed discussion to date appears to be

Alfredo R. Paloyo. 2011. When did we begin to spell “heteros*edasticity” correctly? Ruhr Economic Papers 0300. see here

(a reference I owe to @Andy here in Ten fold chat). I can't do justice to its dense and detailed discussion. What follows is more by nature of an executive summary, modulo a little whimsy.

Modern search facilities make it possible to be confident that homoscedastic(ity) and heteroscedastic(ity) are modern coinages introduced, explicitly or implicitly, by the British statistician Karl Pearson in 1905. (Pearson ranged widely over several disciplines, but in the second half of his life his work was firmly centred on statistics.)

Modifying c to k raises absolutely no statistical issue. The idea is at its simplest that the Greek root being used includes the letter kappa ($\kappa$), whose direct equivalent in English is k, and so that k is the correct spelling.

However, as others have done elsewhere, we note that this suggestion was made particularly by J.H. McCulloch in the journal Econometrica, a journal which failed to follow the same logic by renaming itself Econometrika, nay Ekonometrika. (The roots behind "economics" are also Greek, including the word oikos. Ecologists will want to add that there is a journal Oikos even though, once again, ecology did not call itself oikology.)

Further, it is remarkable that Karl Pearson was no hater of k, as he changed his own name from Carl to Karl and named his own journal Biometrika, in full and conscious recognition of the original Greek words he used when devising that name.

The root question then is purely one of language, and of how faithful it is proper to be to the original words behind a coinage. If you follow up the McCulloch reference, the discussion turns to whether such words came into English directly or via other languages, and so hinges on criteria that may appear to many readers as arbitrary if not arcane. (Note that criteria is another word of Greek origin that escaped the k treatment.) Most language authorities now acknowledge that present spelling can owe much to historical accidents and that any long-established usage eventually can over-turn logic (or more precisely etymology). In total, there is plenty of scope here for scepticism (or skepticism).

In terms of tribal or other preferences, it is my impression that

1. Econometric usage seems to be shifting towards the k form. The McCulloch paper had an effect, indirectly if not directly.

2. British English seems to make more use of c forms over k forms than does American English. The form sceptic is standard in British spelling, for example.

All puns and wordplay here should be considered intentional even when accidental.

• (+1) That was really enjoyable! May 22, 2015 at 12:48
• @AlecosPapadopoulos Thanks! (Although not Greek, I benefited from taking (ancient) Greek in secondary school.) May 22, 2015 at 12:50
• +1 The account that leaves a strong feeling of pleasure. May 22, 2015 at 13:20
• "Oikology" might be confused with the study of obnoxious youths. May 22, 2015 at 14:25
• @NickCox +1 You have a sense of humor in addition to a well rounded education, "All puns and wordplay here should be considered intentional even when accidental" indeed. Nice, I could stand some cheering up about now.
– Carl
Feb 22, 2017 at 22:43

There is a tradition in English language to use special letters to indicate that a word is of Greek origin (and as all language "rules", it is not absolutely observed). Most of the times for example, when you see "ph" in an English word, it indicates that it has a Greek origin, as in, say, "photograph" which is the transcription of a Greek word ("phos" means "light" and "graph" is also a Greek root for "write/draw", so "photograph" $\approx$ "a writing/drawing of light").

The same happens with "c" and "k": the use of "k" indicates that the word has a Greek origin. And it does because "Heteroskedasticity" is a composite word: "Hetero + Skedasis" where "Hetero" is a Greek word that indicates "difference" and "Skedasis" means "dispersion". So "Heteroskedasticity = different dispersion", and so different variance, which is what we want to express with the word.

But as I said previously, language "rules" can be flexible, especially for international languages as English (your remark that in "less international" languages like French or German, the spelling appears to be fixed is to the point)-and so people that had to write the word and perhaps were not sure of its spelling, decided to use "c" which is the "natural" choice. Or they normatively thought that words should be "merged" in the language they are used as much as possible.

As for what Greek statisticians do, I guess even the slightest amount of "national pride" (or national chauvinism), would be enough to make them use "k" instead of "c".

• No offence intended, but you're wrong at all accounts. 1. The "ph" spelling was not introduced into English by English linguists. It was passed down to English from French, from Vulgar Latin, from Classical Latin, from Attic Greek, in that exact order. Romans used "ph" to spell the Greek letter Phi because the latter was pronounced as a double consonant "ph" in ancient Greek and not as an "F" as in post-classical Greek. Jul 11, 2017 at 15:34
• 2. As I explained in my answer, letter K does not indicate anymore Greek origin than letter C does since in classical Latin those two letters made the same sound and C was used exclusively to transliterate Greek words (K was hardly ever used for any reason). The reason K and C are pronounced differently in all Latin and non-Latin languages of Europe today is due to language evolution from classical to vulgar Latin. Jul 11, 2017 at 15:37
• 3. As explained by others, heterosc(k)edasticity is a modern English construct and not a word inherited from Greco-Roman antinquity, that's the only reason it has a dual spelling. Jul 11, 2017 at 15:38
• @Digio I don't see where what I am writing contradicts with what you are writing, honestly. For example, I am writing that "ph indicates Greek origin". You write "Romans used "ph" to spell the Greek letter Phi". Are these conflicting statements? Jul 11, 2017 at 15:42
• @Digio "less international" is a relative characterization. It does not mean not international, obviously. And yes English is currently the "most" "international language", by any reasonable metric. Jul 11, 2017 at 15:44

In Polish it is "heteroskedastyczność", but sometimes "heteroscedastyczność" is used instead. For examples you can check the book by Andrzej Gałecki and Tomasz Burzykowski, who were born and educated in Poland. They use the "c" form in their book written in English. Notice, however, that the forms used by different authors could just reflect editorial policies of journals and publishers, so may not reflect how authors consider how words should be spelled.

Wikipedia (which uses the "c" form) leads to a paper by McCulloch (1985) who argues that the "k" form is appropriate since the word has Greek origin and is written as $\kappa$, which should be transliterated as "k".

McCulloch, J.H. (1985). On Heteros*edasticity. Econometrica, 53(2), 483.

• That is like in German: heteroskedastyczność = Heteroskedastizität. May 22, 2015 at 14:09
• @HorstGrünbusch, yes, but in German one doesn't use "c" to indicate a "k"-sound anyway. May 24, 2015 at 11:29
• In Polish, "k" is used in general for the English hard "c" sound. The Polish "c" is pronounced "tsay", that is, softly, so that Carl would be "tsarl." But, you know that already, don'ca?
– Carl
Dec 17, 2017 at 16:10
• @Carl in English "c" and "k" have both the k sound, the ambiguity in Polish seem to reflect the one in English. The Polish spelling with "c" is strange and seems to be unrelated, it seems to be a direct copy of spelling in English.
– Tim
Dec 17, 2017 at 18:09
• I was not thinking of loan words, wyrazy obce, which can be confusing in Polish. The first time I heard the word "talia" I had no clue what it was, i.e., a misspelling of the French "taille."
– Carl
Dec 17, 2017 at 20:13

The missing explanation is that letter 'C' was always pronounced as the modern English 'K' in classical Latin, while K itself was actually a redundant letter. A Greek word with the letter Kappa borrowed into Latin, in Roman times, would have always been spelled with a C. Later, in Vulgar Latin and by extension in French and English, the pronunciation of C became corrupted and was pronounced as a 'S' or a 'CH' when it came before vowels 'E' and 'I'.

Therefore the objectively correct spelling (by Latin standards) would be with a C, and the fact that an alternate K-spelling exists shows that the word is a modernism and doesn't hail from Roman times. By modern English standards, both spellings are equivalent.

What I'm trying to say is that when Pearson first used the spelling "heteroskedasticity", he made a judgement call to intentionally go against the norm (for his own subjective reasons) and spell it with a K (according to Nick Cox's answer, he did the same with "Biometrika"). There is no linguistic motive behind this spelling other than the fact (perhaps) that he knew that these words would have once been pronounced with a 'k' and found it aesthetically more pleasing to spell them so.

• The assumption that all English neologisms must be derived from Latin (as evidenced by phrases like "real answer" and "objectively correct") does not seem to stand up to the well-reasoned, well-referenced post in this thread contributed by @Nick Cox.
– whuber
Jul 11, 2017 at 18:26
• English is full of exceptions. All English neologism must not derive from Latin, this is why I concluded with "by modern English standards, both spellings are equivalent". However: "Objectively correct spelling" = one that follows the linguistic rules that were passed down over the centuries, as opposed to Pearson's "subjective spelling" that would go against those rules. The day the spelling "skedasticity" was used, an exception was added to the English language. I'm not trying to stand up to anybody's answer, I'm simply stating the facts for those who might be interested. Jul 11, 2017 at 21:35
• I have edited my answer to reflect how it is meant to be complementary to the best chosen answer in the thread. Jul 11, 2017 at 22:22
• Thank you for sharing your knowledge and your willingness to temper the tone of your post. +1.
– whuber
Jul 11, 2017 at 22:23
• Though according to wikipedia it was Francis Edgeworth who insisted that Biometrika should be spelt with a "k" and not a "c". Perhaps (speculation) to simply show that this word was imported directly from Greek and not via Latin. Jul 12, 2017 at 8:07

This is a very interesting discussion.

One problem with using the c instead of k is that in the modern Italianized Latin pronunciation, the combination ce (as with ci) yields a "soft c" (/ch/ sound); moreover, the combination sce yields the soft /sh/ sound. So the word heteroscedastic would be pronounced /hetero-sheh-das-tic/.

The letter c only makes a hard /k/ sound in front of the vowels a, o, and u. To force a c to make the hard /k/ sound in front of soft vowels (e and i), you need to add a modifier. Different Romance languages have handled this in different ways... In Italian, the convention is to use the otherwise silent h after the c: ce = /cheh/, whereas che = /keh/; and sce = /sheh/, whereas sche = /skeh/. In Spanish, the c is replaced by a qu: que = /keh/. There is a similar practice in French and Portuguese.

For this reason, it would be quite rational to favor the k spelling of heteroskedastic. It pays homage to the original Greek transcription and is arguably more phonetically correct in the Romance languages and by extension in English.