X Varies From Y To Z

I have a question about the usage of the pattern "X varies from Y to Z". Does the pattern "X varies from Y to Z" mean that the value of the attribute "X" could change from "Y" to "Z". Or does it mean the value of attribute "X" changes when the value of some other attribute changes from "Y" to "Z"? I found the following the on web:

Example 1:
Iowa nest card data from 65 nests show that nest height varies from ground level to 50 inches, averaging about a foot from the ground.

Example 2:
On a warm summer’s day, for example, air temperatures may vary by 16.5 °C (30 °F) from just above the ground to waist height.

In first example, it seems the value of "nest height" changes from ground level to 50 inches. But, in the second example, it seems the value of the air temperature changes when the value of the height of measurement changes from ground level to waist level".

So, is the usage of the pattern "X varies from Y to Z" wrong in one of the example sentences?

• Both of them look correct to this native speaker. Notice that "from Y to Z" always indicates a beginning point and an end point, whereas by indicates a range but not exact values. Jun 23 '16 at 17:23

Both usages are correct and ordinary, and you've understood them both perfectly.

Most native speakers wouldn't notice the difference, because natives understand X varies from Y to Z to simply denote a range of continuous quantitative variation, regardless of how that range pertains to another mathematical relationship. It can be the range of the dependent variable of a function, the range of an independent variable, or just a range considered without regard to another variable (as in your Example 1). With appropriately clever phrasing, varies can even indicate the corresponding ranges of both the independent and dependent variables of a function (as in your Example 2).

This use of the word varies is actually just a small extension of the common trick in English of using a verb of motion to describe a shape, as explained here. Varying isn't motion, and a scalar range of variation isn't a shape, but the analogy of motion→shape to variation→range follows common analogy-making patterns in English—so much, that many natives won't even see that there is actually a deep mathematical analogy here.

People often use the verb run to indicate a range in this same way:

Show that sin x is strictly increasing as x runs from –π/2 to π/2. [That is, show that sin x is strictly increasing for -π/2 ≤ x ≤ π/2. Source.]

To set up a row vector which runs from zero to one in steps of 1/N, we can use w=0:1/N:1. [Source.]

The grammatical principle is the same: a verb for change, in the simple present tense, can describe the shape or dimensions of a static, unchanging object—even an abstract mathematical object.

• Varies doesn't mean continuous quantitative variation. For example, "pronunciation varies from speaker to speaker" (bl.uk/learning/langlit/sounds/changing-voices/…) Also Opinions varied from 'funereal'... Jun 23 '16 at 19:12
• @ColleenV I hope no one would take the above to mean that varies only means continuous quantitative variation! Even my second example of runs isn't continuous. If you think there is real danger of misunderstanding here, can you suggest something more graceful than, say, "WARNING: THIS ANSWER DOES NOT DESCRIBE ALL USES OF THE WORD 'VARIES'"? Jun 23 '16 at 19:19
• Native speakers do not understand X varies from Y to Z to simply denote a range of continuous quantitative variation. It is neither continuous nor quantitative in typical usage. You're just making stuff up that seems reasonable to you and asserting as fact. Where is this survey that told you how native speakers of all varieties of English understand this pattern? Jun 23 '16 at 19:37
• @ColleenV BTW, the "funereal" example (which is wonderful!) is actually a range of continuous quantitative variation, or at least the author wants you to see it that way, at least in one sense of "quantitative". The meaning of "quantitative", like "varies", varies from context to context. That's what I'm trying to do in this answer: convey some idea, to a non-native, of how we vary the meanings of words to suit different contexts—not to propound a strict rule to be applied regardless of context. Alas, repeating "it's not a rule", "context, context, context" gets tiresome. Jun 23 '16 at 19:45
• @ColleenV Hmm, so you're actually disagreeing with the main idea and not objecting that a reader might overextend it to contexts where it doesn't apply? In that case, we simply disagree, and I invite you to post a competing answer. I don't have any surveys, of course; not only are surveys difficult to find and do, they're not reliable. I'm reporting on common usage that I've encountered a great deal. If you haven't seen it or don't recall it at the moment, I refer to you Ngram and Google Books. Jun 23 '16 at 19:59