# Average Rate of Change isn't/is Statistics

I have the common misconception in my business calculus classes that the Average Rate of Change, say from $x=1$ to $x=5$, is the statistical average of the rates on the four unit intervals $1$ to $2$, $2$ to $3$, etc. So when introducing/reviewing Average Rate of Change, I'm tempted to say "This is not an Average like you learn about in your stats class." Except, in reality, it is. From the perspective at the end of the course, the difference quotient is the result of finding the average value of the derivative. And the idea of the average value of a continuous function is very much like the averages they compute in a stats class.

So my question is: How can I tell them not to compute a statistical average without lying to them about the relation between average rate of change and continuous statistics?

• I am not sure I understand the question: if the intermediate intervals are all unitary (this is critical), then the ARC (of a function of $x$ I guess) is exactly equal to the arithmetic average of the ARC's in the intermediate intervals. What am I missing? Aug 26, 2014 at 19:41
• @AlecosPapadopoulos I am not sure what you mean by "unitary", but the fundamental problem which Aeryk is asking is that understanding the average rate of change as an actual average, you need the fundamental theorem of calculus. Personally, I think the fundamental theorem is such a basic idea that we should be teaching it to students in the first couple of weeks. Aug 26, 2014 at 20:44
• @StevenGubkin By "unitary" I just meant what the OP writes, in the 2nd line of the post about "unit intervals, 1 to 2, 2 to 3 etc", i.e. having length equal to unity. Aug 26, 2014 at 21:52
• This property cannot be used to define the ARC, it does not seem, since it is circular. But since it reduces the problem to smaller subintervals, perhaps the apparent circularity can be removed. It should end up being the Riemann sum definition, and then you need essentially fundamental theorem to prove the equivalence. Aug 26, 2014 at 22:41

$$\frac{1}{4}\int_1^5 f' = \frac{1}{4}\left( \int_1^2 f' + \int_2^3 f' + \int_3^4 f' + \int_4^5 f' \right)$$ The LHS is an average rate of change.

The RHS is a statistical average of four average rates of change.

This requires some differentiability, but I would not treat this as a misconception!

From the point of view of an outsider like me, what the OP describes is the following: there is a function $f()$ and we define the Average Rate of Change of this function between points $a<b$ as

$$ARC_f = \frac {f(b) - f(a)}{b-a} \tag{1}$$

Then the OP considers as an example the interval $[1,5]$, for which we have

$$ARC_f(1,5) = \frac {f(5) - f(1)}{5-1} = \frac 14 [f(5) - f(1)]\tag{2}$$

He then speaks of the "misconception" that students believe that they can recover $ARC_f(1,5)$ by calculating the arithmetic average of the Average Rates of Change in the intermediate unit-length intervals, namely

$$ARC_f(1,5) = ?? \frac 14 \left(\frac {f(2) - f(1)}{2-1}+\frac {f(3) - f(2)}{3-2}+\frac {f(4) - f(3)}{4-3}+\frac {f(5) - f(4)}{5-4}\right) \tag{3}$$

It is obvious that the right-hand-side of $(3)$ is identical to the right-hand side of $(2)$, and so there is no misconception at all, the students are right, as long as we are talking about an interval that can be exactly decomposed into unit-length intervals, and that we take the arithmetic average of the ARCs related to these unit-length intervals.

Rigor? I am more than certain that all involved have in their minds a function that is defined everywhere in $[a,b]$, and any other regularity condition that might be needed -after all, this is business calculus, as the OP informs us.

As many people mentioned already, there is no misconception here. Moreover, there is no such thing as "statistical average" or "calculus/physics/whatever" average. An average is always just an average and it enjoys all the properties of an average and is computed by the same techniques no matter where it comes from.

What differs between various disciplines is the interpretation of that average (the mean or the expected value in statistics, the constant velocity that will give you the same displacement over the given period of time in kinematics, and so on, and so forth). So, I would just be honest with the students here and tell them that the underlying concept is always exactly the same, but the way the result is interpreted is problem specific.

That is the idea about an abstraction in general and it is introduced as early as the third grade when they are doing the word problems like "A dinosaur and a duck ate 17 pine cones together. The dinosaur ate 4. How many pine cones did the duck eat?" and then "Peter took a trip to the village 17 miles away. He went 4 miles by foot and then caught a ride in a car. How many miles did he travel in a car?". The kids should understand that

a) It is exactly the same subtraction problem $$17-4=13$$ from the abstract point of view.

b) Those are different 13's based on the context: in the first problem they are 13 pine cones eaten and in the second one it is 13 miles traveled (so Peter didn't eat 13 pine cones and the duck didn't travel 13 miles in a car).

If they understand both (a) and (b) clearly, you may declare your mission accomplished.

Just my two cents.

Your question is a good one because it relates to a mathematical fact and a pedagogical issue. The mathematical fact is that, for a continuous function, the average rate of change along an interval is equal to the mean of multiple average rates of change along subintervals of equal width. The pedagogical issue is that most of the time, teachers should encourage students to rely on prior skills.

That's why it might seem pedagogically unsound to instruct a business student who is used to finding an average by taking a sum and then dividing by the number of data points that only the first and final values in an interval are needed when finding an average rate of change. But I wouldn't think of it as a lie. I think what's needed is an emphasis using an example of how an average rate of change is distinct from other averages and therefore requires less calculation.

Alice and Bob are given annual census data for the population of Chicago for every year from 1900 to 2000 and asked to find the average annual rate of change in that population from 1900 to 2000. Alice finds 100 changes in population by subtracting the population from one year from the next, and she takes the average of those 100 numbers. Alice has found the average rate of change from the average of 100 annual changes. Bob finds the difference in population between 1900 and 2000 and divides by 100. Bob has found the average rate of change from the total change divided by 100 years.

Alice isn't wrong, but you students definitely want to be like Bob because Bob did less unnecessary calculation.

I think that answers your main question, but there is a little bit more potential for confusion.

In most introductory statistics courses, students learn about "average" as a synonym for "measure of central tendency" in a list of numbers. It's only in that context that the word has the potential to be ambiguous. For example, a typical person buying a home is probably more interested in the average assessed values of single family homes in a neighborhood as a median instead of as a mean. A good introductory statistics course will present the distinction between average-as-median and average-as-mean as part of its pedagogy, regardless of whether the course includes "average rate of change" or not.

I think that genuine potential for ambiguity about the word "average" should not be conflated with anything taught in a calculus course.

• What I think got deemphasized in this discussion is that the "first and final values" you use in computing the average rate of change are the first and final values of a different quantity than the one you are taking the average of (population vs. rate of change of population). In my opinion this is the first thing a student needs to be reminded of if they ask why intermediate values aren't being used. That total population change divided by number of years it took can sensibly be characterized as average population change per year is intuitive. Relating it to the prior notion of average... May 8, 2023 at 19:46
• ...can come later, after more of the machinery of calculus has been developed. That the area under a curve can be calculated simply by subtracting two values stands out as one of the miracles of calculus. The sense of miraculousness somewhat dissipates once it's emphasized that the values come from a different curve and that, given what the two curves represent, it has to be that way. May 8, 2023 at 19:47

I have the common misconception in my business calculus classes that the average rate of change, say from $$x=1$$ to $$x=5$$, is the statistical average of the rates on the four unit intervals $$1$$ to $$2$$, $$2$$ to $$3$$, etc.

This previous answer establishes that Average Unit Cost (AUC) is the average rate of change associated with the Total Cost (TC) function.

The average unit cost from $$q=a$$ to $$q=b$$ units is $$AUC(a,b)=\frac{TC(0)+TC(b)-TC(a)}{b-a}\tag{discrete}$$ $$AUC(a,b)=\frac{TC(0)+\int_a^bMC(q)\,\mathrm dq}{b-a},\tag{continuous}$$ where MC stands for Marginal Cost.

If there is no fixed cost $$TC(0),$$ then $$AUC(a,b)=\frac{TC(b)-TC(a)}{b-a}=\frac{MC(a+1)+\ldots+MC(b)}{b-a}\tag{discrete}$$ $$AUC(a,b)=\frac{\int_a^bMC(q)\,\mathrm dq}{b-a},\tag{continuous}$$ that is, the average unit cost is a genuine statistical mean.

How can I tell them not to compute a statistical average without lying to them ?

In the above example, computing the average rate of change using the statistical method amounts to ignoring the fixed production cost $$TC(0).$$

• I'm no expert in business, but suppose a microchip factory costs \$20 billion and that the marginal cost of the 10 millionth microchip is \$0.20. If I use your formula, I get $AUC(10,\!000,\!000,\ 10,\!000,\!002) = \$10,\!000,\!000,\!000.20$, which doesn't seem like a useful number. Apr 28, 2023 at 20:05 • @WillOrrick Neither the OP's Question nor my response is about usefulness or interesting applications. Incidentally, they've previously created a separate Question regarding this: Application of Minimizing Average Cost. Apr 28, 2023 at 20:16 • We should not teach useless things. If I average over 10,000,000 to 10,000,010 instead of 10,000,000 to 10,000,002, the average unit cost drops wildly. Such numbers are meaningless. Apr 28, 2023 at 20:18 • @WillOrrick You're misreading my point (and the question's motivation): the OP is asking a theoretical question<--this is of interest. I'm (coming from the companion Answer) exhibiting a convenient object for an existence proof. As for what you are interested in, the Question linked in the above comment is apropos. I'm adding no more to this thread. Apr 28, 2023 at 20:49 • The formula for$AUC\$ in your answer does not match anything in the linked question or its answers and, in fact, disagrees with the formulas in that post, so I don't think this discussion would be appropriate over there. Just saying that future readers need to be cautious since the economics appears not to have been thought through. Apr 28, 2023 at 23:27