I teach undergrad stats and every year one student asks "You can't have 1.5 children" (the mean for the dataset). I am flummoxed every time to create a sensical answer. I've tried: "no one person can, but overall the sample can"; I've tried making histograms and plotting it, etc, but it still puzzles some students ... any ideas of how to make the point better?
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26$\begingroup$ “What’s the problem?” you could ask. $\endgroup$– DaveCommented Oct 12, 2023 at 18:36
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14$\begingroup$ This is on-topic here, but also note that there is a stack exchange site specifically for math educators which may already have an answer to this question. matheducators.stackexchange.com $\endgroup$– Sycorax ♦Commented Oct 12, 2023 at 18:44
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9$\begingroup$ Take the Socratic approach "Explain your reasoning". $\endgroup$– AdamOCommented Oct 12, 2023 at 18:59
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8$\begingroup$ Reminds me of Does the Average Person Exist?. I also think the Socratic method often works. $\endgroup$– GalenCommented Oct 12, 2023 at 19:04
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11$\begingroup$ I think you need to explain why the average is useful. And the answer, Imo, is because it helps you reason on totals. Eg you're building a school for 1000 families, how big should you make it. $\endgroup$– seanv507Commented Oct 12, 2023 at 19:42
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15 Answers
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Well, most people have and do take averages of something else.
Since these are college students, they probably know their GPA. Hmmm. Your GPA is 3.415? But no course gives that grade.
Fans of just about any sport will take averages of something. Goals per game, hits per at bat, whatever. Ask what sport they like. Lionel Messi averages 0.79 goals per game. You can't get 0.79 goals!
And so on.
answered Oct 12, 2023 at 19:19
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4$\begingroup$ Yes! Make it as tangible as possible. Sports. Grades. Betting odds. I remember learning about discrete probability distributions by throwing ping pong balls into plastic buckets. $\endgroup$– MikeyCCommented Oct 13, 2023 at 20:31
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3$\begingroup$ It's worth noting that the concept of a GPA is a highly culturally specific one - there is nothing remotely resembling it in the UK, for instance. It also feels like it dodges the question: they may never have thought about it, but now you mention it, how does Lionel Messi manage to score a fractional goal...? $\endgroup$– IMSoPCommented Oct 15, 2023 at 14:33
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1$\begingroup$ Even quantities which we presume are completely continuous have discrete properties limited by quantitation or the universe itself. What is height? What is time? Thus, the average - i.e. arithmetic mean - is almost always a synthetic value. Still useful though! $\endgroup$– AdamOCommented Oct 17, 2023 at 15:48
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$\begingroup$ @AdamO You're right. I mean, we could get down to quantum level. But measurement error is huge, long before then. But I didn't want to get into that sort of thing here. $\endgroup$ Commented Oct 17, 2023 at 16:15
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1$\begingroup$ @PeterFlom Most UK qualifications have a cumulative grade, rather than a rolling average: each module is graded, the grade is weighted, and the weighted grades added up to a total. The overall qualification defines the required modules, and their weighting, and the final grade compares your total against the highest you could have attained. Finally, that percentage is used to give you a banded grade, and nobody cares about the raw number; at undergraduate level, the final result is generally First, 2:1, or 2:2. $\endgroup$– IMSoPCommented Oct 17, 2023 at 16:31
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You’re right. No one can have 1.5 children. Your observation leads to an important point: the mean value does not have to be a possible value.
You could then use that to segue into a discussion of the sampling distribution of the mean, possibly even leading to a discussion of the central limit theorem.
I like the response above for multiple reasons.
It is compassionate. Instead of dismissing the student, you mention that the observation is not only correct but leads to an important point that you have not explicitly taught. Instead of the student being a troublemaker or ignorant, the student is insightful.
It leads to an important point that the mean does not have to be a possible value, leading to more advanced topics that you probably want to cover at some point during the semester and can allude to at that moment.
Finally, if this comes up because you made a comment like, “The average person has 1.5 children,” you can clarify what this slang terminology means, as the literal interpretation, as your students point out, is ridiculous.
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3$\begingroup$ Yes, the mean does not have to be a possible value, but it could also be a possible but nonsensical value: For example, the average human has ~1 testicle. :) $\endgroup$– FrodyneCommented Oct 13, 2023 at 11:06
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1$\begingroup$ @Frodyne That's completely sensible. I have 1 testicle. But you're right. While it's possible, it's kindof an "anti-mode", and not at all representative of the population. $\endgroup$– ArthurCommented Oct 13, 2023 at 13:24
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1$\begingroup$ @Frodyne I think the problem here is the phrase "the average human". Is he a statistical construct which could not actually exist, or does it mean "the typical human"? The typical human has zero or two testicles, depending on sex. A human with one testicle is not typical. $\endgroup$– David42Commented Oct 13, 2023 at 18:18
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1$\begingroup$ @phoog: I'd also use the occasion to introduce median and mode. Where I am, in addition most families have 1 child (slightly more than 50 % of families). I'd therefore likely choose a different distribution where the mode doesn't cover "most of the X". For "the average x has" I always cite a biology prof: "Remember, the average normal human has 1 testicle and 1 ovary" which illustrates extremely well that the mean has no whatsoever need to be something that can occur - I use examples like that when talking about the choice of baseline/controls for classification when I teach machine learning. $\endgroup$ Commented Oct 16, 2023 at 13:16
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An average can be viewed as an "equal allocation". If children were pooled and alloted equally among families, each family would get 1.5 children. There's nothing in the formula for an average that respects the "indivisibility" of items - as far as the mathematical formula is concerned, children can be subdivided just like pizzas or acreage.
The knowledge that children are not, in fact, divisible can be used in the interpretation, rather than the calculation of the average. Finding that the equal allotment requires 1.5 children in each family means that it is impossible to equally allot whole children among families. The observation "but you can't have 1.5 children" gets at this notion - even though we can calculate the numerical average, we cannot in practice equally divide children among families, since we can't actually allot 1.5 children per family. The average is a mathematical exercise of what it would take to do it, not a claim that one could do it in practice.
answered Oct 12, 2023 at 19:41
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$\begingroup$ While this is thoroughly true, it's unlikely to help a student who doesn't understand this in the first place! $\endgroup$ Commented Oct 14, 2023 at 15:55
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2$\begingroup$ @LukeSawczak I think the point is that the student has understood - they are making a valid point that the raw mean doesn't have a real-world meaning in this example, and requires additional interpretation. The next step is then to say why the mean is still useful - e.g. because you can compare "1.5 children" against "1.6 children", and make a meaningful interpretation of the comparison. $\endgroup$– IMSoPCommented Oct 15, 2023 at 15:17
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The OP hasn't made clear exactly what's being said to students to prompt the given pushback.
But there's a linguistic wrinkle that I think has a chance trigger this: a statement like, "The average person has 1.5 children". Well, the average person isn't really a person. This is shorthand for, "The average number of children for all people is 1.5". If necessary, this bit of shorthand should be clarified; I think taken completely literally, as if it were a singular person instead of an average for many, it is legitimately a weird bit of specialized grammar.
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6$\begingroup$ That’s the “slang” to which I refer on my answer. Taken literally, it’s ridiculous, yet such phrasing is frequent and usually understood. $\endgroup$– DaveCommented Oct 13, 2023 at 13:23
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1$\begingroup$ Right, in everyday language, “the average person” suggests real representative individuals, so the intuition is much more like the median or mode. $\endgroup$ Commented Oct 14, 2023 at 5:52
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You could show an example of a simplified version of the problem.
Reduce the sample size to two families; the first with one child and the second with two children.
Ask them to solve for the average number of children.
It is possible that larger numbers temporarily confuse the issue and reducing it to a very simple form may grant the student an "Oh! That's obvious!" moment.
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1$\begingroup$ I would use this approach. It might be obvious the person doesn't already understand means or averages, but a clear simple argument like this will clarify how means/averages work. It's just a metric, not a physical representation of how many children a parent could have. $\endgroup$ Commented Oct 13, 2023 at 19:20
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You could change the scale of the estimate, for ex. instead of saying a (one) person has an average of 1.5 children, you could say that 10 people have an average of 15 children, or that 100 people have an average of 150 children.
answered Oct 13, 2023 at 12:32
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$\begingroup$ It's even more concerning that a statistics student wouldn't realize that those are equivalent ratios. $\endgroup$– AdamOCommented Oct 17, 2023 at 15:50
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The simplest and most concise lay explanation of the mean is a quantity which "balances the scale".
In other words, if the observed sample were physical weights taken along a symmetric number line, the mean is a point which would quite literally balance that line aright.
You can make many other good arguments, like that it has convenient mathematical properties, that it is a precise and well-behaved statistic in most experimental designs, and so on and so forth. But these are kind of practice-based understandings and not motivating examples for students.
Statistics on the whole has suffered because students do not think deeply about their intuition before implementing the methods. So, it would be good for students to struggle with examples. We don't need to be as astute as Fisher to do good stats, but statistics is not a spectator sport either - many students are going to self-select into other disciplines as they get down into brass tacks.
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The mean is a characteristic of a population; it is not a characteristic of an individual (nor is it a characteristic of a 'unit of analysis' sampled from some population of such units).
In the case of 'mean number of children' the population in question is a population of families. This allows us to compare populations (e.g., how does the mean number of children in UK families compare to the mean number of children in German families), and to account for how a population changes over time (e.g., how does the mean number of children in US families this year compare to the mean number of children in US families last year).
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1$\begingroup$ (+1) I found this answer only after I wrote mine and it seems to me we are saying the same thing. Personally, I didn't find the top answers particularity enlightening. $\endgroup$– darioberCommented May 2 at 10:39
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You make a valid point about the common practice of taking averages of values that aren't always naturally occurring numbers. Averages are a useful tool for summarizing data and comparing performance in various fields, and they often involve rounding or interpreting values in a meaningful way.
In the context of college students and their GPAs, it's true that GPA values typically include decimals, even though individual course grades are usually whole numbers. The GPA system uses weighted averages to provide a more nuanced reflection of a student's overall performance.
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The mean is a fraction
The mean value consists of a fraction of the sum of individual observations. When expressed as a decimal, a fraction of the sum of individual observations can maintain the same units as individual observations without reflecting their real world constraint (indivisibility).
$ (1 \text{ apple}+2 \text{ apples}) \div 2= \frac{1 \text{ apple}+2 \text{ apples}}{2} = \frac{3 \text{ apples}}{2} = 1.5 \text{ apples} $
Constraining the mean
A real world constraint can be imposed according to which the solution must be a whole number, $\mathbb W$ (or whole numbers). The constrained solution can then be obtained from the dataset mean, $\bar x$.
$ f(\bar x)= \begin{cases} \bar x&\text{if }\, \bar x \in \mathbb W\\ a \text{ or } b&\text{if }\, \bar x \text{ is a decimal}\\ \end{cases} $
where $a=floor(\bar x)$, $b=ceil(\bar x)$
If $\bar x$ is a decimal, the constrained answer becomes: "On average, people have $a$ or $b$ children". Individual observations in the dataset may or may not correspond to $a$ or $b$.
Constraining predictions from the mean
In a different context, assuming $\bar x=1.136 \text{ children/woman}$, you could simulate drawing groups of $10 \text{ women}$ from a Poisson distribution with $λ=1.136$. You could derive a prediction for the $\text{# of children}$ corresponding to $\text{10 women}$ drawn from the distribution directly from $\bar x$.
$ \text{prediction}_{\text{mean}} = \frac{1.136 \text{ children}}{\require{cancel} \cancel{\text{woman}}} \cdot 10 \require{cancel} \cancel{\text{women}} = 11.36 \text{ children} $
Your prediction will never be correct, because women produce whole children (not children fractions). As before, you can add a real world constraint (prediction or predictions must be $\mathbb W$) to make your prediction relevant.
$ \text{relevant prediction}= \begin{cases} \text{prediction}_{\text{mean}}&\text{if }\, \text{prediction}_{\text{mean}} \in \mathbb W\\ A \text{ or } B&\text{if }\, \text{prediction}_{\text{mean}} \text{ is a decimal}\\ \end{cases} $
where $A=floor(\text{prediction}_{\text{mean}})$, $B=ceil(\text{prediction}_{\text{mean}})$
Assuming a 50 points reward if the prediction is correct, and a points penalty corresponding to the absolute value of the difference if the prediction is incorrect, trends of cumulative points can be obtained over 250 simulation iterations.
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1$\begingroup$ So, "on average, a flipped coin has zero or one head; a yes-no question is answered with either zero or one yeses," and so on. Not terribly useful, is it? ;-) $\endgroup$– whuber ♦Commented Oct 13, 2023 at 20:53
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1$\begingroup$ A couple could ask a reproductive health professional, "Based on your data, how many children will we have?". The dataset could be binary (either 0 or 1 children for each observation). The unconstrained answer could be "On average, people have 0.77 children", and the couple might be puzzled. The answer with a real world constraint would be "On average people have 0 or 1 children". $\endgroup$ Commented Oct 13, 2023 at 21:21
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1$\begingroup$ "... 0 or 1 children" might be correct, but it strikes me as useless. $\endgroup$– whuber ♦Commented Oct 13, 2023 at 21:22
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$\begingroup$ "You can't have 0.77 children"—said the couple. $\endgroup$ Commented Oct 13, 2023 at 21:38
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1$\begingroup$ Out of 4 couples like you, roughly 3 will have 1 child, while 1 stays without child. Sure, details about the distribution can be more useful than just its mean. Naming the possibilities, however, may be less useful than the mean - the more so as people often already know the possibilities. Even more useful in case the couple knows their stats: of 13 couples like you that were included in a study, 10 had one child, 3 stayed without child. $\endgroup$ Commented Oct 15, 2023 at 20:38
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Already plenty of answers, but nevertheless my 2p: I imagine that students don't have a problem understanding where the mean of "1.5 children" comes from. Rather they are puzzled by its interpretation.
I think the puzzle/paradox is resolved by pointing out that the mean is a property of a distribution, or set of numbers, not a property of an individual. In particular, the mean is a measure of central tendency and as such there is nothing odd with being physically impossible for a single measurement.
I imagine that intuition tends to replace the mean for the mode, which is in fact the value observed most frequently. So why is the mean so often used? That may be a different question but I'd say the answer has to do, at least in part, with the mathematical properties of the mean.
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Probably they're not genuinely puzzled
You say this is undergraduate level stats? This kind of question belongs in primary school. In order to be undergraduates, clearly they must understand it.
Consider the possibility that they're messing with you. The reason this comes up every year is that it's mildly amusing, so you can definitely expect someone to make the joke and think they're being original. If you then start floundering and get distracted from what you're supposed to be teaching, the class will find that even more funny, and a few jokers in the group will quite likely push you more.
I suggest not rising to the bait. The answer, as you've said, is "Sure, no one person can, but over the sample (or over several people), that can be the mean". And then move on.
If they keep on asking, tell them you can lend them a primary-school maths book to help explain fractions, with pretty pictures. Push the joke back on them. And then really move on.
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8$\begingroup$ Unfortunately I teach at a community college in CA. There are no math pre-recs in community colleges in CA so unfortunately I get many students who do not know what fractions are; how to calculate a percentage; etc. This question really vexes them. $\endgroup$– RLDavisCommented Oct 13, 2023 at 19:31
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4$\begingroup$ This is probably an overestimation of the capacity of school systems across the world to teach it to all students who reach university or college. See iase-web.org/icots/9/proceedings/pdfs/… for a case study. $\endgroup$– J-J-JCommented Oct 13, 2023 at 19:37
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2$\begingroup$ What @RLDavis says is true for most community colleges across the U.S. (i.e., the colleges that most U.S. students now attend). Exact same situation for me at a community college in NY (where the trend for open-admissions was first initiated, in fact). $\endgroup$ Commented Oct 13, 2023 at 20:16
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3$\begingroup$ I regularly have customers, many of them college-educated, who do not understand how I can forecast daily retail sales of 0.7. After all, you can only sell 0, 1, 2, ... units. I am afraid you are vastly overestimating the mathematical sophistication of undergrads. $\endgroup$ Commented Oct 13, 2023 at 21:44
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4$\begingroup$ So this question has 11 answers and it occurred to no-one (incl. the OP in writing their question) to ask what the background of the students is. (I guess instead everyone assumed their stereotype of "college student".) For me at least the most useful lesson from this thread is that effective scientific communication (communication of any kind really) is designed with the audience in mind. No fancy formulas are going to be help someone who doesn't know what fractions are. $\endgroup$– dipetkovCommented Oct 14, 2023 at 13:39
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It's never too early to teach children about gambling. Here's a good classroom exercise to give some intuition. It does require some coding or spreadsheets from you.
Let's assume the mean is 1.5 kids/woman.
Have students devise "sensible" rules for predicting how many kids there will be in a future sample of 10 women.
Some examples are
- Use the mean of 1.5 -> 15 kids
- Round down to 1 (floor, mode) -> 10 kids
- Round up to 2 (ceiling) -> 20 kids
- Median (Approximate) -> 18.5
Simulate drawing groups of 10 women from a Poisson with $\lambda = 1.5$. Show that all 10 outcomes are positive integers. If the rule gets it exactly right, it wins $50 of imaginary currency. If wrong, the rule loses the absolute value of difference. Keep track of total winnings as you add samples and plot them on the y-axis versus the number of draws. You should get something like this (most of the time):
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Saying there are averages in sports just kicks the explanation further down the field (pun intended).
The confusion comes from thinking the average exists by itself, as a lone parameter. It doesn't, it includes a variance or standard deviation. So the variance will always show real numbers of children are included, no matter how many families you count. If you weren't counting children, and say measuring the height of similar height people, and the mean turns out to be 1.5 m, the variance could feasibly show they are localized around 1.5 m within a few 0.1 m. Something similar couldn't happen with 1.5 children.
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Ask them if the same dataset would represent, say, fuel consumption, would they find a mean of 1.5 strange. If not, point out that it makes things easier that we can calculate the mean without knowing what it represents (and you can drive this home with examples like "mean number of socks sold" that's by their logic should be a multiple of two).
