I kinda already knew this implicitly and even manged to solve the MPG problem (didnt try the other) BUT I never explicitly thought of a mean as the value that can be replaced to get the same result. This is a very useful way of thinking of it.
@@MathTheWorldwhat about people who define the mean as the first moment of a set of data points? How does that translate to your concept of “mean”? Usually we call a “mean” the first moment, the geometric mean is the same as if you took the log of your samples and calculated the first moment. But then you said that the RMD is a “mean,” which is the second moment of your distribution.
In electrical engineering, the harmonic mean is used to find the aggregate value of resistors in parallel. The RMS mean is used to find the DC equivalent of a varying voltage waveform for equal power.
Resistors in parallel is actually more of a harmonic sum. Harmonic mean would be having a bunch of resistors in parallel and asking if you wanted to replace them with the same number of resistors of equal value in parallel, what would that value be.
I typed this out before the end of the video: The next time I see someone try to argue that mean is completely synonymous with "average," I'm going to ask them "harmonic, arithmetic or geometric?"
Nice explanation (from another mathematician)! That said, I think the reason no-one got your answer for Q1 is that it's too ambiguous. I was quite surprised when you said that the mean you wanted was a s.t. (1+a)^3 = (1+x)(1+y)(1+z) - that's not at all obvious from the phrasing. The arithmetic mean would be completely appropriate if, for example, you took all of your profits at the end of each year, starting each financial year with a fresh $1000 investment. Equally, you could read it (as I first did) that the three percentages each refer to different investments, each of which you held for three years. Again, the arithmetic mean would be appropriate there.
@@octopodes7619 I was also a bit confused but the phrasing made it sound like it was talking about the three-year return rate on three separate investments.
The issue with problem 1 is not people not understanding of what an 'average' is. It's with the unclear statement of the problem. It is not explained what a return on an investment is, or how those returns work. Without any further explanation, I could assume that you invest 1000 units of money, then get 133.1% of 1000 over year 1, get 72.3% of 1000 over year 2, and 126.6% of 1000 over year 3.
Exactly my first thought. If the value is not reinvested, it's the arithmetic average of the yearly gain ratios, otherwise it's the geometric one. It's the kind of unclear question that you need to make arbitrary assumptions for. Simultanuously, university (at least in math and science) is teaching you exactly not to do that. Many teachers don't see how counterproductive this inconsistency is in the students learning process.
Additionally, he uses the plural form of 'investment,' which gives the impression that he invested his money in three different assets, rather than indicating that the return on a single investment changed over the three years.
The mathematician in me knows you explained very clearly how to think about what mean you need. The nitpicker points out that the spelling is "reciprocal", where the "re" is the same back-meaning prefix as in "return" and the "pro" is the same forward-meaning prefix as in "promote".
@@matthew8184Surely California is a proper noun and Cal would thus be written with a capital. So Sacramento, California. Lower case cal stands for "calories". The ic is of course the Old English pronoun meaning either "me" or "I". So the whole word "reciprocal" roughly means "[come] back to me with advantageous calories". It's generally used as an interjection when ordering junk food.
current leading hypothesis I can find on the etymology of "reciprocal" is Latin "reciprocus" ("back and forth", "alternating")+ "-alis" (an adjectival ending), with "reciprocus" being "reque proque" ("both back and forward") + "-a/-us/-um" (Latin first and second declension endings, forming a noun or adjective directly from the stem), with "reque proque" being a pretty archaic formation, since "re" as a stand alone adverb does not survive into what records we have of Old Latin, yet alone Classical (and her sisters don't give us anything to reconstruct a stand alone form, only ever the clitic form). "-que" is a clitic reconstructable all the way back to Proto-Indoeuropean as "-kʷe" and is more or less "and" (although as other formations in Latin attest, the meaning was broadened to "any, every" likely through "as" in addition to it's "and" meaning, as seen in e.g. ubique, undique, quoque) [that same particle shows up in English! it comes to Proto-Germanic as "-hw" (which became "-uh" due to epenthesis when suffixed to a word ending in an obstruent), and is seen in the word "though" (which is etymologically "and that", from PIE), "yeah", but was otherwise more productive in Gothic than the West Germanic languages, in which it was the least productive in English.] "reque proque" getting worn down to "reciproc-" is likely due to assimilation (qu+us -> cus) and the confusion of u and i in unstressed syllables (where unstresed very short "e" gets raised/confused with "i").
I think it’s important to state that if you phrased the car questions as: ‘the two vehicles use about the same amount of fuel’ then the answer would be 25, great video
I’m a CS student (minoring in maths). I practically memorised a formulae sheet of different averages, and I agree with your thesis. From experience, I have a bad habit of defaulting to the arithmetic mean when faced with averaging problems. Thank you for reminding us that understanding the problem before crunching numbers ❤️🙏
I remember a few months ago I saw a website explaining the abstract definition of a "mean" by fitting some set of criteria. I regret I haven't been able to find it since then even though I search for it regularly, but I remember two of the defining properties of a mean is that it must generalize to work for any number of elements of a set (that is, it can't just work for 2 values and not for more) and that in the special case of 2 values, it must return a number that is between (inclusive) them (assuming ordering is possible), which the three most popular means obviously fit. Since I've only found one place that even suggested there was a textbook definition for a mean, I'm guessing it's not a very popular idea, but I still find it interesting that someone somewhere did define it fairly succinctly
a mean for a set of numbers must return a value within the set of numbers (or could be undefined, as geometric mean might be with negative values). all the types of average i can think of all work by this mechanism: pick a monotonic function f(x), and for each value in your set, compute f(x). take the normal arithmetic average of those, and compute f-inverse of that. your choice of f(x) will be motivated by what trait you are trying to capture. f(x) = x or logx or 1/x or x^2 generate the 'normal', geometric, harmonic, and RMS averages respectively.
@@theupson The commonly used means also have another useful property: If you multiply all numbers with the same value, the mean will be multiplied with the same value. For example, the mean of 2a and 2b is 2 times the mean of a and b.
A mean (of a finite number of values) is a function mₙ : [a, b]ⁿ→[a, b] (where [a, b] can be extended to ]-∞, b], [a, +∞[ or to the whole ℝ) which satisfies these properties: i) Continuity, we want that a small change of the arguments can only cause a small change of the value ii) Transitivity under the action of Sₙ on its arguments, in other words, invariance of the value when the arguments are permutated iii) Idempotency: mₙ (x, ..., x) = x iv) Monotonicity on each argument v) Invariance under self-composition, as to say for k ≤ n: mₙ(a₁, ..., aₖ, x, y, ...) = = mₙ(mₖ(a₁, ..., aₖ), ..., mₖ(a₁, ..., aₖ), x, y, ...)
@@ciana42Can you share more about where this definition and others like it are sourced from? I've been meaning to dive into a measure-theoretic study of statistics which uses real analysis as the backbone, but a lot of the texts I've been finding seem to be a bit more rigorous than what I'm comfortable with (model theory?!). This definition you've presented here seems fairly palatable and I'd love to get my hands dirty with problems from whichever text you're quoting; it appears to be presented at the level of rigor that I find comfortable at the moment.
I’m a Junior physics mathematics dual major, thanks to this video I can now completely visualize the mean, and why certain calculations are what they like the average of a function, or why we care about the nth centralized moment. Hell even why a convolution is defined how it is, and why a Fourier transform turns a convolution into multiplication, or why the Fourier transform is almost like a weighted average. It’s all just finding the central value, or in the cases above, an analogous distribution. I can go on and on about orthogonality but my point is this gave me the best understanding of the mean that I wish I had before college.
This is the best explanation I have found. I normally just peek at the correct one, but was never able to explain it with such clarity. An important detail if you are solving an exam, most of the time, mean means arithmetic mean, even if it does not work, so be sure to confirm.
@@gcewing Yes, but does it justify usage of root mean squared instead of mean of absolutes? Or is there a deeper meaning here? This thing was on me for sometime and i never got a satifying answer. I know that given a continuous function, it is easier to calculate rms, as we get integral-friendly formula. (which we dont for mean of absolutes). But this still feels like rms isn't giving the best representative, we are looking for.
OOOh but there's even more! You didn't even touch the generalized power mean, where you get an infinite set of different "mean" values based on a power number, p. All you do is raise each value in your data set to the power p, add them together, divide by the number of data points, and raise everything to the power of 1/p. Actually, all of the means you mentioned in this video are generalized power means: Harmonic mean -> p = -1 Geometric mean -> p = 0 (this does involve raising something to the power of 1/0, but the limit as p->0 gives you the geometric mean, so usually p=0 is set as the geometric mean) Arithmetic mean -> p = 1 Root mean square -> p = 2
@@PavanKumar-xv1hg No, @foozlebagel7488 is right. [(a^p+b^p)/2]^(1/p) -> sqrt(a*b) for p->0. This can be proven by calculating the limit of the natural logarithm of the expression using de Hospital's principle.
@@Integer0 Yeah, your right ! I tried it out and got it ! P.M. of (a_1,a_2,.....,a_n) n total values is P.M = L = ({Sum[i=1 to n] (a_i)^p} / n )^(1/p) Taking log on both sides, ln(L) = ln({Sum[i=1 to n] (a_i)^p} / n ) / (p) As limit p tends to 0, lim(p-->0) ln(L) = ln({Sum[i=1 to n] (a_i)^p} / n ) / (p) ln(L) = ln({Sum[i=1 to n] (a_i)^p} ) - ln(n ) / (p) [Just taking out n] Let p = 0 in R.H.S to see if we get it in an indeterminate form ln(1+1+1......+1 ) - ln(n) / 0 [1+1+1... n times] ln(n)-ln(n)/ 0 = 0/0 So we can send the limit to the hospital (Use L'hopital's Theorem lmao) lim(p-->0) ln(L) = ({Sum[i=1 to n] (a_i)^p} )^(-1)*(Sum[i=1 to n ] (a_1)^(p)*ln(a_i) ) / (1) [differentiating the numr and denominator separately] Putting p = 0 in R.H.S ln(L) = ({Sum[i=1 to n] (a_i)^0} )^(-1) * (Sum[i=1 to n ] (a_1)^(0)*ln(a_i) ) [ a_i ^ 0 = 1 ] ln(L) = ({Sum[i=1 to n] (1) )^(-1) * (Sum[i=1 to n ] ln(a_i) ) / (1) [1+1+1... n times = n & sum of logs ln(x) + ln(y) = ln(xy) ] ln(L) = n^(-1) * ln(a_1 * a_2 * a_3 * a_4 * .......*a_n) ln(L) = ln((a_1 * a_2 * a_3 * a_4 * .......*a_n)^(1/n)) or just, L = P.M(p-->0) = (a_1 * a_2 * a_3 * a_4 * .......*a_n)^(1/n) = G.M of (a_1,a_2,.....,a_n) So it follows what O.P. said. P.M(p) = ({Sum[i=1 to n] (a_i)^p} / n )^(1/p) Following this definition, P.M(2) = R.M.S P.M(1) = A.M P.M(p--->0) = G.M P.M(-1) = H.M P.M(-2) = anti H.M. square ?? P.M(-2) and P.M(2) has kinda like the same vibes as A.M and H.M [Personal thoughts feel free to ignore] We also know P.M(k)>=P.M(k-1)>=P.M(k-2)>=..... >=R.M.S>=A.M>=G.M>=>H.M [I don't know if H.M>=P.M(-2)>=P.M(-3)... and so on is true or not tho] Its my first time writing a proof on yt so tell me if some part is unclear to you guys(I tried just 2 terms and extended it to n terms), Thanks !
Isn't this just the definition of L-p norm spaces for sequences? A sequence x = x_n is in the space L-p if the norm ||x|| = (sum |x_n|^p )^1/p is finite
I appreciate that you made a point of using mean and average quite interchangeably. Some people insist that "average" must only refer to the arithmetic mean, which never sat right with me...
I was also recently discussing means in the classroom, for me in the context of heat transfer. For heat transfer in a pipe, the logarithmic mean of temperature is an important quantity. It's not the type of quantity that people would normally think of as a mean, even compared to things like the geometric and harmonic means. This actually inspired a series of posts in my blog about the subject, where I ultimately discus how to use principles from calculus to derive a fairly general class of means. I don't think I can link it here, but my blog is in my bio.
Thank you to point out. This problem can be very harmful in reality and those mistakes happen a lot in real work environment with grade students in stats, data science, data analysts and comp. science, also happens in scientific publications as well. This problem is most underestimated in real life and in academia - it leads to wrong decision and strategies in business and data manipulation in academic publications.
So I've been doing engineering and product management for 20 years and this is the best explanation I have ever seen.. I also realise some really bad "means" I've applied in the past
A mean can be just any number between the min and max, so the general definition is using the probability density function (doing integral that kind of thing). Thank you so much for pointing out one very inspiring way of thinking about means! But this way still does not cover all ways of thinking about means
Glad to find a better explanation of this! I encountered all of these in AP stats, but I only memorized where to use them, rather than properly understanding why they were different and where they should be used
people don't understand percenatages, the first one they get, but when there are 3 terms this would be +33.1% of X, -27.7% of Y, and +26.6% of Z so with an initial capital of 1,000, that would mean X =1,000, Y= 1,331 and Z= 962.313 the gross interest would be 21.8288% based on 1,000. but if you want that over the 3 years it would be ∛1.218288, which is 1.06803, so around 6.8% PA. you can't average out the percentages, as these percentages are of different values. for the second one you have 1/15 + 1/35 = (7+3)/105 = 10/105 = 2/21 so 2 vehicles each with 1 gallon of fuel, 21 miles for each vehicle. so that would be 2/42 combining the milage or basically 1/21 for each vehicle. what was with all that other stuff in the video? this is basic adding of fractions. it seems that people also don't understand fractions either, including "Math the World" I know technically it is a ratio being described as a fraction. but the rule still applies, you need a common denominator. but the 1/21 is correct, for 1 gallon of fuel total the total milage would be 21 miles for the two vehicles.
There's kind of a trap in the commonly used phrase "adding a percentage". It's not an addition, it's a multiplication, and many things become clearer when you think of it that way. I don't remember anyone pointing that out to me in school, though.
@@gcewing a percentage is just a 100th of a number. You can add or multiply these fractions, convert it to a decimal, etc. but you can absolutely add them.
The problems themselves I did not find particularly complicated (except I found the first problem a bit vague, but your solution was one of the options). What I liked is that for the first time did I understand that there's a fundamental approach to understanding which mean you need to calculate.
I love the quasi-arithmetic mean and how it generalizes different means. It utilizes the idea that you have some function which is used to find the total value, and to find the mean you have to use inverse function of it
For the mpg problem the units (dimensions) of the required solution (miles per gallon) told us the approach to use. Find total miles, find corresponding total gallons used, and divide one by the other. In other words: for any specific distance, d, travelled by each car, calculate the total miles that were travelled (2d), and how many gallons were used (d/35 + d/15), then find mpg = miles/gallon. I used d=105 for convenience because it is the lowest common multiplier of 15 and 35. Then mpg = 2*105/(105/35 + 105/15) = 21mpg. (As it's a ratio, any distance can be chosen.) As a follow up I plugged the values into the formula for the harmonic mean and got the same answer. (Edit: I've just noticed that if you cancel 105 from numerator and denominator you actually have the formula for the harmonic mean: HM = n/(Σ(1/x)) .)
Thanks for an explanation of this that I (as a mathematician) can use to explain these concepts to non-mathematicians! I have a concept I call, "transformed mean," which is not mine originally, but is almost never introduced when different kinds of mean are discussed. Basically, for any function f:R→R, which is strictly monotonic (increasing or decreasing) over some interval, the f-transformed mean, M(f; xj) of a set of n numbers x1...xn, all within that interval, is M(f; xj) = f⁻¹([∑ f(xj)]/n) which just consists of taking the simple, arithmetic average (mean) of the transformed numbers, then "untransforming" that result. Under this definition, when f is any (non-constant) linear function, the f-transformed mean (FTM) is just, trivially, the arithmetic mean. When f is a logarithmic function, the FTM is the geometric mean. When f is the reciprocal function, the FTM is the harmonic mean. When f is the square function, the FTM is the RMS. Etc. And of course, each of these means has a corresponding transformed sum. Just remove the division by n in that definition. The important part of your presentation here, IMO, is the emphasis on just what operation is being used to combine the numbers; and that this is what determines how to take the right kind of mean. Fred
The MPG problem is an excellent example of where the harmonic mean (HM) is called for. Another class of perennials for using the HM, is rate-of-work problems.
The funny thing is that if the returns are small then the result of that will be very close to correct, e.g. averaging +1%, +2%, +3% the stupid way gives +2% while the correct answer is approximately +1.997%
👏 wow!!! just brilliant how you explained everything. I understood all these concepts as I'm an IT engineer but in a very haphazard and complicated way. Just see the 1 whole page I used for the first problem to get the correct answer. You explained it so beautifully and I wish I had you as my teacher and mentor growing up. Keep up the brilliant work. I'm going to become a patron if I can as u just swept me off my feet with how beautifully and brilliantly you explained it, just wow!!!
Ohhh I rememebr this now! We had this in primary school! We were taught about the arithmetic mean, but then I encountered the task where there were 2 vehicles with different speed, one was used for the first section of the path, the other for the second, with sections not being equal. Their speed and the distances were given and the task was to calculate the -average- mean speed. My brain hurt bad while trying to understand why the arithmetic mean of the speeds doesn't give the same value as when dividing the total distance by the total time spent.
@MathTheWorld Thank you for your work! This may be a young channel, but I think you guys will become my favorite math TH-cam channel. Most other math channels seem to go for exotic and mind-blowing stuff, or normal but extra hard stuff; in both scenarios being removed from the needs and realities of using math. A math channel than can actually demonstrate why math is useful/powerful is something that has been sorely lacking in TH-cam. I look forward to watching your videos for many years to come. Ganbatte kudasai!
I majored in math and physics and minored in statistics and probably would’ve done the same as the students for the first question. I learned of different “means” and applied them to data in varying contexts, but many texts explicitly state that the average is the arithmetic mean, so I’d assume that’s what the question is asking for. Other comments have also explained why that could make sense given the description of the “investments”. But, often the lectures provide further context. The instructor may have done example problems that imply a certain interpretation of the plural investments, or, more likely, that in this course “average” means “mean” and not necessarily “arithmetic mean”.
Also, the term I know for situations where an appropriate mean is to be applied is "effective". For example, you can ask for the "effective interest", which is the interest you'd need to get the same result.
you could do it "the way you learned in school" on the average returns but you have to understand that each year has a different starting value. the return values given were not the normalized returns based on the initial 1000.
I've been wondering how to think about the mean for ages and finally got a satisfactory answer. I learned all these types of means but never knew what fundamentally made them a 'mean'
Fun fact: for any list of postivive reals, the arithmetic mean is always at least the geometric mean, which is in turn at least the harmonic mean. There are some uses for this nice classical inequality chain. This is part of the so-called "mean inequality chain" QM-AM-GM-HM.
You know what's interesting is that I know all of this and have used many of these various calculations. Depending on the situation though. (Though I did major in Chemistry and do use math/statistics very frequently.) ( don't really know the difference by names like harmonic, arithmetic or geometric(i'm also not native english), I just calculate what makes sense in context) Mean, Average. Weighted means, standard deviantion , relative standard deviation. Population, Sampled data, extrapolation, calculating statistical outliers and if they should be "disregarded" from the mean. I just always ask myself ......the mean of what? What is it that I want to know? Statistics can be tricky to read, but also tricky to present clearly. I often read news artikels and they talk about means as if that even says anything by itself. I hate mean / median abuse by the media. (what where your deviations? Do you have a normal distribution?) Sometimes you read: Increase of 100% in people that get X disease (sure it tells you it dubbeled). ( But what was the original %? if it was 0.01%, thats just a 0.01% increase really on the total population, if it was 25% then yeah a 25% increase is alot on the total population.)....I mean seriously be clear.
I would argue a much easier way to solve the second problem is to set it up like this: Imagine the family has to drive 105 miles. A number easily divisible by both cars' miles/gallon. The first car will use 105/15 = 7 gallons to drive the distance, whereas the second will use 105/35 = 3 gallons. So, the average miles/gallon for your entire family will simply be the total distance covered for both vehicles, divided by the total number of gallons used, ie. 210/10 = 21 mpg.
that would mean, no pun intended, that both cars take 10 gallons to get there, yet the average is double that. You mixed up miles/gallon and gallons/mile if im not mistaken
@@undozan4180 The mean is the single number that could be used to replace each of the individual values and yield the same result. So if the family used those two cars to travel 105 miles in each car, they would use 7 gallons in one car (15mpg) and 3 gallons in the other (35mpg) for a total of 10 gallons. But if the family used two identical cars, each giving 21mpg, they would use 5 gallons in each car for a total of 10 gallons, Same result. That shows that if the cars travel equal distances, the mean fuel consumption of 15mpg and 35mpg is clearly 21mpg. You were mistaken.
I don't see how your method is really any different. His solution is the same methodology but while having each car drive 1 mile. Instead of 7 and 3 gallons, he gets 1/35 and 1/15 gallons respectively. All you really did was add an additional step by multiplying by the least common multiple (105 = 3 * 5 * 7).
9:30 This is my biggest gripe with all these sorts of websites and has raised my respect for my teachers who actually took time to explain and encourage us to read the explanations of the terms we are using instead of just blindly applying formulas that we rote learn.
Great video, I figured out the meaning halfway thru since I’ve always assumed mean and average were similar enough without thinking enough about what a mean really is. Thanks
Here, some types of means are discussed that share the property that if that mean replaces each actual parameter value, we get the same outcome. This is a valuable insight you convey to your students. But note that means can have other purposes, where those means don't have that property, like the modal value. The purpose is still the same, but thinking of them in terms of problem solving is much different, I think.
The mode also works with his definition though? For example, the mode of 1, 2, 3, 4, 4 is 4 (most common number) and the mode of 4, 4, 4, 4, 4 is also 4.
I think the issue people are having with the first question is that it says "investments" meaning multiple. But when you invest, you are technically only investing from a single pot of money 1: 100 + 33.1% = 133.1 (at the end of year 1) 2: 133.1 -36.868 (-27.7%) = 96.232 (at the end of year 2) 3: 96.232 + 26.6 (26.6%) = 121.83 (end of year 3). Furthermore, I think people are also getting confused is because you used arithmetic mean to find the average annual rate when we should of used geometric mean to account for the compounding effects over time. I got : (121.83 / 100)^1/3 -1 (1.2183)^1/3 -1 1.06803 - 1 .06803 or *100 = 6.8% average return per year for 3 years. 100 *.068 = 106.8 106.8 *.068 = 7.2624 + 106.8 = 114.0624 114.0624 * .068 = 7.7562432 + 114.0624 = 121.8186432
The other problem is when people confuse average/mean of a very skewed distribution (like wealth distribution) with the median or the mode. "The average salary is X" puts the idea that that salary is earn by the most people (which would be the mode), some people understand it as the salary that mark 50% of the population (that is the median), and almost nobody think about how it is a useless number, because if you eat 9 chickens, he eats 1, and I eat 0, the average chicken per person is 3, but the median, which is closer to reality, is 1 (well, technically depends on how you define the median of a discrete distribution, but alas). This is why these "subtleties" are not just for fun and giggles, but are fundamental tools that everyone should be able to command properly, or else we are getting lied constantly.
Here's a head-scratcher: what's the mean direction of n equal-magnitude vectors on the same surface, intersecting at their origin, each at angle 2pi/n rads from the next?
Firstly, just want to add my +1 to people saying that the first question is disastrously explained, and this was almost certainly done in order to be misleading. For shame, sir. Secondly, at 1:27 you say "assuming simple interest in this problem..." and then IMMEDIATELY start using compound interest. That makes it seem like your attempt to be misleading even got *you* a bit confused. The rest of the video is awesome.
I'm curious what about the first problem is misleading, since to me it was straight forward and I'm unsure how else to interpret it. I agree that "simple interest" is confusing phrasing, but I don't think he confused himself. I think he just meant that annual returns are like interest that is applied once, rather than interest that would compound multiple times within a year.
@@didles123 It wasn't explicitly said that those numbers where the yearly return of the investment and it wasn't explicitly said with which number the years started. Investing 1.000 in year 1 -> success, therefore investing 2.000 in year 2 -> failure, investing only 1.000 in year 3 again would be weird but possible. Same with investing 1.000 each year and withdrawing the win/loss after each year. For the correct solution you must assume the most standard and reasonable procedure. Therefore most people should get it, but you can if you are nitpicky and assume weird stuff you are not contradicting the question and get another answer.
The overall issue imo is that the "of" before the percentages is too close to the word "investments," which suggests multiple investment with three separate returns each over three years. Ending the sentence after "investments" and starting a new sentence with something like "Year 1 had 33.1%, year 2 had, -27.7%, and..." would have been better.
So... I was able to pause the video on my phone, solve both problems correctly on a Julia terminal I had open, and unpause the video before my 1 minute, company mandated screensaver kicked in on my phone. I think it took me about 45 seconds (although I didn't actually time myself). These were not hard problems. I was able to just reason my way through them. The fact that your university students couldn't get the correct answer leads me to feel like there's a failure in the educational system here. A parallel video about the training in the educational system that allowed these failures in reasoning to occur would be absolutely fascinating. The distribution of reasoning failures across backgrounds, majors, and classes that had been taken would also be very interesting and potentially worth writing a paper on.
Don't know how you're going to do this in the end and, in spite of having a pretty good mathematics education at a prestigious university, I was never formally taught anything other than the arithmetic mean. I've learned of the others since, but not had the practice required to really learn their usefulness. That said, I figured out both problems correctly fairly easily by A) determining what the rate had been over 3 years and taking the cube root because whatever average would need to be cubed and B) by figuring out how much gas would be used to drive 1 mile, adding, then dividing 2 by that number because the two cars would have driven 2 miles total. Of course, those are going to be some mean or other.
I had this exact doubt. I had to ask ChatGPT and understand the difference b/w Arithmetic Mean and Harmonic Mean. This video just tops it off very well. Thank you so much! ❤
There's a very nice generalization of these means you can see (amongst other places) on the "mean" wikipedia page : generalized f-mean. In short, if you have a monotonous function f, the f mean of a and b is the number m such that 2f(m)=f(a)+f(b). f(x) = x gives the arithmetic mean f(x) = 1/x the harmonic mean f(x) = ln(x) the geometric mean f(x) = x^2 the quadratic mean And it generalizes for instance to any f(x) = x^n (of which the limit when n goes to 0 is f(x) =ln(x), the geometric mean) And there a very nice way all the means will always be sorted, according to the n value they correspond to: Harmonic mean < geometric mean < arithmetic mean < quadratic mean, corresponding to following n values : -1 < 0 < 1 < 2 < n if n greater than 2
Thanks for sharing this. I saw it while I was doing research for this video, but I didn't think it would be a good fit for our target audience. This is the kind of idea that I'm always astounded by. A mathematician can take such separate things, and see a unifying idea across all of them.
I have thought a lot about the first question in the past. Because using the arithmetic mean on interest rates gives you the expected interest rate. So if you want to use it for one period the arithmetic mean would be best. If you want something in between it will get more complex.
I managed to get the gas milage one right, but only because A.) I knew the arithmetic mean would be wrong since you were challenging assumptions, and 2.) I was aware that more civilized parts of the world used fuel/distance instead and figured that was worth trying. Great video, I certainly learned a lot, and you earned a subscriber! Looking forward to more math videos :)
1. what a great video - this is the kind of math education thats different from just writing down formulas (it needs people that "think math" and have a way to put it into some perspective for students to get it), very good! 2. let's change the wikipedia page for means?! :)
Very helpful. I recently learned all about these different means because of a project I'm working on. This gave me a much deeper understanding of _why_ they exist, _how_ they're calculated, and _which_ one to use for a given purpose. PS: Despite my lack of mathematical prowess, I got the gas mileage question right and almost got the ROI question right too. Instead of taking the cube root, i divided by 3. Doh! I was close though.
Lovely video!! For the first problem I thought you were going to talk about weighted sums since it was not mentioned that you take ALL the money you have left from the first year investment to the next year So my first thought was "we can't know since rates aren't enough to find the average" But now I get what you mean
If you need to use a weighted average with the harmonic mean, for instance if the two vehicles travelled different distances, you can just put the weights in the numerator. For example: 100 mi / 15 mpg + 150 mi / 35 mpg = 250 mi / x.
Nice content, with great examples. I had never considered them like this together. these three rhyms - just mean of sums - reciprocal of mean of sums of reciprocals - square root of mean of sums of squares this is a bit different - Nth root of products
I am wondering, what conditions do we need on a symmetric function F: R^n -> R, such that there exist means for F, i.e. for any numbers x_1, ... , x_n, there exists a unique m for which F(m, ..., m)=F(x_1, ..., x_n)
Being continuous and strictly monotonic suffices for two arguments, I suspect it also suffices for more than 2 arguments. Strict monotonicity is not a strong condition for continuous real functions: for a single-argument function it is equivalent to it being injective, hence for multi-argument functions it is equivalent to being injective in each individual argument for any combination for fixed values for the other arguments.
@@fractalfan As expected! So then another question is: sometimes you may want to have a domain that's not all of ℝ, e.g. the geometric mean is defined on the positive reals. Presumably the result generalizes for F : K^n -> ℝ where K is some convex subset of ℝ?
@@MatthijsvanDuin I managed to show it for any hypercube, i.e. an n-fold set product of an interval (possibly infinite). I would need to think about what complications could arise for a generic convex subset, especially since the point (m, ..., m) may not lie in the convex hull of (x_1, ..., x_n) and its images under all the permutations
I already knew the difference between the arithmetic mean and the geometric mean, but I don't remember ever learning when to use the geometric except for problems stated as "find the geometric mean of...". Nice video!
This video is great ! I found it to be a good recap on means AND an even better proof that wisely chosen paractical exemples can be critical in teaching ... To be absolutely clear about that second point, as a french, I feeled that harmonic mean was suddenly out of my reach when trying to wrap my head around miles and gallons. ^^
for the investment returns: first turn them into multipliers by adding 100% and then dividing my 100. e.g. 33% turns into 1.33. Then multiply the three to get the total return, and then cube root it. The resulting number is the average (when converted back to a %) for the car one, simply invert the ratio, assume 100 miles (or some other convenient number) and calculate for each car how many gallons they need for that. Then add them, divide by two to get the average, and invert back to miles per gallon.
This is a great summary of the arithmetic, geometric and harmonic mean. I still don't understand the purpose/ application of the root mean square, and I have a degree in physics. The RMS seems to give a value close to the arithmetic mean of the magnitude of the values. Definitely missing the why. I can calculate the SD for various experiments all day.
You should check out the Generalized Power Mean, which ties together all means and makes it quite clear which is bigger than the other. The Power Mean of order P is the P-th root of the arithmetic mean of the P-th powers of the terms. When P = 1, this is just the normal arithmetic mean. When P = 2, this is the Root Mean Square. When P = -1, this is the Harmonic Mean The limit as P approaches 0 is the Geometric Mean, which makes it a removable discontinuity, therefore we set it such that when P = 0, this is the Geometric Mean. The limit as P approaches (- Infinity) is the minimum value of the terms. The limit as P approaches (Infinity) is the maximum value of the terms. Therefore we have that: MAX >= RMS >= AM >= GM >= HM >= MIN
I think of it this way. The arithmatic mean is the middle of two numbers on an arithmatic scale, wheras the geometric mean is the number in between two numbers on a logarithmic scale.
Super good video! Really made me see means in a new light. Perhaps another issue with mean education is that examples are often used without an associated action for totalling. For example, average height of a group of people. I think this is the kind of thing most people think of when they think of the mean. There isn’t a lot of meaning (no pun intended) to adding up everyone’s heights. Also, for mpg, it could be you want the average mpg achieved for a fixed number of gallons, then the arithmetic mean would make more sense. So, you can’t exactly assign a mean to a type of rate. The context matters.
I was just thinking about this a while ago when I remembered learning the geometric mean in school. I realized I had no idea when or why you'd actually want to use the geometric mean so I looked it up and didn't find a great explanation other than just some example cases (which did include yearly returns). I had never heard of the second kind of mean in the video! But I'd like to think I'd notice that the arithmetic mean wouldn't work without having known it was a tricky question ahead of time.
I once taught seniors from a math textbook that went over the (arithmetic) mean as an introduction to statistics. They gave velocities as an example. One of the given problems was: "Someone is traveling from point A to point B with a mean velocity of 60 km/h. How long does the return travel take if you know that the mean velocity for the full round trip was 90 km/h?" The answer given the book though was 120 km/h. Now I ask students to discuss the depth of my dissapointment with the writers of that textbook. Also, I ask them to calculate the mean velocity of the return trip.
Before even considering the mean, the question asks for a time, not a rate. Further,surely this cannot be known as neither a time or distance between A and B is specified. If i am following, i think 180km/h mean velocity B to A
They do not teach the concept of average value but simply enumerate all possible ways we have so far come up with. The difference between application and theory.
A video that proves that many of us need to restudy mathematics or delve into topic that maths teachers did not mention in class. So far, I am please to have been presented. with four kinds of means. If University students have difficulty in solving questions of averages or means, we should not expect people with minimum mainstream education to be the compensatory mathematical experts overnight.
In the second problem if you give same gallons qty to both vehicles and calculate how many miles they traveled and then calculate the average pe gallon you will obtain 25, it depends on the context.
Great video. I finally understand the fundamental idea of a mean. There's one missing idea that wasn't explicitly stated I think. The "total" used for miles per gallon isn't clear. Do we use the total miles as a total? Or the total gas used? Or does it not matter?
Above average conceptual discussion on the mean. One minus: the answer of the first question says investments, it does not say that the amount goes in in year one and does not change until year 3. Therefore it would be valid to interpret as separate each year.
Great work! Can you please make a video about different types of norms. There must again something be in common which can be described more intuitivly then the standard definition.
very clear explanation! i have a follow up question, though: how exactly does this tie into real-valued random variables? in these contexts, one often only talks about the expected value E[X] = int(xp(x)dx), which ostensibly only generalises the arithmetic mean. what do the other means look like in this context? do they correspond simply to changing the probability measure?
We're all kind of average if you know what I mean.
that's kind of mean
@@bloom945I mean, to be fair, we're all kind of mean, on average
damn the series of comments on means in a sequence
am i an average math student? because i dont understand what do you mean
I mean, you ain’t wrong
I kinda already knew this implicitly and even manged to solve the MPG problem (didnt try the other) BUT I never explicitly thought of a mean as the value that can be replaced to get the same result. This is a very useful way of thinking of it.
Thanks!
I agree, it's a very helpful way to think from first principles!
The mean of values and the equivalent representative value are not the same concept. I feel the questions asked the former and expected the latter.
@@frederf3227 In what way do they differ?
@@MathTheWorldwhat about people who define the mean as the first moment of a set of data points? How does that translate to your concept of “mean”?
Usually we call a “mean” the first moment, the geometric mean is the same as if you took the log of your samples and calculated the first moment. But then you said that the RMD is a “mean,” which is the second moment of your distribution.
In electrical engineering, the harmonic mean is used to find the aggregate value of resistors in parallel.
The RMS mean is used to find the DC equivalent of a varying voltage waveform for equal power.
Resistors in parallel is actually more of a harmonic sum. Harmonic mean would be having a bunch of resistors in parallel and asking if you wanted to replace them with the same number of resistors of equal value in parallel, what would that value be.
I heard him say reciprocal of a reciprocal and I immediately wondered about that! Thank you!
I typed this out before the end of the video: The next time I see someone try to argue that mean is completely synonymous with "average," I'm going to ask them "harmonic, arithmetic or geometric?"
Oh sorry, I meant the root mean square
@@diribigalerror or voltage?
@@donwald3436huuuuh... Let's say, error.
@@donwald3436rms velocity 💪
...or logarithmic, or factorial
Nice explanation (from another mathematician)! That said, I think the reason no-one got your answer for Q1 is that it's too ambiguous. I was quite surprised when you said that the mean you wanted was a s.t. (1+a)^3 = (1+x)(1+y)(1+z) - that's not at all obvious from the phrasing.
The arithmetic mean would be completely appropriate if, for example, you took all of your profits at the end of each year, starting each financial year with a fresh $1000 investment. Equally, you could read it (as I first did) that the three percentages each refer to different investments, each of which you held for three years. Again, the arithmetic mean would be appropriate there.
I also read it that way. I thought those were three seperate investments.
What's the point of investing if you ignore compounding?
@@octopodes7619 I was also a bit confused but the phrasing made it sound like it was talking about the three-year return rate on three separate investments.
@@octopodes7619 The point of investing without compounding is passive income.
@@octopodes7619 there's none but the question made it seem like 3 seperate investments gave him back 3 return values.
The issue with problem 1 is not people not understanding of what an 'average' is. It's with the unclear statement of the problem. It is not explained what a return on an investment is, or how those returns work. Without any further explanation, I could assume that you invest 1000 units of money, then get 133.1% of 1000 over year 1, get 72.3% of 1000 over year 2, and 126.6% of 1000 over year 3.
Exactly my first thought. If the value is not reinvested, it's the arithmetic average of the yearly gain ratios, otherwise it's the geometric one. It's the kind of unclear question that you need to make arbitrary assumptions for. Simultanuously, university (at least in math and science) is teaching you exactly not to do that. Many teachers don't see how counterproductive this inconsistency is in the students learning process.
Additionally, he uses the plural form of 'investment,' which gives the impression that he invested his money in three different assets, rather than indicating that the return on a single investment changed over the three years.
Exactly, this is the classic example of a smug teacher who is just bad at his job
@@victorthaler9495 this
You don't have to all seethe about getting the math wrong. It clearly says it is three years of return on the same set
I cannot thank you enough for helping folks concretize maths. This is a big deal. Please don't stop. You're appreciated.
The mathematician in me knows you explained very clearly how to think about what mean you need. The nitpicker points out that the spelling is "reciprocal", where the "re" is the same back-meaning prefix as in "return" and the "pro" is the same forward-meaning prefix as in "promote".
cal stands for california
@@matthew8184Surely California is a proper noun and Cal would thus be written with a capital. So Sacramento, California. Lower case cal stands for "calories".
The ic is of course the Old English pronoun meaning either "me" or "I".
So the whole word "reciprocal" roughly means "[come] back to me with advantageous calories". It's generally used as an interjection when ordering junk food.
@@furbyfubar thanks 👍
Also “operation” has only one P haha. It’s pretty on brand to have spelling errors as a mathematician though.
current leading hypothesis I can find on the etymology of "reciprocal" is Latin "reciprocus" ("back and forth", "alternating")+ "-alis" (an adjectival ending), with "reciprocus" being "reque proque" ("both back and forward") + "-a/-us/-um" (Latin first and second declension endings, forming a noun or adjective directly from the stem), with "reque proque" being a pretty archaic formation, since "re" as a stand alone adverb does not survive into what records we have of Old Latin, yet alone Classical (and her sisters don't give us anything to reconstruct a stand alone form, only ever the clitic form). "-que" is a clitic reconstructable all the way back to Proto-Indoeuropean as "-kʷe" and is more or less "and" (although as other formations in Latin attest, the meaning was broadened to "any, every" likely through "as" in addition to it's "and" meaning, as seen in e.g. ubique, undique, quoque) [that same particle shows up in English! it comes to Proto-Germanic as "-hw" (which became "-uh" due to epenthesis when suffixed to a word ending in an obstruent), and is seen in the word "though" (which is etymologically "and that", from PIE), "yeah", but was otherwise more productive in Gothic than the West Germanic languages, in which it was the least productive in English.]
"reque proque" getting worn down to "reciproc-" is likely due to assimilation (qu+us -> cus) and the confusion of u and i in unstressed syllables (where unstresed very short "e" gets raised/confused with "i").
I think it’s important to state that if you phrased the car questions as: ‘the two vehicles use about the same amount of fuel’ then the answer would be 25, great video
I’m a CS student (minoring in maths). I practically memorised a formulae sheet of different averages, and I agree with your thesis. From experience, I have a bad habit of defaulting to the arithmetic mean when faced with averaging problems. Thank you for reminding us that understanding the problem before crunching numbers ❤️🙏
I remember a few months ago I saw a website explaining the abstract definition of a "mean" by fitting some set of criteria. I regret I haven't been able to find it since then even though I search for it regularly, but I remember two of the defining properties of a mean is that it must generalize to work for any number of elements of a set (that is, it can't just work for 2 values and not for more) and that in the special case of 2 values, it must return a number that is between (inclusive) them (assuming ordering is possible), which the three most popular means obviously fit. Since I've only found one place that even suggested there was a textbook definition for a mean, I'm guessing it's not a very popular idea, but I still find it interesting that someone somewhere did define it fairly succinctly
a mean for a set of numbers must return a value within the set of numbers (or could be undefined, as geometric mean might be with negative values). all the types of average i can think of all work by this mechanism: pick a monotonic function f(x), and for each value in your set, compute f(x). take the normal arithmetic average of those, and compute f-inverse of that. your choice of f(x) will be motivated by what trait you are trying to capture. f(x) = x or logx or 1/x or x^2 generate the 'normal', geometric, harmonic, and RMS averages respectively.
@@theupson The commonly used means also have another useful property: If you multiply all numbers with the same value, the mean will be multiplied with the same value. For example, the mean of 2a and 2b is 2 times the mean of a and b.
A mean (of a finite number of values) is a function mₙ : [a, b]ⁿ→[a, b] (where [a, b] can be extended to ]-∞, b], [a, +∞[ or to the whole ℝ) which satisfies these properties:
i) Continuity, we want that a small change of the arguments can only cause a small change of the value
ii) Transitivity under the action of Sₙ on its arguments, in other words, invariance of the value when the arguments are permutated
iii) Idempotency: mₙ (x, ..., x) = x
iv) Monotonicity on each argument
v) Invariance under self-composition, as to say for k ≤ n:
mₙ(a₁, ..., aₖ, x, y, ...) =
= mₙ(mₖ(a₁, ..., aₖ), ..., mₖ(a₁, ..., aₖ), x, y, ...)
@@ciana42Can you share more about where this definition and others like it are sourced from? I've been meaning to dive into a measure-theoretic study of statistics which uses real analysis as the backbone, but a lot of the texts I've been finding seem to be a bit more rigorous than what I'm comfortable with (model theory?!). This definition you've presented here seems fairly palatable and I'd love to get my hands dirty with problems from whichever text you're quoting; it appears to be presented at the level of rigor that I find comfortable at the moment.
I’m a Junior physics mathematics dual major, thanks to this video I can now completely visualize the mean, and why certain calculations are what they like the average of a function, or why we care about the nth centralized moment. Hell even why a convolution is defined how it is, and why a Fourier transform turns a convolution into multiplication, or why the Fourier transform is almost like a weighted average. It’s all just finding the central value, or in the cases above, an analogous distribution. I can go on and on about orthogonality but my point is this gave me the best understanding of the mean that I wish I had before college.
Great video! I've always wondered the practical reason why we have so many different types of mean, and this makes it crystal clear.
This is the best explanation I have found. I normally just peek at the correct one, but was never able to explain it with such clarity. An important detail if you are solving an exam, most of the time, mean means arithmetic mean, even if it does not work, so be sure to confirm.
Root Mean Square is probably used more by electricians as it tells you how much equivalent dc voltage a given ac system would convert to.
It’s used whenever deviations from the central tendency can be negative as well as positive (ie there is directionality in the error)
@@darwinbodero7872for the said problem, we could have used absolute operator. Why does it have to be square? square amplifies error values.
@@Salty0 Squares are algebraically easier to deal with than absolute values.
@@Salty0in the case for voltage/current, its because v^2/r = i^2r = power. Since we are primarily looking at power, we use Vrms and Irms.
@@gcewing
Yes, but does it justify usage of root mean squared instead of mean of absolutes? Or is there a deeper meaning here? This thing was on me for sometime and i never got a satifying answer.
I know that given a continuous function, it is easier to calculate rms, as we get integral-friendly formula. (which we dont for mean of absolutes).
But this still feels like rms isn't giving the best representative, we are looking for.
OOOh but there's even more! You didn't even touch the generalized power mean, where you get an infinite set of different "mean" values based on a power number, p.
All you do is raise each value in your data set to the power p, add them together, divide by the number of data points, and raise everything to the power of 1/p.
Actually, all of the means you mentioned in this video are generalized power means:
Harmonic mean -> p = -1
Geometric mean -> p = 0 (this does involve raising something to the power of 1/0, but the limit as p->0 gives you the geometric mean, so usually p=0 is set as the geometric mean)
Arithmetic mean -> p = 1
Root mean square -> p = 2
Geometric mean is more like 1/p where p is the number of terms you wanna take the mean of. G.M. of (a,b,c,d,e,f)
G.M. = (a*b*c*d*e*f)^(1/5) p =5 here
@@PavanKumar-xv1hg No, @foozlebagel7488 is right.
[(a^p+b^p)/2]^(1/p) -> sqrt(a*b) for p->0. This can be proven by calculating the limit of the natural logarithm of the expression using de Hospital's principle.
@@Integer0 Yeah, your right ! I tried it out and got it !
P.M. of (a_1,a_2,.....,a_n) n total values is P.M = L = ({Sum[i=1 to n] (a_i)^p} / n )^(1/p)
Taking log on both sides,
ln(L) = ln({Sum[i=1 to n] (a_i)^p} / n ) / (p)
As limit p tends to 0,
lim(p-->0)
ln(L) = ln({Sum[i=1 to n] (a_i)^p} / n ) / (p)
ln(L) = ln({Sum[i=1 to n] (a_i)^p} ) - ln(n ) / (p) [Just taking out n]
Let p = 0 in R.H.S to see if we get it in an indeterminate form
ln(1+1+1......+1 ) - ln(n) / 0 [1+1+1... n times]
ln(n)-ln(n)/ 0 = 0/0
So we can send the limit to the hospital (Use L'hopital's Theorem lmao)
lim(p-->0)
ln(L) = ({Sum[i=1 to n] (a_i)^p} )^(-1)*(Sum[i=1 to n ] (a_1)^(p)*ln(a_i) ) / (1) [differentiating the numr and denominator separately]
Putting p = 0 in R.H.S
ln(L) = ({Sum[i=1 to n] (a_i)^0} )^(-1) * (Sum[i=1 to n ] (a_1)^(0)*ln(a_i) ) [ a_i ^ 0 = 1 ]
ln(L) = ({Sum[i=1 to n] (1) )^(-1) * (Sum[i=1 to n ] ln(a_i) ) / (1) [1+1+1... n times = n & sum of logs ln(x) + ln(y) = ln(xy) ]
ln(L) = n^(-1) * ln(a_1 * a_2 * a_3 * a_4 * .......*a_n)
ln(L) = ln((a_1 * a_2 * a_3 * a_4 * .......*a_n)^(1/n))
or just,
L = P.M(p-->0) = (a_1 * a_2 * a_3 * a_4 * .......*a_n)^(1/n) = G.M of (a_1,a_2,.....,a_n)
So it follows what O.P. said.
P.M(p) = ({Sum[i=1 to n] (a_i)^p} / n )^(1/p)
Following this definition,
P.M(2) = R.M.S
P.M(1) = A.M
P.M(p--->0) = G.M
P.M(-1) = H.M
P.M(-2) = anti H.M. square ??
P.M(-2) and P.M(2) has kinda like the same vibes as A.M and H.M [Personal thoughts feel free to ignore]
We also know P.M(k)>=P.M(k-1)>=P.M(k-2)>=..... >=R.M.S>=A.M>=G.M>=>H.M
[I don't know if H.M>=P.M(-2)>=P.M(-3)... and so on is true or not tho]
Its my first time writing a proof on yt so tell me if some part is unclear to you guys(I tried just 2 terms and extended it to n terms),
Thanks !
Isn't this just the definition of L-p norm spaces for sequences?
A sequence x = x_n is in the space L-p if the norm ||x|| = (sum |x_n|^p )^1/p is finite
@@nestorv7627 well yes all Norms are means (but not all means are Norms)
I appreciate that you made a point of using mean and average quite interchangeably. Some people insist that "average" must only refer to the arithmetic mean, which never sat right with me...
doesn't it also refer to mode and median?
I was also recently discussing means in the classroom, for me in the context of heat transfer. For heat transfer in a pipe, the logarithmic mean of temperature is an important quantity. It's not the type of quantity that people would normally think of as a mean, even compared to things like the geometric and harmonic means. This actually inspired a series of posts in my blog about the subject, where I ultimately discus how to use principles from calculus to derive a fairly general class of means. I don't think I can link it here, but my blog is in my bio.
I loved reading your Blog Posts!
@@julioaurelio Thank you! I'm behind on posting some of my newest musings, but I hope to publish some more soon.
Thank you to point out. This problem can be very harmful in reality and those mistakes happen a lot in real work environment with grade students in stats, data science, data analysts and comp. science, also happens in scientific publications as well. This problem is most underestimated in real life and in academia - it leads to wrong decision and strategies in business and data manipulation in academic publications.
So I've been doing engineering and product management for 20 years and this is the best explanation I have ever seen.. I also realise some really bad "means" I've applied in the past
A mean can be just any number between the min and max, so the general definition is using the probability density function (doing integral that kind of thing). Thank you so much for pointing out one very inspiring way of thinking about means! But this way still does not cover all ways of thinking about means
Glad to find a better explanation of this! I encountered all of these in AP stats, but I only memorized where to use them, rather than properly understanding why they were different and where they should be used
yeah stats to me was harder than complex analysis i took in college lol
people don't understand percenatages, the first one they get, but when there are 3 terms this would be +33.1% of X, -27.7% of Y, and +26.6% of Z
so with an initial capital of 1,000, that would mean X =1,000, Y= 1,331 and Z= 962.313
the gross interest would be 21.8288% based on 1,000. but if you want that over the 3 years it would be ∛1.218288, which is 1.06803, so around 6.8% PA.
you can't average out the percentages, as these percentages are of different values.
for the second one you have 1/15 + 1/35 = (7+3)/105 = 10/105 = 2/21 so 2 vehicles each with 1 gallon of fuel, 21 miles for each vehicle. so that would be 2/42 combining the milage or basically 1/21 for each vehicle. what was with all that other stuff in the video? this is basic adding of fractions. it seems that people also don't understand fractions either, including "Math the World"
I know technically it is a ratio being described as a fraction. but the rule still applies, you need a common denominator.
but the 1/21 is correct, for 1 gallon of fuel total the total milage would be 21 miles for the two vehicles.
There's kind of a trap in the commonly used phrase "adding a percentage". It's not an addition, it's a multiplication, and many things become clearer when you think of it that way. I don't remember anyone pointing that out to me in school, though.
@@gcewing a percentage is just a 100th of a number. You can add or multiply these fractions, convert it to a decimal, etc. but you can absolutely add them.
The problems themselves I did not find particularly complicated (except I found the first problem a bit vague, but your solution was one of the options). What I liked is that for the first time did I understand that there's a fundamental approach to understanding which mean you need to calculate.
I love the quasi-arithmetic mean and how it generalizes different means. It utilizes the idea that you have some function which is used to find the total value, and to find the mean you have to use inverse function of it
For the mpg problem the units (dimensions) of the required solution (miles per gallon) told us the approach to use. Find total miles, find corresponding total gallons used, and divide one by the other. In other words: for any specific distance, d, travelled by each car, calculate the total miles that were travelled (2d), and how many gallons were used (d/35 + d/15), then find mpg = miles/gallon. I used d=105 for convenience because it is the lowest common multiplier of 15 and 35. Then mpg = 2*105/(105/35 + 105/15) = 21mpg. (As it's a ratio, any distance can be chosen.) As a follow up I plugged the values into the formula for the harmonic mean and got the same answer. (Edit: I've just noticed that if you cancel 105 from numerator and denominator you actually have the formula for the harmonic mean: HM = n/(Σ(1/x)) .)
Thanks for an explanation of this that I (as a mathematician) can use to explain these concepts to non-mathematicians!
I have a concept I call, "transformed mean," which is not mine originally, but is almost never introduced when different kinds of mean are discussed.
Basically, for any function f:R→R, which is strictly monotonic (increasing or decreasing) over some interval, the f-transformed mean, M(f; xj) of a set of n numbers x1...xn, all within that interval, is
M(f; xj) = f⁻¹([∑ f(xj)]/n)
which just consists of taking the simple, arithmetic average (mean) of the transformed numbers, then "untransforming" that result.
Under this definition, when f is any (non-constant) linear function, the f-transformed mean (FTM) is just, trivially, the arithmetic mean.
When f is a logarithmic function, the FTM is the geometric mean.
When f is the reciprocal function, the FTM is the harmonic mean.
When f is the square function, the FTM is the RMS.
Etc.
And of course, each of these means has a corresponding transformed sum. Just remove the division by n in that definition.
The important part of your presentation here, IMO, is the emphasis on just what operation is being used to combine the numbers; and that this is what determines how to take the right kind of mean.
Fred
The MPG problem is an excellent example of where the harmonic mean (HM) is called for.
Another class of perennials for using the HM, is rate-of-work problems.
thank you fred, very cool! ill look more into it later
So many finance videos calculate the average rate of return by doing the simple average on percentages and it really bothers me.
The funny thing is that if the returns are small then the result of that will be very close to correct, e.g. averaging +1%, +2%, +3% the stupid way gives +2% while the correct answer is approximately +1.997%
A.M. is always greater or equal than G.M
0:21 can we talk about the difficulty in correctly spelling difficulty?
👏 wow!!! just brilliant how you explained everything. I understood all these concepts as I'm an IT engineer but in a very haphazard and complicated way. Just see the 1 whole page I used for the first problem to get the correct answer.
You explained it so beautifully and I wish I had you as my teacher and mentor growing up. Keep up the brilliant work. I'm going to become a patron if I can as u just swept me off my feet with how beautifully and brilliantly you explained it, just wow!!!
i have never understood the harmonic mean, always found it esoteric and strange, but now I have an hint of an intuition
Ohhh I rememebr this now! We had this in primary school! We were taught about the arithmetic mean, but then I encountered the task where there were 2 vehicles with different speed, one was used for the first section of the path, the other for the second, with sections not being equal. Their speed and the distances were given and the task was to calculate the -average- mean speed. My brain hurt bad while trying to understand why the arithmetic mean of the speeds doesn't give the same value as when dividing the total distance by the total time spent.
@MathTheWorld Thank you for your work! This may be a young channel, but I think you guys will become my favorite math TH-cam channel. Most other math channels seem to go for exotic and mind-blowing stuff, or normal but extra hard stuff; in both scenarios being removed from the needs and realities of using math.
A math channel than can actually demonstrate why math is useful/powerful is something that has been sorely lacking in TH-cam.
I look forward to watching your videos for many years to come.
Ganbatte kudasai!
I majored in math and physics and minored in statistics and probably would’ve done the same as the students for the first question. I learned of different “means” and applied them to data in varying contexts, but many texts explicitly state that the average is the arithmetic mean, so I’d assume that’s what the question is asking for. Other comments have also explained why that could make sense given the description of the “investments”.
But, often the lectures provide further context. The instructor may have done example problems that imply a certain interpretation of the plural investments, or, more likely, that in this course “average” means “mean” and not necessarily “arithmetic mean”.
Also, the term I know for situations where an appropriate mean is to be applied is "effective". For example, you can ask for the "effective interest", which is the interest you'd need to get the same result.
Anyone asking for averages of percentages should immediately raise a red flag.
I only solved the first problem. I intuitively had the idea down but you put it into words beautifully.
you could do it "the way you learned in school" on the average returns but you have to understand that each year has a different starting value. the return values given were not the normalized returns based on the initial 1000.
I've been wondering how to think about the mean for ages and finally got a satisfactory answer. I learned all these types of means but never knew what fundamentally made them a 'mean'
Fun fact: for any list of postivive reals, the arithmetic mean is always at least the geometric mean, which is in turn at least the harmonic mean. There are some uses for this nice classical inequality chain. This is part of the so-called "mean inequality chain" QM-AM-GM-HM.
Expected probability formula for a generalised log linear model should do the trick.😊
You know what's interesting is that I know all of this and have used many of these various calculations. Depending on the situation though. (Though I did major in Chemistry and do use math/statistics very frequently.) ( don't really know the difference by names like harmonic, arithmetic or geometric(i'm also not native english), I just calculate what makes sense in context)
Mean, Average. Weighted means, standard deviantion , relative standard deviation. Population, Sampled data, extrapolation, calculating statistical outliers and if they should be "disregarded" from the mean.
I just always ask myself ......the mean of what? What is it that I want to know? Statistics can be tricky to read, but also tricky to present clearly.
I often read news artikels and they talk about means as if that even says anything by itself. I hate mean / median abuse by the media. (what where your deviations? Do you have a normal distribution?)
Sometimes you read: Increase of 100% in people that get X disease (sure it tells you it dubbeled). ( But what was the original %? if it was 0.01%, thats just a 0.01% increase really on the total population, if it was 25% then yeah a 25% increase is alot on the total population.)....I mean seriously be clear.
Best video on the topic anywhere! 🎉😊
I would argue a much easier way to solve the second problem is to set it up like this:
Imagine the family has to drive 105 miles. A number easily divisible by both cars' miles/gallon.
The first car will use 105/15 = 7 gallons to drive the distance, whereas the second will use 105/35 = 3 gallons.
So, the average miles/gallon for your entire family will simply be the total distance covered for both vehicles, divided by the total number of gallons used, ie. 210/10 = 21 mpg.
that would mean, no pun intended, that both cars take 10 gallons to get there, yet the average is double that. You mixed up miles/gallon and gallons/mile if im not mistaken
@@undozan4180 the average is for the entire family, not each vehicle of the family.
@@undozan4180 The mean is the single number that could be used to replace each of the individual values and yield the same result.
So if the family used those two cars to travel 105 miles in each car, they would use 7 gallons in one car (15mpg) and 3 gallons in the other (35mpg) for a total of 10 gallons.
But if the family used two identical cars, each giving 21mpg, they would use 5 gallons in each car for a total of 10 gallons, Same result.
That shows that if the cars travel equal distances, the mean fuel consumption of 15mpg and 35mpg is clearly 21mpg. You were mistaken.
I don't see how your method is really any different. His solution is the same methodology but while having each car drive 1 mile. Instead of 7 and 3 gallons, he gets 1/35 and 1/15 gallons respectively. All you really did was add an additional step by multiplying by the least common multiple (105 = 3 * 5 * 7).
@@didles123 Intuition matters. This video is not about solving simple math problems, but discovering the logic of why the operation makes sense.
9:30 This is my biggest gripe with all these sorts of websites and has raised my respect for my teachers who actually took time to explain and encourage us to read the explanations of the terms we are using instead of just blindly applying formulas that we rote learn.
It is a big beef of mine as well!
Great video, I figured out the meaning halfway thru since I’ve always assumed mean and average were similar enough without thinking enough about what a mean really is. Thanks
Here, some types of means are discussed that share the property that if that mean replaces each actual parameter value, we get the same outcome. This is a valuable insight you convey to your students. But note that means can have other purposes, where those means don't have that property, like the modal value. The purpose is still the same, but thinking of them in terms of problem solving is much different, I think.
The mode also works with his definition though? For example, the mode of 1, 2, 3, 4, 4 is 4 (most common number) and the mode of 4, 4, 4, 4, 4 is also 4.
@@cube2fox you are absolutely right. I probably had thought of the median.
@@danielschwegler5220 But the median of 1, 2, 3, 10, 20 is 3 (middle number) while the median of 3, 3, 3, 3, 3 is also 3.
I think the issue people are having with the first question is that it says "investments" meaning multiple. But when you invest, you are technically only investing from a single pot of money
1: 100 + 33.1% = 133.1 (at the end of year 1)
2: 133.1 -36.868 (-27.7%) = 96.232 (at the end of year 2)
3: 96.232 + 26.6 (26.6%) = 121.83 (end of year 3).
Furthermore, I think people are also getting confused is because you used arithmetic mean to find the average annual rate when we should of used geometric mean to account for the compounding effects over time.
I got :
(121.83 / 100)^1/3 -1
(1.2183)^1/3 -1
1.06803 - 1
.06803 or *100 = 6.8% average return per year for 3 years.
100 *.068 = 106.8
106.8 *.068 = 7.2624 + 106.8 = 114.0624
114.0624 * .068 = 7.7562432 + 114.0624 = 121.8186432
The other problem is when people confuse average/mean of a very skewed distribution (like wealth distribution) with the median or the mode.
"The average salary is X" puts the idea that that salary is earn by the most people (which would be the mode), some people understand it as the salary that mark 50% of the population (that is the median), and almost nobody think about how it is a useless number, because if you eat 9 chickens, he eats 1, and I eat 0, the average chicken per person is 3, but the median, which is closer to reality, is 1 (well, technically depends on how you define the median of a discrete distribution, but alas).
This is why these "subtleties" are not just for fun and giggles, but are fundamental tools that everyone should be able to command properly, or else we are getting lied constantly.
This was so satisfying. I learned this at uni but never understood why this mattered.
Whoa! Thanks for opening up my understanding of means. I've always wanted to understand the why and what of harmonic means
Here's a head-scratcher: what's the mean direction of n equal-magnitude vectors on the same surface, intersecting at their origin, each at angle 2pi/n rads from the next?
Thank you for defined the means, I was wondering for 40 years at least why so many means. Thank you
You are so welcome!
Firstly, just want to add my +1 to people saying that the first question is disastrously explained, and this was almost certainly done in order to be misleading. For shame, sir.
Secondly, at 1:27 you say "assuming simple interest in this problem..." and then IMMEDIATELY start using compound interest. That makes it seem like your attempt to be misleading even got *you* a bit confused.
The rest of the video is awesome.
I'm curious what about the first problem is misleading, since to me it was straight forward and I'm unsure how else to interpret it.
I agree that "simple interest" is confusing phrasing, but I don't think he confused himself. I think he just meant that annual returns are like interest that is applied once, rather than interest that would compound multiple times within a year.
@@didles123 It wasn't explicitly said that those numbers where the yearly return of the investment and it wasn't explicitly said with which number the years started.
Investing 1.000 in year 1 -> success, therefore investing 2.000 in year 2 -> failure, investing only 1.000 in year 3 again would be weird but possible. Same with investing 1.000 each year and withdrawing the win/loss after each year.
For the correct solution you must assume the most standard and reasonable procedure. Therefore most people should get it, but you can if you are nitpicky and assume weird stuff you are not contradicting the question and get another answer.
The overall issue imo is that the "of" before the percentages is too close to the word "investments," which suggests multiple investment with three separate returns each over three years.
Ending the sentence after "investments" and starting a new sentence with something like "Year 1 had 33.1%, year 2 had, -27.7%, and..." would have been better.
So... I was able to pause the video on my phone, solve both problems correctly on a Julia terminal I had open, and unpause the video before my 1 minute, company mandated screensaver kicked in on my phone. I think it took me about 45 seconds (although I didn't actually time myself). These were not hard problems. I was able to just reason my way through them. The fact that your university students couldn't get the correct answer leads me to feel like there's a failure in the educational system here.
A parallel video about the training in the educational system that allowed these failures in reasoning to occur would be absolutely fascinating. The distribution of reasoning failures across backgrounds, majors, and classes that had been taken would also be very interesting and potentially worth writing a paper on.
This is a wonderful wonderful wonderful video and truly connects the dots to what is meant by an "average" or mean. Truly enjoyed this!
Don't know how you're going to do this in the end and, in spite of having a pretty good mathematics education at a prestigious university, I was never formally taught anything other than the arithmetic mean. I've learned of the others since, but not had the practice required to really learn their usefulness. That said, I figured out both problems correctly fairly easily by A) determining what the rate had been over 3 years and taking the cube root because whatever average would need to be cubed and B) by figuring out how much gas would be used to drive 1 mile, adding, then dividing 2 by that number because the two cars would have driven 2 miles total. Of course, those are going to be some mean or other.
I had this exact doubt. I had to ask ChatGPT and understand the difference b/w Arithmetic Mean and Harmonic Mean. This video just tops it off very well. Thank you so much! ❤
There's a very nice generalization of these means you can see (amongst other places) on the "mean" wikipedia page : generalized f-mean.
In short, if you have a monotonous function f, the f mean of a and b is the number m such that 2f(m)=f(a)+f(b).
f(x) = x gives the arithmetic mean
f(x) = 1/x the harmonic mean
f(x) = ln(x) the geometric mean
f(x) = x^2 the quadratic mean
And it generalizes for instance to any f(x) = x^n (of which the limit when n goes to 0 is f(x) =ln(x), the geometric mean)
And there a very nice way all the means will always be sorted, according to the n value they correspond to:
Harmonic mean < geometric mean < arithmetic mean < quadratic mean,
corresponding to following n values : -1 < 0 < 1 < 2 < n if n greater than 2
Thanks for sharing this. I saw it while I was doing research for this video, but I didn't think it would be a good fit for our target audience.
This is the kind of idea that I'm always astounded by. A mathematician can take such separate things, and see a unifying idea across all of them.
I have thought a lot about the first question in the past. Because using the arithmetic mean on interest rates gives you the expected interest rate. So if you want to use it for one period the arithmetic mean would be best. If you want something in between it will get more complex.
Best video to means I have ever seen!
Thank you!
Really good video! I hope you get more subscribers, you deserve it!
I managed to get the gas milage one right, but only because A.) I knew the arithmetic mean would be wrong since you were challenging assumptions, and 2.) I was aware that more civilized parts of the world used fuel/distance instead and figured that was worth trying.
Great video, I certainly learned a lot, and you earned a subscriber! Looking forward to more math videos :)
1. what a great video - this is the kind of math education thats different from just writing down formulas (it needs people that "think math" and have a way to put it into some perspective for students to get it), very good!
2. let's change the wikipedia page for means?! :)
Very helpful. I recently learned all about these different means because of a project I'm working on. This gave me a much deeper understanding of _why_ they exist, _how_ they're calculated, and _which_ one to use for a given purpose.
PS: Despite my lack of mathematical prowess, I got the gas mileage question right and almost got the ROI question right too. Instead of taking the cube root, i divided by 3. Doh! I was close though.
Lovely video!!
For the first problem I thought you were going to talk about weighted sums since it was not mentioned that you take ALL the money you have left from the first year investment to the next year
So my first thought was "we can't know since rates aren't enough to find the average"
But now I get what you mean
If you need to use a weighted average with the harmonic mean, for instance if the two vehicles travelled different distances, you can just put the weights in the numerator. For example: 100 mi / 15 mpg + 150 mi / 35 mpg = 250 mi / x.
Nice content, with great examples. I had never considered them like this together.
these three rhyms
- just mean of sums
- reciprocal of mean of sums of reciprocals
- square root of mean of sums of squares
this is a bit different
- Nth root of products
I am wondering, what conditions do we need on a symmetric function F: R^n -> R, such that there exist means for F, i.e. for any numbers x_1, ... , x_n, there exists a unique m for which F(m, ..., m)=F(x_1, ..., x_n)
Being continuous and strictly monotonic suffices for two arguments, I suspect it also suffices for more than 2 arguments.
Strict monotonicity is not a strong condition for continuous real functions: for a single-argument function it is equivalent to it being injective, hence for multi-argument functions it is equivalent to being injective in each individual argument for any combination for fixed values for the other arguments.
@MatthijsvanDuin I was able to prove that strict monotonicity and continuity are enough for any number of variables.
@@fractalfan As expected! So then another question is: sometimes you may want to have a domain that's not all of ℝ, e.g. the geometric mean is defined on the positive reals. Presumably the result generalizes for F : K^n -> ℝ where K is some convex subset of ℝ?
@@MatthijsvanDuin I managed to show it for any hypercube, i.e. an n-fold set product of an interval (possibly infinite). I would need to think about what complications could arise for a generic convex subset, especially since the point (m, ..., m) may not lie in the convex hull of (x_1, ..., x_n) and its images under all the permutations
Good explanation. I think I would have got the right answers because of my engineering background, but you definitely added clarity.
dang, I've studied math a lot and this is by far the best explanation for what, why, and how we have different means. kudos!
Wow this was great! I lived my whole life as sum/n without ever thinking there could be different ways to take the average or why. Thanks you!
I already knew the difference between the arithmetic mean and the geometric mean, but I don't remember ever learning when to use the geometric except for problems stated as "find the geometric mean of...". Nice video!
Thanks!
This video is great !
I found it to be a good recap on means AND an even better proof that wisely chosen paractical exemples can be critical in teaching ...
To be absolutely clear about that second point, as a french, I feeled that harmonic mean was suddenly out of my reach when trying to wrap my head around miles and gallons. ^^
for the investment returns: first turn them into multipliers by adding 100% and then dividing my 100. e.g. 33% turns into 1.33. Then multiply the three to get the total return, and then cube root it. The resulting number is the average (when converted back to a %)
for the car one, simply invert the ratio, assume 100 miles (or some other convenient number) and calculate for each car how many gallons they need for that. Then add them, divide by two to get the average, and invert back to miles per gallon.
This is a great summary of the arithmetic, geometric and harmonic mean.
I still don't understand the purpose/ application of the root mean square, and I have a degree in physics. The RMS seems to give a value close to the arithmetic mean of the magnitude of the values. Definitely missing the why.
I can calculate the SD for various experiments all day.
You should check out the Generalized Power Mean, which ties together all means and makes it quite clear which is bigger than the other.
The Power Mean of order P is the P-th root of the arithmetic mean of the P-th powers of the terms.
When P = 1, this is just the normal arithmetic mean.
When P = 2, this is the Root Mean Square.
When P = -1, this is the Harmonic Mean
The limit as P approaches 0 is the Geometric Mean, which makes it a removable discontinuity, therefore we set it such that when P = 0, this is the Geometric Mean.
The limit as P approaches (- Infinity) is the minimum value of the terms.
The limit as P approaches (Infinity) is the maximum value of the terms.
Therefore we have that: MAX >= RMS >= AM >= GM >= HM >= MIN
I think of it this way. The arithmatic mean is the middle of two numbers on an arithmatic scale, wheras the geometric mean is the number in between two numbers on a logarithmic scale.
Super good video! Really made me see means in a new light. Perhaps another issue with mean education is that examples are often used without an associated action for totalling. For example, average height of a group of people. I think this is the kind of thing most people think of when they think of the mean. There isn’t a lot of meaning (no pun intended) to adding up everyone’s heights. Also, for mpg, it could be you want the average mpg achieved for a fixed number of gallons, then the arithmetic mean would make more sense. So, you can’t exactly assign a mean to a type of rate. The context matters.
I was just thinking about this a while ago when I remembered learning the geometric mean in school. I realized I had no idea when or why you'd actually want to use the geometric mean so I looked it up and didn't find a great explanation other than just some example cases (which did include yearly returns). I had never heard of the second kind of mean in the video! But I'd like to think I'd notice that the arithmetic mean wouldn't work without having known it was a tricky question ahead of time.
I once taught seniors from a math textbook that went over the (arithmetic) mean as an introduction to statistics. They gave velocities as an example. One of the given problems was:
"Someone is traveling from point A to point B with a mean velocity of 60 km/h. How long does the return travel take if you know that the mean velocity for the full round trip was 90 km/h?"
The answer given the book though was 120 km/h.
Now I ask students to discuss the depth of my dissapointment with the writers of that textbook. Also, I ask them to calculate the mean velocity of the return trip.
Before even considering the mean, the question asks for a time, not a rate. Further,surely this cannot be known as neither a time or distance between A and B is specified. If i am following, i think 180km/h mean velocity B to A
I came looking for copper and found gold. Thank you, Math The World!
They do not teach the concept of average value but simply enumerate all possible ways we have so far come up with. The difference between application and theory.
Great video. Gave me a different and more holistic understanding of means 🎉
thanks, this was very intuitive. I am a Data Science student, quite good in math, but I didn't realize i didn't know this
Glad it was helpful!
Bro, I knew these means from the AM - GM - HM inequality, but still didn't have the slighest idea for what they were for...
I'm feeling really proud that I was able to arrive at the right answer in the first try , gave me a confidence boost thanks bro
That is awesome! Keep it up!
Awesome and interesting content, amene explanatio, clear and meanigful examples, comprehensive selection of concepts ❤ +1 subscriber
Thanks. Learned something new today.
Glad to hear it!
This was really enlightening! Thank you.
A video that proves that many of us need to restudy mathematics or delve into topic that maths teachers did not mention in class. So far, I am please to have been presented. with four kinds of means. If University students have difficulty in solving questions of averages or means, we should not expect people with minimum mainstream education to be the compensatory mathematical experts overnight.
Sounds a lot like work versus time problems. Good food for thought, thanks!
In the second problem if you give same gallons qty to both vehicles and calculate how many miles they traveled and then calculate the average pe gallon you will obtain 25, it depends on the context.
Thanks for the clear explanation. Could you also explain when it's appropriate to use the median and when the mode?
Thank you for your explanation - I studied statistics, but as that was almost 30 years ago, understandably I'd forgotten some ideas :)
Great video. I finally understand the fundamental idea of a mean. There's one missing idea that wasn't explicitly stated I think. The "total" used for miles per gallon isn't clear. Do we use the total miles as a total? Or the total gas used? Or does it not matter?
This channel is a gem
I've been waiting for this video for at least few years
Yeah! Are there any other videos you are waiting for?
Above average conceptual discussion on the mean.
One minus: the answer of the first question says investments, it does not say that the amount goes in in year one and does not change until year 3. Therefore it would be valid to interpret as separate each year.
Great work! Can you please make a video about different types of norms. There must again something be in common which can be described more intuitivly then the standard definition.
very clear explanation! i have a follow up question, though: how exactly does this tie into real-valued random variables? in these contexts, one often only talks about the expected value E[X] = int(xp(x)dx), which ostensibly only generalises the arithmetic mean. what do the other means look like in this context? do they correspond simply to changing the probability measure?
Very interesting useful video! I have shared it with some of my engineering friends
Thanks for sharing!
The AC voltage value is the root mean square (RMS) of the voltage in time (for example, 110V in the US).