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On the chance that somebody here really does like math and understands mathematical terms well, I have a question about three of those terms.
In a nutshell, what is the difference between average, mean and median? Why would you use one over the other--in other words, why would you use an average over the mean?
If this is too complicated to discuss here, even though it's basic math, just ignore this thread altogether.
Also, I think I've got the three terms right, but, if I represented them not quite-on-target, just modify them. I just remember learning about average, mean, and median. It could have been average, mean and mode, but I don't think so...
Average, of course, was easy. Just a sum and then division. Good for making estimates for future predictions. Say, take a bunch of good recipes, compare ingredients, and devise a new recipe based on averages. That kind of thing. Or certainly grades in a grade book to find average grade.
But the other two--mean and median (if that's the term; could be mean and mode?)--I don't use them and suspect I probably should.
Anyway, please forgive if this is just to un-word-related.
Best regards,
WW
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stranger
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Here you go. Calling each number in the data-set a "term"
mode is the term that occurs most frequently.
median is the "middle" term. Arrange the terms in order in a line, and the median is the one in the middle of the line.
mean is the usual average. For the mean of n separate terms, add all the terms to get their sum, and then divide that sum by the number of terms (that is, divide by n). More exactly, that is the arithmetic mean.
For the geometric mean of n separate terms, multiply all the terms to get their product, and then take the nth root of that product.
All these will come to pretty much the same number if your data is distributed in the usual bell curve, but if it's not, the mean can paint a badly misleading picture of the data.
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old hand
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Good Q WW
I suffered my way through statistics and probability theory at uni but think there's still a bit of unscarred grey matter that can assist with the answers.
Average and mean are the same; derived from the sum of the elements (ie pieces of data) divided by the number of elements. I forget why mathematicians call it the mean (rather than the avaerage), maybe it was the alliterative qualities of the word - considering it is always used in conjunction with "mode" and "median". (BTW, "conjunction" is a mathematical term also!)
The median value of a data set is the middle value (once all the elements have been put in order). This is more useful to statisticians as far flung elements at either end of the data set tend to cancel each other out. Hence you get an "unskewed" value for the median - one that is unaffected by data which may be spurious or, if not spurious, irrelevant to the data set as a whole.
The "modal value" (or "mode") of a data set is the most common value - this is handy as it is not a function of the highest or lowest value in the data set.
I proffer this in the absence of Bean - she's the guru on this stuff - with a gift for presenting the answer so that we all can understand it!
stales
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Ah, I see now. Thanks very much for the definitions. I see that I was very wrong in that mean and average were one and the same. I think mode is the most interesting of the three--the ones that appears most frequently. I can't begin to imagine its practical application, but there must be many.
Thanks again to you both,
WW
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stranger
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They've answered your question about what does mean mean?
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Carpal Tunnel
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When teaching anger management (as I do so often) I ask participants to record the time of day and what it was that got them *upset. From this collected group information we determine our "mean mean meaning". 
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It seems in your time-of-day for anger data collecting, it would be interesting to note the most frequently occuring term, the mode, as I learned today. And wouldn't it be a mean surprise if that time of day was that falling in your anger management class! Ha!
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stranger
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>> From this collected group information we determine our "mean mean meaning".
You mean that by that means you determine our "mean mean meaning"?
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veteran
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I was on vacation and missed posting to this question. Everything Stales said is correct and I've nothing of substance to add - only a small thought.
We have a bunch of raw data - and the data is a pretty good reflection of itself. But looking at a big stack of numbers we often "miss the forest for the trees" and statistics is partly about trying to figure out some way of thinking about the data without thinking about all the components of the data - which we often aren't even capable of doing.
No single number like mean or mode or median is going to give us everything we need to know about the data. There are other things like standard deviation, skew, sample size - all important - but even these don't always cover it. In every case I know or can think of having a large sample size is good. Unfortunately, it's not always practical or possible - and this is where I get a little fuzzy with things and it starts looking like a black art.
k
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Pooh-Bah
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In reply to:
On the chance that somebody here really does like math and understands mathematical terms well...
Mathematics ... so unappreciated! There are so many wonderful mathematical terms for wordlovers to enjoy, either because the concepts that they express are so interesting, or else the words themselves are cool, such as "zeroth," the ordinal version of "zero." Not a bad word to have handy in a game of Scrabble, I imagine. Then there's "zepto," a prefix meaning 10 raised to the (-21st) power. (Perhaps not eligible in Scrabble as a mere prefix, but might merge well with "gram" or "meter.")
Then there's the lowly "e," perhaps the most common letter in English writing, and a mere 1 point in Scrabble, but in math it denotes the number equal to about 2.718281828, which is mostly notably the number for which the equation f(x)=a^x is equal to its own derivative. Granted, that sounds dry at first, but it means that functions expressed in those terms can go through derivations as easily as a ambassador with diplomatic priveliges can go through Customs. And how fascinating that e=(1/0!)+(1/1!)+(1/2!)+(1/3!)+(1/4!)+....
Of course that leads us to "!" which has nothing to do with surprise or alarm, but rather the factorial of a number, where 4!=1 x 2 x 3 x 4, and 5!=1 x 2 x 3 x 4 x 5 and so on. I admit that I can hardly see the term 6! without thinking "SIX!" before I remember, "oh, um..I mean, uh, six factorial."
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old hand
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So if "mode" occurs most frequently, it will have the biggest impact on the mean/average, yes?
Please someone - goat (welcome, btw!) or someone! - help me out with a "concrete" example....Say we have a little math test, and these are the results:
MG: 33%
WW: 61% (I'm guessing, okay?! but not with regard to my own mark! I did get that once, on a math test!)
wow: 66%
wwh: 75%
dodyskin: 75%
consuela: 75%
etaoin (hope I spelled that right): 77%
goat: 88%
stales: 95%
inselpeter: 98%
There, that's 10 scores. What is the mean/average, and what is the mode? (I'm guessing the latter is 75%, unless I've got hold of the wrong end of the stick entirely.)
help help, math was never my strong point but I would like to understand these terms, they have always confused me. Maybe that's why I got 52% in Grade 12 math - gaahhhh! (33 1/3% on the Christmas exam, 36 1/3 % on the June exam - I am sure the teacher fudged something to give me a bare pass so he'd never have to see me again!)
Let us go in peace to love and serve the board.
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Carpal Tunnel
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the problem with math(s) is, we start kids off with algerba, and while it is a place to start, we keep at it..for ages.. i liked math so much better when i started to learn other branches... non-euclidian geometery, geometery, numbers (and bases binary, octal, hex, and all the other ones) fibonacci numbers, golden ratio's... there are so many wonderful ideas.. and we get get bogged down doing long division to 5 places!--
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So if "mode" occurs most frequently, it will have the biggest impact on the mean/average, yes?
Clearly 1 had a bigger impact than any of the other numbers.
You'd think so. And it's true - except. Say we have a list of 53 values:
6, 6, 6, 601, 602, 603, 604, 605, 606, ... ,650.
The mean (average) is close to 600. In many real problems you might get data kinda similar to this (readings from a sensor). In this case, our mode is not very descriptive of the data at all. We might be tempted to just call them 'outliers' and throw them away. But we have to be careful. It's often a judgement call - is it real data or an equipment malfunction (or maybe just an quirk of the equipment or procedure).
MG: 33%
WW: 61% (I'm guessing, okay?! but not with regard to my own mark! I did get that once, on a math test!)
wow: 66%
wwh: 75%
dodyskin: 75%
consuela: 75%
etaoin (hope I spelled that right): 77%
goat: 88%
stales: 95%
inselpeter: 98%
Let me line the data up so I can read it:
33 61 66 75 75 75 77 88 95 98
If I punched into my calculator correctly, the average is 74.3%
(743/10). Note that 74.3% is not a value in your list. The mode
is 75 and the median is also 75. Frankly, I don't use mode very
much (I don't know why - it's just never come up). Well, I guess
I do sorta use it implicitly in histograms, but I don't normally
(ahem) think about it. Then again, I'm not a professional statistician.
Btw, you can remember etaoin's name, which you did spell correctly, as the most common letters in the English language in descending order, etaoin shrdlu, which I believe Bill, at least, already hinted at.
k
All that and I miscounted. I modified my example and forgot to recount. Say 53 (or so items, with a few duplications even, but no n-tuples with n greater than 2).
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Pooh-Bah
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In reply to:
the problem with math(s) is, we start kids off with algerba, and while it is a place to start, we keep at it..for ages.. i liked math so much better when i started to learn other branches... non-euclidian geometery, geometery, numbers (and bases binary, octal, hex, and all the other ones) fibonacci numbers, golden ratio's... there are so many wonderful ideas.. and we get get bogged down doing long division to 5 places!--
You're right, the higher levels of math are a lot more fun than the pedestrian arithmetic of grammar school and junior high. One of my mathematics professors in college wasn't very good at arithmetic at all, and he would frequently point out that math and arithmetic were not the same things. Once a friend of mine expressed a combination of horror and wonder that I was majoring in math. He said, "Man, long division, that's what did me in!" I got the impression that he thought college math was just longer and longer division, with the occasional word problem (in Swahili) thrown in.
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Carpal Tunnel
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lucky me! i had "new math" in the early '60s, and so i was learning binary and octal when LBJ was president (still in elementary school, some new math even as a preteen..
I remember clearly 2 lessons, one on pi, and one one plotting circles, elipes, parabulas and hyperbalas..
these same fomulas came up in news reports, for space flights, and i remember thinking how cool it was, that i could understand what the engineers where saying.. this was fun math, and the crap about the shop keeper who was selling mixed dried fruit, that was 10% raisens, and 15% prunes, and so many figs, and prunes cost so much, and figs cost so much, and how much did he would he charge for his mix dried fruits...Or problem that involved the total time it took a train to complete its journey, if it went this speed for this many miles, and this speed for an other few miles, and stopped this long at this station..
Who cared? but plotting the arc that the space capsule traveled? or finding the orbit? that was exciting! that was math of the future, not math of the past!
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stranger
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Math riddle coming. What's the difference between an actuary and a sicilian actuary?
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The goat asks, "Math riddle coming. What's the difference between an actuary and a sicilian actuary?"
Google replies, "A traditional actuary can tell you how many people are going to die in the next year. The Sicilian actuary can give you their names and addresses."
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veteran
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One of my mathematics professors in college wasn't very good at arithmetic at all, and he would frequently point out that math and arithmetic were not the same things.
"[Not being] very good" at arithmetic is a relative term. A person who fails to learn arithmetic and algebra is doomed to a life of wading in the mathematical shallows.
I suspect the problem is not with the student or with the subject in most cases, but with the teacher.
k
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the problem with math(s) is, we start kids off with algerba, and while it is a place to start, we keep at it..for ages.. i liked math so much better when i started to learn other branches... non-euclidian geometery, geometery, numbers (and bases binary, octal, hex, and all the other ones) fibonacci numbers, golden ratio's... there are so many wonderful ideas.. and we get get bogged down doing long division to 5 places!--
I have to disagree. The problem is that we cover so many different things and none of them well. This is one of my primary gripes with the Standards and Practices adopted by the NCTM. Throughout their document they shut off contrary views by simply stating, "Well, there's no supporting evidence for that opinion." And then, for one of their major points, all of the data points away from what they're saying and suddenly their defense is, "Well, there's no reason to think otherwise!"
Here's the exact point - according to their data! Other advanced countries pretty consistently do better at math than we do, but they teach it differently. They cover topics at a slower rate, but in much more depth. The advantage is that they really learn the subject before proceding. This means when they come back to the subject later, they don't have to spend all their time reviewing what students didn't learn in the first place. We cover many more topics than they do in the other countries; however, we spend a lot of time reviewing, etc. By the time kids graduate, the kids in the other countries have covered more topics in more detail - because they learned the topic before going on. Does the National Council of Teachers of Mathematics infer that possibly they're doing something right? No way! Their response is "well, there's no reason to think it wouldn't work just as well our way!" (Well, other than the fact that we're already doing it our way and they're smoking us - according to their own documentation.)
When I was about 5 or 7 my dad swam me out to the middle of a lake and dropped me off. He was so sure I'd just start swimming, but I sank like a rock - was absolutely terrified of the water for years later. Eventually, someone took the time to teach me and I've loved it ever since. I'm better now than I ever was as a kid. And I've been sure to teach my kids - first spending a long time getting them accustomed to the water and then getting them in classes and then a team. This August they start with a more advanced trainer.
It would have been easy for me to have settled for the shallow end. There are all sorts of things I can do and admire without swimming a stroke and -- those things are worthwhile things to do. I could just play marco polo and toss balls and "dive" for things. I could watch the other people swim and do real diving. Life would have been okay for me. But I would not have learned to swim. To my mind, doing math without mastering algebra is like playing in the water without learning to swim.
I think we have very different experiences with the new math. I was a victim at the tail end of it. I failed the second grade and when I was in the third grade they would walk me to the far end of the school to take math with the first graders - quite a humiliating walk, though it may have been for the best.
I oscillated from thinking that I was a complete idiot and wondering if just maybe the teachers didn't understand much of what they were teaching. Already, I had some inkling that when I said I understand something I had a very different thing in mind than what other students meant when they said the same thing for themselves. They felt that if they could copy whatever the teacher was doing that they understood it.
Fortunately in the fifth grade I got my first real teacher. My worst subject (math) became my best over night. They quit fluttering around half-teaching things and started to really get into how things fit together.
In retrospect, I can see several problems. 1) Teachers - particularly in the lower grades - often don't know much more about a subject than exactly what they're teaching. 2) In some cases (hopefully small, but I'm not sure), teachers only know the mechanics of what they're doing. They don't even know their own subject very well. 3) Possibly the most important, they very seldom have much practical experience with the kinds of math they're teaching, so they hem and haw around it when students ask for examples or more detailed explanations. 4) Another important one - PARENTS pass on their math-o-phobic tendencies to their kids, not through genetics of course, but through their actions, sometimes subtle and sometimes overt. 5) Current testing regimens allow teachers to use "teaching to the test" as an excuse for covering too many different subjects far too shallowly. 6) (not sure how important this really is) The NCTM's standard justifies teaching shallow knowledge, while recommending preventing students who ARE capable of taking advanced math from doing do. 7) Modern textbooks are written to be flashy and pretty while carefully concealing the important information - almost as an aside. 8) Very often work-sheets that teachers have students do are nonsensical or just silly, indicating to me that the people who make these worksheets (from workbooks the teachers have) don't really understand the subject very well.
I've witnessed "poor students," math-o-phobes to a Tee, actually get excited about learning the boring stuff and ask perfect questions - only to be rebuffed by a teacher who couldn't admit he didn't know the answer. It happens a lot.
(There should be no problem saying "I don't know" and if a teacher doesn't say that a few times a week, then she probably isn't pushing her students hard enough.)
These are what I think are the real problems. I realize there are people who believe otherwise and they're welcome to believe that way. No one can do otherwise than learn from his own experiences. But if I'm the only person to maintain it, then so be it. We could just as well say that students should not have to spend so much time learning spelling and grammar. In fact, I think we could make more of a case for this, since linguistic redundancy ensures that people can usually read text in the face of ambiguity, contradiction, and damned-near unparsability. Or we might maintain that one needn't read a single literary classic - as the Cliff Notes contain everything one really needs to know. We could maintain any of these things, but I think they're all mistaken - and for the same reason - we're settling for the wading pool.
k
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Fallible,
Hear! Hear! Your observations on math teachers are so right! I hated math in sixth grade--Mrs. Smith--yecch! We worked with division of fractions by fractions, and I didn't want to learn it because it made no sense to me why we were inverting the numerator and denominator--that seemed to me to be a huge mental operation--one like turning the world upside down. I refused to just memorize it. I bugged her to death to explain how it came to be--why we were doing so. She ignored me. I remember my dad saying, "Just do it!" And I balked. Seems silly now, but I couldn't make myself do something I couldn't understand. I refused to do homework--refused to take quizzes. Just sat on my tuffet and wouldn't do anything at all. Mrs. Smith threatened to hold me back a year because I wouldn't do the work in division of fractions. My dad was probably angrier at me at that point in my life more than at any other, but I was stalwart.
Can't remember much in seventh grade, but we moved to Fairfax County in Virginia in eighth grade where they were teaching the "new math." I took to it like a fish to water. What an awakening it was!
And everything in math was smooth, easy, and great fun till calculus. When I hit calculus, I zoned out and haven't looked back since.
I'd expect there are lots of horror stories from math students.
WW (which is "Worries" squared)
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Pooh-Bah
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In reply to:
And everything in math was smooth, easy, and great fun till calculus. When I hit calculus, I zoned out and haven't looked back since.
I took everything up to calculus in high school because it was expected of me as a college-bound student and the son of teachers, but I can't say I really had any fun with it. I did what I had to do to get by. In college when I took a calculus class for pre-med, I found I really loved it. It was fascinating and beautiful, and I was actually eager to do my homework after class. The other thing was I read a biography of an Indian mathematician named Ramanujan (http://www.amazon.com/exec/obidos/ASIN/0671750615/qid=1026250088/sr=8-1/ref=sr_8_1/103-0172822-0908611) , which was also fascinating and made me see the creative aspects of the science. That's how I ended up choosing math as a major.
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