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How would you explain the concept of mean, median, and mode of a list of numbers and why they are important to somebody with only basic arithmetic skills? Let's not mention skewness, CLT, central tendency, their statistical properties, etc.

I have explained to somebody that mean is just a quick and dirty way to "summarize" a list of numbers. But looking back, this is hardly illuminating.

Any thoughts or real world examples?

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  • $\begingroup$ They are "central tendency", aka "most likely outcome" in different domains. Intensity, order, and frequency, in particular. The real world also has variation - so things like standard deviation, inter-quartile (or quantile) range, and inter-modal range are also quite useful because they indicate "tendency of variation" or "typical variation in outcomes". $\endgroup$ – EngrStudent Mar 6 '16 at 23:36
  • $\begingroup$ You can give an example that there's a machine generating numbers at random. You collect all the numbers it generates within a list. You now want to present it to your friends without citing every number in the list. Thus you look for measures that can help you describe it. The Mean/Median/Mode are three similar measures that provide insight into basic properties of the machine. $\endgroup$ – Kevin Pei Mar 6 '16 at 23:46
  • $\begingroup$ @KevinPei But what does "mean" mean in this case? Mean/Median/Mode doesn't explain much in a contrived, self-contained example. $\endgroup$ – Dombey Mar 6 '16 at 23:58
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    $\begingroup$ Finding mean is a problem of finding the pivot point balancing the seesaw after the kids (of the same weight) have embarked on it in arbitrary number and on arbitrary positions on the beam. Finding median is the same task, only the kids are said to cluster tight in just two positions either on "this" side or "that" side. $\endgroup$ – ttnphns Mar 7 '16 at 9:05
  • $\begingroup$ You can't explain this without the notion of distribution. With only basic arithmetic skills you have to draw pictures. $\endgroup$ – Aksakal Mar 30 '16 at 15:15
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Thank you for this simple-yet-profound question about the fundamental statistical concepts of mean, median, and mode. There are some wonderful methods /demonstrations available for explaining and grasping an intuitive -- rather than arithmetic -- understanding of these concepts, but unfortunately they are not widely known (or taught in school, to my knowledge).

Mean:

1. Balance Point: Mean as the fulcrum

The best way to understand the concept of mean it to think of it as the balance point on a uniform rod. Imagine a series of data points, such as {1,1,1,3,3,6,7,10}. If each of these points are marked on a uniform rod and equal weights are placed at each point (as shown below) then the fulcrum must be placed at the mean of the data for the rod to balance.

enter image description here

This visual demonstration also leads to an arithmetic interpretation. The arithmetic rationale for this is that in order for the fulcrum to balance, the total negative deviation from the mean (on the left side of the fulcrum) must equal to the total positive deviation from the mean (on the right side). Hence, the mean acts as the balancing point in a distribution.

This visual allows an immediate understanding of the mean as it relates to the distribution of the data points. Other property of the mean that becomes readily apparent from this demonstration is the fact that the mean will always be between the min and the max values in the distribution. Also, the effect of outliers can be easily understood – that a presence of outliers would shift the balancing point, and hence, impact the mean.

2. Redistribution (fair share) value

Another interesting way to understand the mean is to think of it as a redistribution value. This interpretation does require some understanding of the arithmetic behind the calculation of the mean, but it utilizes an anthropomorphic quality – namely, the socialist concept of redistribution – to intuitively grasp the concept of the mean.

The calculation of the mean involves summing up all values in a distribution (set of values) and dividing the sum by the number of data points in the distribution.

$$ \bar{x} = (\sum_{i=1}^n{x_i})/n $$

One way to understand the rationale behind this calculation is to think of each data point as apples (or some other fungible item). Using the same example as before, we have eight people in our sample: {1,1,1,3,3,6,7,10}. The first person has one apple, the second person has one apple, and so on. Now, if one wants to redistribute the number of apples such that it is “fair” to everyone, you can use the mean of the distribution to do this. In other words, you can give four apples (i.e., the mean value) to everyone for the distribution to be fair/equal. This demonstration provides an intuitive explanation for the formula above: dividing the sum of a distribution by the number of data points is equivalent to partitioning the whole of the distribution equally to all of the data points.

3. Visual Mnemonics

These following visual mnemonics provide the interpretation of the mean in a unique way:

enter image description here

This is a mnemonic for the leveling value interpretation of the mean. The height of the A's crossbar is the mean of the heights of the four letters.

enter image description here

And this is another mnemonic for the balance point interpretation of the mean. The position of the fulcrum is roughly the mean of the positions of the M, E, and doubled N.

Median

Once the interpretation of mean as the balancing point on a rod is understood, the median can be demonstrated by an extension of the same idea: the balancing point on a necklace.

Replace the rod with a string, but keep the data markings and weights. Then at the ends, attach a second string, longer than the first, to form a loop [like a necklace], and drape the loop over a well-lubricated pulley.

enter image description here

Suppose, initially, that the weights are distinct. The pulley and loop balance when the same number of weights are to each side. In other words, the loop ‘balances’ when the median is the lowest point.

Notice that if one of the weights is slid way up the loop creating an outlier, the loop doesn’t move. This demonstrates, physically, the principle that the median is unaffected by outliers.

Mode

The mode is probably the easiest concept to understand as it involves the most basic mathematical operation: counting. The fact that it’s equal to the most frequently occurring data point leads to an acronym: “Most-often Occurring Data Element”.

The mode can also be thought of the most typical value in a set. (Although, a deeper understanding of ‘typical’ would lead to the representative, or average value. However, it’s appropriate to equate ‘typical’ with the mode based on the very literal meaning of the word ‘typical’.)


Sources:

  • The Median is a balance point -- Lynch, The College Mathematics Journal (2009)
  • Making Statistics Memorable: New Mnemonics and Motivations -- Lesser, Statistical Education, JSM (2011)
  • On the Use of Mnemonics for Teaching Statistics -- Lesser, Model Assisted Statistics and Applications, 6(2), 151-160 (2011)
  • What does the mean mean? – Watier, Lamontagne and Chartier, Journal of Statistics Education, Volume 19, Number 2 (2011)
  • Typical? Children's and Teachers' Ideas About Average – Russell and Mokros, ICOTS 3 (1990) OVERALL REFERENCE: http://www.amstat.org/publications/jse/v22n3/lesser.pdf
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I have to wonder whether your criteria are achievable as you seem to want maximal effectiveness and explanatory power with minimal materials. But a simple example such as

1 1 2 2 2 3 3 4 5 6 15

allows immediate calculation of the mode (2), the median (3) and the mean (44/11) = 4 and thus shows that they can be different.

You could then explain that the ideas of the most common value, the value in the middle and the mean are different. And introduce complications by

  1. changing values to show the mode can be ambiguous

  2. using an example with an even number of values to explain the convention for calculating the median

  3. varying values in the tails to emphasise what happens to the mean, and why and why not that may be desirable.

  4. using simpler examples in which two or three of mean, median, mode coincide.

I have not mentioned central tendency in my teaching except to say that it's a term in various literatures. I prefer to talk about level and how it may be quantified. Conversely, I don't think any serious data analysis is possible unless people have a minimal feeling for skewness as more usual than symmetry.

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  • $\begingroup$ Yes, adjusting the values will change the summary statistics but still what does "mean" mean by itself? $\endgroup$ – Dombey Mar 7 '16 at 1:27
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    $\begingroup$ calculation $\neq$ understanding $\endgroup$ – shadowtalker Mar 7 '16 at 3:52
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    $\begingroup$ What is red? We don't always need to know definitions to use ideas. A sound understanding of red probably requires physics, physiology and psychology, but I've never needed it. I know a lot about how the mean works, but at one fundamental level its definition is just its formula. $\endgroup$ – Nick Cox Mar 7 '16 at 8:18
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    $\begingroup$ @NickCox very fair and very true. but my experience in college is still pretty recent, and i remember too many problems where i blindly calculated an answer without understanding what i calculated or why i did it $\endgroup$ – shadowtalker Mar 7 '16 at 15:43
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    $\begingroup$ @ssdecontrol That doesn't ever stop happening completely... $\endgroup$ – Nick Cox Mar 7 '16 at 16:33
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This is how I explain them:

The (arithmetic) mean is the point that takes the entire data set into account, and settles somewhere "in the middle." Have them think of a point cloud, or a blob, in space: the mean is the center of mass of that point cloud.

The median is the point that has "the same number of points on all sides" (where obviously the concept of a "side" isn't well-defined in 2+ dimensions). This represents another kind of "middle," and in fact a more intuitive kind in some sense. Thinking of that same blob in space, it is clear that if the blob is lopsided then the mean will be shifted. But this lopsidedness can be achieved in one of two ways: either you add more points in one area, or you increase the dispersion of points in that area. If you increase the dispersion of points in one area without increasing the number of points, then the median still has the same number of points "on all sides" and will not shift commensurate with the mean.

You can demonstrate this with two very trivial "blobs": $y = (1, 2, 3, 4, 5)$ and $y' = (1, 2, 3, 4, 99)$. $\operatorname{mean}(y) = \operatorname{median}(y)$, whereas $\operatorname{mean}(y') > \operatorname{median}(y')$. But I recommend starting with the geometric/visual "blob-based" explanation first: in my experience it's easier to start with a hand-waving graphical demonstration, then move to concrete toy examples. I find that most people (myself included) aren't naturally number-oriented, and starting with a numerical explanation is a recipe for confusion. You can always go back and teach more precise definitions later.

The mode is the point that, if points are randomly sampled from that blob, is most likely to appear (recognizing that this is a fudge for continuous data). This can be, but doesn't have to be, located near the mean or median.

Once you've explained these concepts, then you can move onto a more "statistical-looking" demo:

demo

The solid line is the mean. The dashed line is the median. The dotted line is the mode. The mean represents the positions of the data points along the x axis, while the median reflects only the number of data points on either side. The mode is just the point of greatest probability, which is different from both the mean and the median.

R code:

set.seed(47730)
y <- rgamma(100, 2, 2)
d <- density(y)
plot(d)
rug(y)
abline(v = mean(y), lty = 1)
abline(v = median(y), lty = 2)
abline(v = d$x[which.max(d$y)], lty = 3)
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  • $\begingroup$ Good explanations, but really this assumes a lot more than "basic arithmetic skills": geometric thinking, elementary mechanics, random sampling, probability theory (including density function) are all invoked here. That is a comment, not intended as a knock-down criticism, as I think the question is a tall order. $\endgroup$ – Nick Cox Mar 7 '16 at 7:54
  • $\begingroup$ @NickCox good points for sure. But now that I'm thinking about it, I stand by using them, because they in turn can all be explained without math (e.g. the "see-saw" explanation by ttnphns in the comments on the main question), or they are already understood intuitively on some level by a lot of people. The density is a bit of a reach, but I don't think you ever need to go there $\endgroup$ – shadowtalker Mar 7 '16 at 15:40
  • $\begingroup$ (@ttnphns: tagging you in case you want to weigh in. it won't let me tag you both in one comment) $\endgroup$ – shadowtalker Mar 7 '16 at 15:41
  • $\begingroup$ Density is not that abstruse. Most people should recall density from physics and population density from geography, or just general knowledge. $\endgroup$ – Nick Cox Mar 7 '16 at 16:35
  • $\begingroup$ @NickCox I thought that's what you meant by referring to elementary mechanics. And other than the density demo I don't see how random sampling is needed here, either. If anything, I imagined that the sticking point would be getting a non-technical student comfortable with the idea of a point cloud. Maybe take this to chat? $\endgroup$ – shadowtalker Mar 7 '16 at 18:25
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The "mean", "median" and "mode" are "central tendency", aka "most likely outcome" in different domains. They are all "best bets" in different "games".

Probability and Statistics is a field that was, in part, built by gamblers (link, link) . When you go to the horse races, or the poker table, you want to know some science that helps you win. They did too, and wrote about it, so you don't have to invent it yourself.

In a horse race, you want to pick a winner. You don't have future information, but you do know some past information. You know how fast each horse ran in the past few races. If you want to make an estimate of how fast they are likely to run in their next race, you can compute and compare the mean, aka the average, race-times.

Another central tendency is the "median" - which is the center of a sorted list. What if I put a horrible typo on your list of race times, and the value was 1000x longer than all the others. It would mess up your estimate. You might not bet on the winning horse. How do you address that? You could manually look for that one value, or you might use the "median".

What if you are playing cards, like "blackjack", and you are trying to figure out if you need another card given the previous cards. The card you are looking for is not a 3.14 because cards numbers are integer values. How do you figure out what your best bet is when "average" or median is not meaningful? In this case, you want to bet on the "mode" - the most likely card to come out of the dealers stack.

In all three cases, the central tendency is just another way of saying "best bet".

If you want to account not only for central tendency in your betting, that is to say if you want to bet so that you are able to reduce the impacts of a loss while maximizing winnings, then you must look at "tendencies of variation". Things like standard deviation, inter-quantile-ranges, or alternative modes and their frequencies, are all used to minimize the maximum losses while maximizing the likely winnings.

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I think it's useful to explain this concept when considering multiple means, medians, and modes. These values don't exist by themselves in a vacuum.

For example, here's how I would explain mean.

Let's say you have 2 crates of watermelons (crate 1 and 2). It's sealed off so you can't see the watermelons inside and thus you don't know their sizes. However, you do know the total weights of the watermelons in each crate and each contains the same number of watermelons. From that, you can compute the mean weights of each crate of watermelons (M1 and M2).

Now that you have two different mean values M1 and M2, you can do a rough comparison of the individual contents. If M1 > M2, then a randomly selected watermelons from crate 1 may probably be heavier than one picked from crate 2.

Of course, I would love comments on this perspective.

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