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161

This answer may have a slightly more mathematical bent than you were looking for. The important thing to recognize is that all of these means are simply the arithmetic mean in disguise. The important characteristic in identifying which (if any!) of the three common means (arithmetic, geometric or harmonic) is the "right" mean is to find the "additive ...


107

Part of the issue is that the frequentist definition of a probability doesn't allow a nontrivial probability to be applied to the outcome of a particular experiment, but only to some fictitious population of experiments from which this particular experiment can be considered a sample. The definition of a CI is confusing as it is a statement about this (...


102

You can mechanically check that the expected value does not exist, but this should be physically intuitive, at least if you accept Huygens' principle and the Law of Large Numbers. The conclusion of the Law of Large Numbers fails for a Cauchy distribution, so it can't have a mean. If you average $n$ independent Cauchy random variables, the result does not ...


65

You can use a t-test to assess if there are differences in the means. The different sample sizes don't cause a problem for the t-test, and don't require the results to be interpreted with any extra care. Ultimately, you can even compare a single observation to an infinite population with a known distribution and mean and SD; for example someone with an IQ ...


62

This depends on your loss function. In many circumstances it makes sense to give more weight to points further away from the mean--that is, being off by 10 is more than twice as bad as being off by 5. In such cases RMSE is a more appropriate measure of error. If being off by ten is just twice as bad as being off by 5, then MAE is more appropriate. In any ...


62

Mean = mode doesn't imply symmetry. Even if mean = median = mode you still don't necessarily have symmetry. And in anticipation of the potential followup -- even if mean=median=mode and the third central moment is zero (so moment-skewness is 0), you still don't necessarily have symmetry. ... but there was a followup to that one. NickT asked in comments ...


55

The mean and variance are defined in terms of integrals. What it means for the mean or variance to be infinite is a statement about the limiting behavior for those integrals For example, for a continuous density the mean is $\lim_{a,b\to\infty}\int_{-a}^b x f(x)\ dx$ (considered as a Riemann integral, say). This can happen, for example, if the tail is "...


54

This is another illustration of Jensen's inequality $$\mathbb E[\log X] < \log \mathbb E[X]$$ (since the function $x\mapsto \log(x)$ is strictly concave] and of the more general (anti-)property that the expectation of the transform is not the transform of the expectation when the transform is not linear (plus a few exotic cases). (Most of my undergraduate ...


53

Consider two values symmetrically placed around $0.5$ - like $0.4$ and $0.6$ or $0.25$ and $0.75$. Their logs are not symmetric around $\log(0.5)$. $\log(0.5-\epsilon)$ is further from $\log(0.5)$ than $\log(0.5+\epsilon)$ is. So when you average them you get something less than $\log(0.5)$. Similarly, if you take a teeny interval around a collection of ...


51

Here is a more practical (and not mathematical) answer: The SD (standard deviation) quantifies scatter — how much the values vary from one another. The SEM (standard error of the mean) quantifies how precisely you know the true mean of the population. It takes into account both the value of the SD and the sample size. Both SD and SEM are in the same ...


44

Let $\theta$ be your parameter of interest for which you want to make inference. To do this, you have available to you a sample of observations $\mathbf{x} = \{x_1, \ldots, x_n \}$ along with some technique to obtain an estimate of $\theta$, $\hat{\theta}(\mathbf{x})$. In this notation, I have made explicit that $\hat{\theta}(\mathbf{x})$ depends on $\mathbf{...


43

Expanding on @Brandon 's excellent comment (which I think should be promoted to answer): The geometric mean should be used when you are interested in multiplicative differences. Brandon notes that geometric mean should be used when the ranges are different. This is usually correct. The reason is that we want to equalize the ranges. For example, suppose ...


42

Benefits of using the mean to summarise central tendency of a 5 point rating As @gung mentioned I think there are often very good reasons for taking the mean of a five-point item as an index of central tendency. I have already outlined these reasons here. To paraphrase: the mean is easy to calculate The mean is intuitive and well understood The ...


42

If the population is known to be normal, a 95% confidence interval based on a single observation $x$ is given by $$x \pm 9.68 \left| x \right| $$ This is discussed in the article "An Effective Confidence Interval for the Mean With Samples of Size One and Two," by Wall, Boen, and Tweedie, The American Statistician, May 2001, Vol. 55, No.2. (pdf)


41

Answer added in response to @whuber's comment on Michael Chernicks's answer (and re-written completely to remove the error pointed out by whuber.) The value of the integral for the expected value of a Cauchy random variable is said to be undefined because the value can be "made" to be anything one likes. The integral $$\int_{-\infty}^{\infty} \frac{x}{\...


41

Clearly it's possible, but it's not clear that it could ever be a good idea. Let's spell out several ways in which this is a limited or deficient solution: In effect you are saying that the outlier value is completely untrustworthy, to the extent that your only possible guess is that the value should be the mean. If that's what you think, it is likely to ...


40

Because Alan Turing was born after Ronald Fisher. In the old days, before computers, all this stuff had to be done by hand or, at best, with what we would now call calculators. Tests for comparing means can be done this way - it's laborious, but possible. Tests for quantiles (such as the median) would be pretty much impossible to do this way. For ...


38

As far as I know, the "mean" of a cluster and the centroid of a single cluster are the same thing, though the term "centroid" might be a little more precise than "mean" when dealing with multivariate data. To find the centroid, one computes the (arithmetic) mean of the points' positions separately for each dimension. For example, if you had points at: (-1,...


37

I wouldn't call 'exponential' particularly highly skew. Its log is distinctly left-skew, for example, and its moment-skewness is only 2. 1) Using the t-test with exponential data and $n$ near 500 is fine: a) The numerator of the test statistic should be fine: If the data are independent exponential with common scale (and not substantially heavier-tailed ...


35

While the above answers are valid explanations of why the Cauchy distribution has no expectation, I find the fact that the ratio $X_1/X_2$ of two independent normal $\mathcal{N}(0,1)$ variates is Cauchy just as illuminating: indeed, we have $$ \mathbb{E}\left[ \frac{|X_1|}{|X_2|} \right] = \mathbb{E}\left[ |X_1| \right] \times \mathbb{E}\left[ \frac{1}{|X_2|}...


34

Suppose you start $\{x_i\}$ with mean $m_1$ and non-zero standard deviation $s_1$ and you want to arrive at a similar set with mean $m_2$ and standard deviation $s_2$. Then multiplying all your values by $\frac{s_2}{s_1}$ will give a set with mean $m_1 \times \frac{s_2}{s_1}$ and standard deviation $s_2$. Now adding $m_2 - m_1 \times \frac{s_2}{s_1}$ ...


32

To complete the answer to the question, Ocram nicely addressed standard error but did not contrast it to standard deviation and did not mention the dependence on sample size. As a special case for the estimator consider the sample mean. The standard error for the mean is $\sigma \, / \, \sqrt{n}$ where $\sigma$ is the population standard deviation. So in ...


32

Christoph Hanck has not posted the details of his proposed example. I take it he means the uniform distribution on the interval $[0,\theta],$ based on an i.i.d. sample $X_1,\ldots,X_n$ of size more than $n=1.$ The mean is $\theta/2$. The MLE of the mean is $\max\{X_1,\ldots,X_n\}/2.$ That is biased since $\Pr(\max < \theta) = 1,$ so $\operatorname{E}({\...


31

I'll try to boil it down to 3-4 rules of thumb and provide some more examples of the Pythagorean means. The relationship between the 3 means is HM < GM < AM for non-negative data with some variation. They will be equal if and only if there's no variation at all in sample data. For data in levels, use the AM. Prices are a good example. For ratios, ...


30

Of course, why not? Here's an example (one of dozens I found with a simple google search): (Image source is is the measuring usability blog, here.) I've seen means, means plus or minus a standard deviation, various quantiles (like median, quartiles, 10th and 90th percentiles) all displayed in various ways. Instead of drawing a line right across the plot,...


30

Here is an intuitive argument with light math. Let's say we have a $d$ claiming to be minimizing the MAE of points $x_i$. And, let's say we have $n_l$ and $n_r$ points on its left and right. If we move $d$ slightly left, i.e. an amount of $\Delta$, then all the absolute differences on the left will decrease by $\Delta$, and all the absolute differences on ...


30

The letters that derive from $\mu$ include the Roman M and the Cyrillic М. Hence considering that the word "mean" starts with an $m$ the choice seems relatively straightforward given an already existing tradition to use greek letters in mathematical abbrevation. To satisfy certain individuals craving for actual historical research and assuming that the ...


29

No doubt you have been told otherwise, but mean $=$ median does not imply symmetry. There's a measure of skewness based on mean minus median (the second Pearson skewness), but it can be 0 when the distribution is not symmetric (like any of the common skewness measures). Similarly, the relationship between mean and median doesn't necessarily imply a similar ...


29

I think the confusion can be resolved by considering that the concept of "regression to the mean" really has nothing to do with the past. It's merely the tautological observation that at each iteration of an experiment we expect the average outcome. So if we previously had an above average outcome then we expect a worse result, or if we had a below average ...


29

The arithmetic mean is related to the geometric mean through the Arithmetic-Mean-Geometric-Mean (AMGM) inequality which states that: $$\frac{x_1+x_2+\cdots+x_n} n \geq \sqrt[n]{x_1 x_2\cdots x_n},$$ where equality is achieved iff $x_1=x_2=\cdots=x_n$. So probably your data points are all very close to each other.


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