The arithmetic and geometric averages provide a summary of historical performance. They are both unique in calculation and contain biases (as stated by my lecturer).  My question is where exactly are these underlying biases with respect to calculation and under which situation (with example) would a statistician prefer to use the arithmetic average over the geometric and vise versa.  Thank you
11514
So we have arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM). Their mathematical formulation is also well known along with their associated stereotypical examples (e.g., Harmonic mean and it's application to 'speed' related problems).  However, a question that has always intrigued me is "how do I decide which mean is the most appropriate to use in a given context?" There must be at least some rule of thumb to help understand the applicability and yet the most common answer I've come across is: "It depends" (but on what?).  This may seem to be a rather trivial question but even high-school texts failed to explain this -- they only provide mathematical definitions!  I prefer an English explanation over a mathematical one -- simple test would be "would your mom/child understand it?"
29975
For example, I have historical loss data and I am calculating extreme quantiles (Value-at-Risk or Probable Maximum Loss). The results obtained is for estimating the loss or predicting them? Where can one draw the line? I am confused.
60707
Take the case of book ratings on a website. Book A is rated by 10,000 people with an average rating of 4.25 and the variance $\sigma = 0.5$. Similarly Book B is rated by 100 people and has a rating of 4.5 with $\sigma = 0.25$.   Now because of the large sample size of Book A the 'mean stabilized' to 4.25. Now for 100 people, it may be that if more people read Book B the mean rating may fall to 4 or 4.25.   how should one interpret the comparison of means from different samples and what are the best conclusions one can/should draw?    For example - can we really say Book B is better than Book A.
92134
I'm new to statistics in general (but a very seasoned developer). I'm trying to grasp why it seems like there's a lot of consideration given to intercepts, at least where it comes to models.  For example, the formulas below were taken from :  Fuel ~ 0 + Weight + Displacement      # formula with no intercept term Fuel ~ -1 + Weight + Displacement     # formula with no intercept term   Whether you omit the intercept terms or set them to 0, -1, etc.., it's just a superficial adjustment, right? In the end, does the intercept have no more importance than to just represent a vertical translation of the model?
18489
Let's say I perform linear regression on some data that produces the following $R^2$: $\text{RSS} = 1966815.13$ $\text{TSS} = 2145213.91$ $R^2 =  0.083$ Now let's say I bucket (take the average of successive equal size windows) the data and perform the exact same linear regression. This time the $R^2$ is: $\text{RSS} = 1187.56$ $\text{TSS} = 4758.32$ $R^2 =  0.75$ What have I just done here? I clearly haven't improved the explanatory power of the model over the original data, yet I've hacked the $R^2$ by averaging. What's the bias / variance argument of why this occurs? A graphical representation:  Edit: I've reduced the total variance of the data, but the bias remains unaffected. When (if ever) is this a valid statistical approach?
38124
I've taken up self-study of math. (How smart can that be?) I've just about finished a course in real analysis which spent a lot of time on metric spaces and some time revisiting calculus.  I was thinking of trying abstract algebra. I would appreciate any book recommendations.  Thanks in advance.  Andrew
59606
I am creating a component that aims to calculate the average and variance of a metric associated with events happening during time but with a limited internal memory.  Imagine that the events are visitors entering in a shop and the metric is their age.  During time, my component receives events with the age of each visitor. I don't want my component to memorize the history of each ages. Ideally, I would like a light component storing only: the average A, the variance V and the number of events N.  After each event with age E, I want to update those three values :  N&lt;=N+1 A&lt;=(A*N+E)/(N+1) V&lt;=???   What for V? I am thinking of something like :  V&lt;=(V*N+(E-A)^2)/(N+1)   I know it is not exact as my previous V is using the old A which is no more the average.   Q1 - Is there an exact formula? Q2 - If not, is my proposal a good estimate? Is it biased? Will it converge correctly when N increases? Q3 - Is there a better formula?
3976
Here is the question:  You roll a fair 6-sided dice iteratively until the sum of the dice rolls is greater than or equal to M. What is the mean and standard deviation of the sum minus M when M=300?  Should I write a code to answer these kind of questions?  Please give me some hints on that. thanks!
97356
I'm having some difficulty solving this problem :     Given the sequence $(a_n)$ with $a_n \geq a_{n+1} \geq 0$ show that if $$\sum \sqrt{a_n a_{n+1}} &lt; \infty \implies \sum a_n &lt; \infty.$$    I first thought of using the Arithmetic Mean-Geometric Mean Inequality:   $$\forall{a,b} &gt; 0: \frac{a+b}{2} \geq \sqrt{ab}.$$  Therefore, since $a_n \geq a_{n+1} \geq 0$ we have: $$\frac{a_n+a_{n+1}}{2} \geq \sqrt{a_n a_{n+1}} \;\; \forall{n} \in \mathbb{N}.$$  but I can't use the Direct Comparaison Test here since it will only work to show the other side of the implication $\sum a_n &lt; \infty \implies \sum \sqrt{a_n a_{n+1}} &lt; \infty$.
108714
I have a feeling this may have been asked elsewhere, but not really with the type of basic description I need.  I know non-parametric relies on the median instead of the mean to compare... something.  I also believe it relies on "degrees of freedom"(?) instead of standard deviation.  Correct me if I'm wrong, though.  I've done pretty good research, or so I'd thought, trying to understand the concept, what the workings are behind it, what the test results really mean, and/or what to even do with the test results; however, no one seems to ever venture into that area.  For the sake of simplicity let's stick with the Mann-Whitney U-test, which I've noticed is quite popular (and also seemingly misused and overused too in order to force one's "square model into a circle hole").  If you'd like to describe the other tests as well feel free, although I feel once I understand one, I can understand the others in an analogous way towards various t-tests etc.  Let's say I run a non-parametric test with my data and I get this result back:  2 Sample Mann-Whitney - Customer Type         Test Information         H0: Median Difference = 0        Ha: Median Difference ≠ 0         Size of Customer    Large   Small Count                    45    55 Median                    2     2  Mann-Whitney Statistic: 2162.00  p-value (2-sided, adjusted for ties):   0.4156     I'm familiar with other methods, but what is different here?  Should we want the p-value to be lower than .05?  What does the "Mann-Whitney statistic" mean?  Is there any use for it?  Does this information here just verify or not verify that a particular source of data I have should or should not be used?  I have a reasonable amount of experience with regression and the basics, but am very curious about this "special" non-parametric stuff - which I know will have it's own shortcomings.  Just imagine I'm a fifth grader and see if you can explain it to me.
82643
Back in April, I attended a talk at the UMD Math Department Statistics group seminar series called "To Explain or To Predict?". The talk was given by  who teaches at UMD's Smith Business School. Her talk was based on research she did for a paper titled , and a follow up working paper titled .   Dr. Shmueli's argument is that the terms predictive and explanatory in a statistical modeling context have become conflated, and that statistical literature lacks a a thorough discussion of the differences. In the paper, she contrasts both and talks about their practical implications. I encourage you to read the papers.   The questions I'd like to pose to the practitioner community are:   How do you define a predictive exercise vs an explanatory/descriptive one? It would be useful if you could talk about the specific application. Have you ever fallen into the trap of using one when meaning to use the other? I certainly have. How do you know which one to use?
41027
I'm new to statistics and reading stats books I found different definitions for Variance.  Definition1: $ s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 $  Definition2: $ s^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2 $  I would like to know why is used different definition for same thing?
70792
Take as an example the well-known . In the original paper, the regression model was used to model the amount people were willing to pay for improved air quality.  But the 'people' whose behaviour was being modelled was never specified. Presumably, it was not limited to 'residents of Greater Boston'. Only implausibly would it be 'humans anywhere on the planet'. Nor, 'all US residents'.   So what inference are we invited by the authors to draw about the scope they intend for their theory? Perhaps 'all residents of large US metro areas'. But - what about other First World metro areas? The first two sections of the Boston paper scrupulously avoid any reference which might suggest that they had a particular 'population' to which their 'sample' was to apply.  There is also the dimension of time - the data came from the 70s, whereas the theory is has no time limitations spelt out.  Plus - going back to first principles - how could the Boston data be treated as a sample of a population including, say, the Chicago metro area when no data from the Chicago area was considered?   Besides, lots of the covariates in the Boston regression weren't samples at all, but totals - so, for example, the proportion of blacks was supplied from census stats.  My question: how can we conceive the Boston data as supporting the geographically unlimited propositions in the paper? Is there a mathematical way of showing equivalence or approximation of the Boston data analysis to other cities? Or is it just assumed as a matter of common sense - such that the paper's authors would be amazed to think anyone could have any doubts on the matter?
79684
Calculating absolute values is much more efficient than calculating squares. Is there any advantage, then, to using the latter as a cost function over the former? Squares are easier to treat analytically, but in practice that doesn't matter.
38752
I'm researching non-Euclidean geometry.  Now I am looking for a good source for it.  Please suggest me something.Thanks.
72308
I've been reading over this , trying to make sense of how the mean and variance of a gaussian conditional was derived. I've come to accept that unless I allocate a dozen or so hours to refreshing my linear algebra knowledge, it's out of my reach for the time being. that being said, I'm looking for a conceptual explanation for that these equations represent: $$\mu_{1|2} = \mu_1 + \Sigma_{1,2} * \Sigma^{-1}_{2,2}(x_2 - \mu_2)$$ I read the first as &quot;Take $\mu1$ and augment it by some factor, which is the covariance scaled by the precision (measure of how closely $X_2$ is clustered about $\mu_2$, maybe?) and projected onto the distance of the specific $x_2$ from $mu_2$.&quot; $$\Sigma_{1|2} = \Sigma_{1,1} - \Sigma_{1,2} * \Sigma^{-1}_{2,2} * \Sigma_{1,2}$$ I read the second as, &quot;take the variance about $\mu_1$ and subtract some factor, which is covariance squared scaled by the precision about $x_2$.&quot; In either case, the precision $\Sigma^{-1}_{2,2}$ seems to be playing a really important role. A few questions:  Am I right to treat precision as a measure of how closely observations are clustered about the expectation? Why is the covariance squared in the latter equation? (Is there a geometric interpretation?) So far, I've been treating $\Sigma_{1,2} * \Sigma^{-1}_{2,2}$ as a ratio, (a/b), and so this ratio acts to scale the (second) $\Sigma_{1,2}$, essentially accounting for/damping the effect of the covariance; I don't know if this is valid. Anything else you'd like to add/clarify?
34040
Is there a way to incrementally calculate (or estimate) the average of a vector (a set of numbers) without knowing their count in advance?    For example you have a = [4 6 3 9 4 12 4 18] and you want to get an estimate of the average but you don't have all the values at hand so you want to have a running average without keeping all the previous values available.