Ace Your Reading and interpreting data Tests with Plainmath's Expert Help and Detailed Resources

How to interpret a too small chi-square ${\mathrm{\chi <!--\; \chi \; -->}}^2$ value?
I use the Chi-square (i.e., ${\mathrm{\chi <!--\; \chi \; -->}}^2$) goodness-of-fit test to measure the distance between chunks of data and a theoretical distribution.
For most of the data being tested the results make sense. I got from time to time unusual results, which correspond to very small values (i.e., between 10 and 100). The sample size is not the problem, there is at least 5 elements for each symbol.
After a careful analysis of the observed data, the frequency of each symbol in the data being tested is very close to the theoretical distribution as the ${\mathrm{\chi <!--\; \chi \; -->}}^2$ value suggest.
But it seems to be too perfect for me. It is unlikely according to the chi-square distribution function that a given value would occur (i.e., $z_{0.0001}$).
Should I reject the null hypothesis because the data is too good to be true ?
I did not find clear explanations or I missed something.
How to interpret unlikely values in the left tail of the chi-square distribution ?

How to compute Bias and Variance for the given scenarios?
I'm currently studying the "Learning from data" course - by Professor Yaser Abu, and I do not get the "bias-variance tradeoff" part of it. Actually, the concepts are fine − the math is the problem.
In the lecture 08, he defined bias and variance as follows:
$\text{Bias}={\mathbb{E}}_{\mathbf{x}}[(\stackrel{¯<!--\; ¯\; -->}{g}(\mathbf{x})-<!--\; -\; -->f(\mathbf{x}){)}^2]$, where $\stackrel{¯<!--\; ¯\; -->}{g}(\mathbf{x})={\mathbb{E}}_{\mathcal{D}}[g^{(\mathcal{D})}(\mathbf{x})]$
$\text{Var}={\mathbb{E}}_{\mathbf{x}}\left[{\mathbb{E}}_{\mathcal{D}}[(g^{(\mathcal{D})}(\mathbf{x})-<!--\; -\; -->\stackrel{¯<!--\; ¯\; -->}{g}(\mathbf{x}){)}^2]\right]$
To clarify the notation:
D means the data set $({\mathbf{x}}_1,y_1),\cdots <!--\; \cdots \; -->,({\mathbf{x}}_n,y_n)$.
g is the function that approximates $f$; i.e., I'm estimating $f$ by using g. In this case, g is chosen by an algorithm $\mathcal{A}$ in the hypothesis set $\mathcal{H}$.
After that, he proposed an example that was stated in the following manner:
Example: Let $f(x)=\sin <!--\; \; -->(\mathrm{\pi <!--\; \pi \; -->}x)$ and a data set $\mathcal{D}$ of size N=2. We sample x uniformly in [−1,1] to generate $({\mathbf{x}}_1,y_1)$ and $({\mathbf{x}}_2,y_2)$. Now, suppose that I have two models, ${\mathcal{H}}_0$ and ${\mathcal{H}}_1$.
${\mathcal{H}}_0:h(x)=b$
${\mathcal{H}}_1:h(x)=ax+b$
${\mathcal{H}}_0:h(x)=b$
${\mathcal{H}}_1:h(x)=ax+b$
For ${\mathcal{H}}_0$, let $b=\frac{y_1+y_2}{2}$. For ${\mathcal{H}}_1$, choose the line that passes through $({\mathbf{x}}_1,y_1)$ and $({\mathbf{x}}_2,y_2)$.
Simulating the process as described, he states that:
Looking for ${\mathcal{H}}_0$, $\text{Bias}\approx <!--\; \approx \; -->0.50$ and $\text{Var}\approx <!--\; \approx \; -->0.25$.
Looking for ${\mathcal{H}}_1$, $\text{Bias}\approx <!--\; \approx \; -->0.21$ and $\text{Var}\approx <!--\; \approx \; -->1.69$.
Here is my main question: How can one get these results analytically?
I've tried to solve the integrals (it didn't work) that came from the $\mathbb{E}[\cdot <!--\; \cdot \; -->]$, but I'm not sure if
I'm interpreting in the right way which distribution is which. For example, how to evaluate ${\mathbb{E}}_{\mathcal{D}}[g^{(\mathcal{D})}(\mathbf{x})]$ (it is the same as evaluating ${\mathbb{E}}_{\mathcal{D}}\left[b\right]$ or ${\mathbb{E}}_{\mathcal{D}}[ax+b]$, for ${\mathcal{H}}_0$ and ${\mathcal{H}}_1$, respectively, right?)? The random variable which has uniform distribution over [−1,1] is x, right? Thus
${\mathbb{E}}_{\mathbf{x}}[\cdot <!--\; \cdot \; -->]$ is evaluated with respect to a random variable that follows $U[-<!--\; -\; -->1,1]$] distribution, right?
If anyone could help me to understand at least one of the two scenarios, by achieving the provided numbers for the Bias and Var quantities; it would be extremely helpful.
Thanks in advance,
André

Three Questions on Interpreting the Outcome of the Probability Density Function
I have three questions: What is the interpretation of the outcome of the Probability Density Function (PDF) at a particular point? How is this result related to probability? What we exactly do when we maximize the likelihood?
To better explain my questions:
(i) Consider a continuous random variable X with a normal distribution such that $\mathrm{\mu <!--\; \mu \; -->}=1.5$ and ${\mathrm{\sigma <!--\; \sigma \; -->}}^2=2$. If we evaluate the PDF at a particular point, say 3.4, using the formula:
$f(X)=\frac{1}{\sqrt{2\mathrm{\pi <!--\; \pi \; -->}{\mathrm{\sigma <!--\; \sigma \; -->}}^2}}{e}^{\frac{-<!--\; -\; -->{\left(X-<!--\; -\; -->\mathrm{\mu <!--\; \mu \; -->}\right)}^2}{2{\mathrm{\sigma <!--\; \sigma \; -->}}^2}},$
we get $f(3.4)=0.1144$. How we interpret this value?
(ii) I previously read that the result of 0.1144 is not necessarily the probability that X takes the value of 3.4. But how the result is related to probability concept?
(iii) Consider a sample of the continuous random variable X of size N=2.5, such that $X_1=2$ and $X_2=3.5$. We can use this sample to maximize the log-likelihood:
$max\ln <!--\; \; -->L(\mathrm{\mu <!--\; \mu \; -->},\mathrm{\sigma <!--\; \sigma \; -->}|X_1,X_2)=\ln <!--\; \; -->f(X_1)+\ln <!--\; \; -->f(X_2)$
If f(X) is not exactly a probability, what are we maximizing? Some texts detail that "we are maximizing the probability that a model (set of parameters) reproduces the original data". Is this phrase incorrect?

Meaning of the word "axiom"
One usually describes an axiom to be a proposition regarded as self-evidently true without proof.
Thus, axioms are propositions we assume to be true and we use them in an axiomatic theory as premises to infer conclusions, which are called "theorems" of this theory.
For example, we can use the Peano axioms to prove theorems of arithmetic.
This is one meaning of the word "axiom". But I recognized that the word "axiom" is also used in quite different contexts.
For example, a group is defined to be an algebraic structure consisting of a set G, an operation $G\times <!--\; \times \; -->G\to <!--\; \to \; -->G:(a,b)\mapsto <!--\; \mapsto \; -->ab$, an element $1\in <!--\; \in \; -->G$ and a mapping $G\to <!--\; \to \; -->G:a\mapsto <!--\; \mapsto \; -->a^{-<!--\; -\; -->1}$ such that the following conditions, the so-called group axioms, are satisfied:
$\mathrm{\forall <!--\; \forall \; -->}a,b,c\in <!--\; \in \; -->G.(ab)c=a(bc)$
$\mathrm{\forall <!--\; \forall \; -->}a\in <!--\; \in \; -->G.1a=a=a1$ and
$\mathrm{\forall <!--\; \forall \; -->}a\in <!--\; \in \; -->G.a{a}^{-<!--\; -\; -->1}=1=a^{-<!--\; -\; -->1}a$
Why are these conditions (that an algebraic structure has to satisfy to be called a group) called axioms? What have these conditions to do with the word "axiom" in the sense specified above? I am really asking about this modern use of the word "axiom" in mathematical jargon. It would be very interesting to see how the modern use of the word "axiom" historically developed from the original meaning.
Now, let me give more details why it appears to me that the word is being used in two different meanings:
As peter.petrov did, one can argue that group theory is about the conclusions one can draw from the group axioms just as arithmetic is about the conclusions one can draw from the Peano axioms. But in my opinion there is a big difference: while arithmetic is really about natural numbers, the successor operation, addition, multiplication and the "less than" relation, group theory is not just about group elements, the group operation, the identity element and the inverse function. Group theory is rather about models of the group axioms. Thus: The axioms of group theory are not the group axioms, the axioms of group theory are the axioms of set theory.
Theorems of arithmetic can be formalized as sentences over the signature (a. k. a. language) $\{0,s,+,\cdot <!--\; \cdot \; -->\}$, while theorems of group theory cannot always be formalized as sentences over the signature $\{\cdot <!--\; \cdot \; -->,1{,}^{-<!--\; -\; -->1}\}$. Let me give an example: A typical theorem of arithmetic is the case n=4 of Fermat's last theorem. It can be formalized as follows over the signature $\{0,s,+,\cdot <!--\; \cdot \; -->\}$:
$\mathrm{\neg <!--\; \neg \; -->}\mathrm{\exists <!--\; \exists \; -->}x\mathrm{\exists <!--\; \exists \; -->}y\mathrm{\exists <!--\; \exists \; -->}z(x\ne 0\wedge <!--\; \wedge \; -->y\ne 0\wedge <!--\; \wedge \; -->z\ne 0\wedge <!--\; \wedge \; -->x\cdot <!--\; \cdot \; -->x\cdot <!--\; \cdot \; -->x\cdot <!--\; \cdot \; -->x+y\cdot <!--\; \cdot \; -->y\cdot <!--\; \cdot \; -->y\cdot <!--\; \cdot \; -->y=z\cdot <!--\; \cdot \; -->z\cdot <!--\; \cdot \; -->z\cdot <!--\; \cdot \; -->z).$
A typical theorem of group theory is Lagrange's theorem which states that for any finite group G, the order of every subgroup H of G divides the order of G. I think that one cannot formalize this theorem as a sentence over the "group theoretic" signature $\{\cdot <!--\; \cdot \; -->,1{,}^{-<!--\; -\; -->1}\}$; or can one?

What skills should I learn before going for a PhD in numerical analysis, numerical linear algebra and scientific computing?
I have two years in my hand after which I would be applying for graduate school most probably in the US. I already have a masters degree in pure mathematics and have very limited computer skills. For some reason, eventually I got interested in applied mathematics.
So, now I am going for a second masters (which has not yet started because of the pandemic) in applied mathematics which is heavy on the above-mentioned topics. As of computer skills, I have started learning C++ but I don't know what else should I learn. It would be very helpful if someone can give a roadmap which would prepare me to finally embark for a PhD in one of these areas i.e., in and around numerical analysis.
What math and computer science courses should I take? What software should I learn? Any general advice you would like to give is also welcome. Book recommendations would also be very helpful for me.Thanks!

Data that follows a distribution with non-finite expectation
I find it difficult to get around the idea of some random variables following some distributions (such as the Cauchy Distribution) not to have finite means.
How does one actually interpret data from such random variables? Do we give up and not investigate sample means or ...? Please help clarify this.

I recently had an idea for an app that I would like to start developing for personal use and development, that attempts to present you with recipe idea's for lunch/dinner etc and by recording your responses learns your preferences. I was thinking it would do this by recording very specific details off each recipe such as carbcount caloriecount proteincount and etc, (factors which might act as determinants for our preference). Then this program would run an OLS regression with prob of being chosen as the dependent variable (for which we will have data on as we know what our user rejected (recipe) and what he accepted and how many times). We will then have various independant variables with which we will try and create an unbiased estimator. We can then run all recipe's to be presented under this regression and rank the recipes in order of probability to be chosen, highest to lowest.
Would this be a viable thing to do? If no, why not and what could perhaps be better?

How to interpret the imaginary part of an inverse fourier transform
The fourier transform of arbitrary real data can (usually will) result in complex data. If the real data represents samples in time, then the complex FT data represent frequencies with the magnitudes representing the amplitude and the phase angles representing the phase.
If I perform some arbitrary manipulation on the complex FT data prior to applying an inverse fourier transform, the result of the IFT can also be complex. My question is, how do I interpret the imaginary part of the IFT result? My guess is that a complex IFT result is non-physical, and that this must imply that my original arbitrary manipulation was also therefore non-physical.
Is this correct? If so, are there equations which describe whether such a manipulation would be non-physical.

I am reading a study in which the mean age of a population is 41 years. The mode and median are however both 35 years. I do not have access to the raw data.
How I can interpret the difference between the mean and the mode/median?

How was the conditional probability obtained in this equation?
I'm reading about inverse problems and the Bayesian approach and I'm having troubles understanding how the following equations were obtained.
We consider the following equation initially.
y=G(u)+$\mathrm{\eta <!--\; \eta \; -->}$
y is a set of measured data. G is a mathematical model and u are the parameters of the model. The noise that is present in the observed data is given by $\mathrm{\eta <!--\; \eta \; -->}$ (0 mean noise).
The authors then go on to describe the Bayesian approach and they say that the likelihood function, that is the probability of y given u is given by:
$\mathrm{\rho <!--\; \rho \; -->}(y|u)=\mathrm{\rho <!--\; \rho \; -->}(y-<!--\; -\; -->G(u))$
How did they derive this relation above?

Why are zeros/roots (real) solutions to an equation of an n-degree polynomial?
I can't really put a proper title on this one, but I seem to be missing one crucial point. Why do roots of a function like $f(x)=a{x}^2+bx+c$ provide the solutions when $f(x)=0$. What does that y=0 mean for the solutions, the intercept at the x axis? Why aren't the solutions at $f(x)=403045$ or some other arbitrary n?
What makes the x-intercept special?

The limit of gaussian distribution on curve manifold
I am reading the paper. In introduction, it said
In many applications, even simple estimation problems involving angular data are often considered as traditional linear or nonlinear estimation problems and handled with classical techniques such as the Kalman Filter [1], the extended Kalman Filter (EKF), or the unscented Kalman Filter (UKF) [2]. In a circular setting, most traditional approaches to filtering suffer from assuming a Gaussian probability density at a certain point. They fail to take into account the periodic nature of the problem and assume a linear vector space instead of a curved manifold. This shortcoming can cause poor results, in particular when the angular uncertainty is large. In certain cases, the filter may even diverge.
My background is engineering, and I understand what Gaussian distribution in N-dimension means and the role of Gaussian distribution in Kalman filter. But I don't understand why Gaussian distribution fail to estimate angular data in its nature. It will be great if there are mathematical explanation and reference. Thank you.

How to interpret parameter estimates in factor prediction ( in R )
So I have some data set in a .csv file and there are three factor levels, 1 , 2 , 3, (there are fifteen of each) and each has a corresponding score.
Here are some details.
so the data is contained in a simple csv file, the first column is labelled Team, and the second column is labelled Score.
The first column consists of fifteen 1's, followed by fifteen 2's , followed by fifteen 3's.
The R code I used was
data.source<-"http.www.. " ( the data set)
SportScores<-read.csv(file=data.source)
I set x such that x prints 1 1 1.... 1 2 2 2 ... 2 3 3 3 .... 3 Levels 1 2 3
names(sportScores)
y<-SportScores$Scores
So using lm I get parameter estimates in R as
Intercept (35.800)
x2 (0.066)
x3 (12.40)
the t value are very large for intercept and x3, but very small for x2, ie it indicated to me that we cannot reject the null in this case, but what is the null?
${\mathrm{\beta <!--\; \beta \; -->}}_0=35.8$
${\mathrm{\beta <!--\; \beta \; -->}}_1^c=0.06667$
${\mathrm{\beta <!--\; \beta \; -->}}_2^c=12.40$
But how do I interpret this? I want to see any differences in scores between the 3 levels, etc. I mean, what even is the test being conducted? For example
${\mathrm{\beta <!--\; \beta \; -->}}_1^c=0.06667$
has a small t value, so the null hypothesis is not rejected, but what even is the null hypothesis in this case? Moreover, from the code output itself, how can I know the associated individual standard errors of the estimated means?

Olympiad Inequality $\underset{cyc}{\sum <!--\; \sum \; -->}\frac{x^4}{8x^3+5y^3}⩾<!--\; ⩾\; -->\frac{x+y+z}{13}$
$x,y,z>0$ , prove$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3}⩾<!--\; ⩾\; -->\frac{x+y+z}{13}$
Note: Often Stack Exchange asked to show some work before answering the question. This inequality was used as a proposal problem for National TST of an Asian country a few years back. However, upon receiving the official solution, the committee decided to drop this problem immediately. They don't believe that any students can solve this problem in 3 hour time frame.
Update 1: In this forum, somebody said that BW is the only solution for this problem, which to the best of my knowledge is wrong. This problem is listed as "coffin problems" in my country. The official solution is very elementary and elegant.
Update 2: Although there are some solutions (or partial solution) based on numerical method, I am more interested in the approach with "pencil and papers." I think the approach by Peter Scholze in here may help.
Update 3: Michael has tried to apply Peter Scholze's method but not found the solution yet.
Update 4: Symbolic expanding with computer is employed and verify the inequality. However, detail solution that not involved computer has not been found. Whoever can solve this inequality using high school math knowledge will be considered as the "King of Inequality".

On integration over distribution.
When I read various journal articles related to machine learning, I often face integrals over distribution.
In an article I am reading now, for example, a risk function associated with distribution $\mathrm{\varphi <!--\; \varphi \; -->}$ is defined by
$R_i(\mathrm{\theta <!--\; \theta \; -->})=\int <!--\; \int \; -->f_L(f_{\mathrm{\theta <!--\; \theta \; -->}}(x),y)d\mathrm{\varphi <!--\; \varphi \; -->}(x,y),$,
where
X and Y are a feature space and a label space, respectively.
$f_{\mathrm{\theta <!--\; \theta \; -->}}:\mathcal{X}\to <!--\; \to \; -->\mathcal{Y}$ is a given model parameterized by $\mathrm{\theta <!--\; \theta \; -->}\in <!--\; \in \; -->\mathrm{\Theta <!--\; \Theta \; -->}$,
$f_L:\mathcal{X}\times <!--\; \times \; -->\mathcal{Y}\to <!--\; \to \; -->{\mathbb{R}}_{\ge <!--\; \ge \; -->0}$ is a loss function, and
$\mathrm{\varphi <!--\; \varphi \; -->}$ is the data generating distributions.
In addition to the above case and the others (even not related to this field), I have seen many times integration formulas over distributions. However, whenever I encountered them, I couldn't grasp what it is.
Rather, I am familiar with the following equation:
$\int <!--\; \int \; -->f_L(x)p_X(x)dx,$
where fL(x) is a cost (or reward) function achieved by an event x, and pX(x) is a probability that an event x occurred.
Can someone please let me know what the integral over a distribution means?

Intuition behind optimzation problem of ridge regression?
In one of the texts that I am reading it is given that regularization parameter restricts the choice of functions in case data given is not sufficient for processing of signal. It is given that lambda acts as relative trade-off between norm and loss function. I have two questions:
Why we are choosing norm as the criterion in this optimization problem?
How lambda is acting as a trade-off?
My understanding says that as I increase lambda my weight has to be reduced much more as compared to when lambda is low,which happens only when we restrict lower limit of the optimization problem.So how lambda is a trade-off when both norm and loss-function reduces with increase of lambda.
Please shed some light on this.

Two-sigma notation k-means clustering
I am looking at a k-means clustering algorithm, where the problem is the following:
${\displaystyle {\text{argmin}}_{C_1,..C_k;c_1{c}_1,..c_k{c}_k}\underset{i=1}{\overset{k}{\sum <!--\; \sum \; -->}}\underset{x\in <!--\; \in \; -->C_i}{\sum <!--\; \sum \; -->}||xx-<!--\; -\; -->c_i{c}_i|{|}^2}$
where x is a data vector, e.g. $x_1{x}_1=(1,2,3{)}^{\prime }$, $c_i{c}_i$ is the centroid of $C_i$ cluster. I conceptually understand what the above means, namely we find such centroids for each cluster that minimize the total squared distance between each centroid in a cluster and datapoints in that cluster. But I fail to understand how I would expand out the above expression.

Calculating R-squared with duplicate data
I have the following question regarding the proper usage of R-squared value.
Say I have an equation, that predicts energy consumption for the month of a building. One of the input variables accounts for time, so 1 would be for January, 2 would be for February, etc. and 12 would be for December
For one year, I would have 12 numbers for predicted data points. Say also at the end of the year I am able to get the exact energy consumption from my Electric company. So now I have 12 predicted data points and 12 exact data points which were measured without any error.
Would it be an accurate analysis to take these data points, and perform a regression analysis using excel? Could I then interpret the R-squared value as the telling me how similar the two sets of data are, and overall tell me whether or not the equation that predicts energy consumption is a good enough approximation of the true energy consumption?
Also, if there's a some statistics genius who could tell me the proper way to interpret R-squared, I would really appreciate it. Thanks

Sheaf of rings on a discrete set.
I was reading through some notes for an exam and one exericse asks me to prove the following
There is a unique sheaf of rings making a topological set X with discrete topology a ringed space.
I tried doing it but I feel I'm missing something, using the definition of presheaf and than of sheaf doesn't seem to bring me any result. How can I solve such a problem? I leave you my definitions of presheaf and sheaf.
A presheaf $\mathfrak{F}$ (of rings) on a topological space X consists of the data:
for every open set $U\subset <!--\; \subset \; -->X$ a ring $\mathfrak{F}(U)$ (think of this as the ring of functions on U),
for every inclusion $U\subset <!--\; \subset \; -->V$ of open sets in X a ring homomorphism ${\mathrm{\rho <!--\; \rho \; -->}}_{V,U}:\mathfrak{F}(V)\to <!--\; \to \; -->\mathfrak{F}(U)$ called the restriction map (think of this as the usual restriction of functions to a subset), such that
$\mathfrak{F}(\mathrm{\varnothing <!--\; \varnothing \; -->})=0$
${\mathrm{\rho <!--\; \rho \; -->}}_{U,U}$ is the identity map of $\mathfrak{F}(U)$ for all U,
for any inclusion $U\subset <!--\; \subset \; -->V\subset <!--\; \subset \; -->W$ of open sets in X we have ${\mathrm{\rho <!--\; \rho \; -->}}_{V,U}\circ <!--\; \circ \; -->{\mathrm{\rho <!--\; \rho \; -->}}_{W,V}={\mathrm{\rho <!--\; \rho \; -->}}_{W,U}$
The elements of $\mathfrak{F}(U)$ are usually called the sections of F over U, and the restriction maps ρV,U are written as $\mathrm{\phi <!--\; \phi \; -->}\to <!--\; \to \; -->\mathrm{\phi <!--\; \phi \; -->}|U$
A presheaf $\mathfrak{F}$ is called a sheaf of rings if it satisfies the following gluing property:
if $\subset <!--\; \subset \; -->X$ is an open set, $\{U_i:i\in <!--\; \in \; -->I\}$ an arbitrary open cover of U and ${\mathrm{\phi <!--\; \phi \; -->}}_i\in <!--\; \in \; -->\mathfrak{F}(U_i)$ sections for all i such that ${\mathrm{\phi <!--\; \phi \; -->}}_i{|}_{U_i\cap <!--\; \cap \; -->U_j}={\mathrm{\phi <!--\; \phi \; -->}}_j{|}_{U_i\cap <!--\; \cap \; -->U_j}$ for all i,$i,j\in <!--\; \in \; -->I$, then there is a unique $\mathrm{\phi <!--\; \phi \; -->}\in <!--\; \in \; -->\mathfrak{F}(U)$ such that $\mathrm{\phi <!--\; \phi \; -->}{|}_{U_i}={\mathrm{\phi <!--\; \phi \; -->}}_i$ for all i.
EDIT: This is what is given as the definition of a K-ringed space:
A ringed spaces equipped with a sheaf of rings such that the elements of ${\mathfrak{O}}_X(U)$ are actual functions from U to a fixed ring K;
EDIT: It turns out that the actual definition is
A ringed spaces equipped with a sheaf of rings such that the elements of ${\mathfrak{O}}_X(U)$ are actual functions from U to a fixed ring K and ${\mathfrak{O}}_X(U)$ is not only a subring of the ring of functions from U→K but a sub−K−algebra of it;
What does this change?

What does it mean for the "mean" to be a range
$\begin{array}{|cccc|}\hline \text{Component}& \text{Disk}& \text{Node}& \text{Rack}\\ \text{MTTF}& 10-<!--\; -\; -->50\text{ years}& 4.3\text{ months}& 10.2\text{ years}\\ \hline\end{array}$
Now reading this, it looks like the mean value for a disk failure ranges between 10-50 years!If I remember correctly, the mean or average is a single value and not a range? How should I interpret this?Does it mean that the statistical data available isn't reliable or there is a big difference between different samples or reports?