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75

From a statistical point of view the average of no sample points should not exist. The reason is simple. The average is an indication of the centre of mass of the distribution. Clearly, for no observations there can be no way to prefer one location vs. another as their centre of mass since the the empty set is translation invariant. More mathematically, ...


29

The Fréchet mean generalizes the concept of mean to arbitrary metric spaces. It is the point which minimizes the sum-of-squared distances between elements of the dataset $X$: $$\text{arg}\min_\bar{x} \sum_{x\in X} d(\bar{x},x)^2$$ In the case that $X=\varnothing$, the summation is the empty sum and hence $0$, thus there is no minimizer and the mean is ...


18

The correct answer is "Error: Cannot compute the average without any numbers. Please enter at least one number." $0$ is incorrect, because division by $0$ is undefined, not $0$: $\frac00\neq 0$. If you have $0$ elements, you simply cannot compute their average. NaN is slightly better, but still kind of wrong. It's a special value of the IEEE floating point ...


14

It cannot be done. The average of the $\frac{b_i}{u_i}$ cannot be recovered from the average of the $b_i$ and the average of the $u_i$. For suppose that $b_1=b_2=b_3=b_4=6$ and $u_1=u_2=u_3=u_4=3$. Then $$\frac{b_1}{u_1}+\frac{b_2}{u_2}+\frac{b_3}{u_3}+\frac{b_4}{u_4}=8\tag{1}.$$ Suppose now that $b_1=b_2=b_3=b_4=6$, and $u_1=u_2=2$ and $u_3=u_4=4$. Then ...


13

The average of an empty collection of numbers is clearly undefined, as is the centroid of the empty set (or a set of measure zero, for that matter). Therefore the value $0$ given by one of your computers is wrong. What a clever computer should say in such a case depends on the implementation. I'd expect at least some sort of error message, but certainly not ...


7

The average of $n$ numbers is their sum divided by $n$. If $n=0$ then the sum of $0$ numbers is $0$. But dividing by $0$ will result in a computation error. Either answer can be taken as correct, depending on your needs. If you want the average of $0$ numbers to be defined, make it $0$ (since it's the only sane choice), and if you want it to be ...


3

For IEEE 754-1985, the correct result would either be ±infinity with the Division by zero error flag set if you view the average as sum / count. The other option, if you view the empty set to be the correct answer, would be to return 0 with the Underflow error flag set indicating that the answer is incorrect, but 0 is the nearest value. ...


3

From my experience so far in statistics, I have more often heard "average" when discussing samples and in nonparametric statistics. I have first seen the definition of the expected value in a frequentist parametric statistic context, and we understood the expected value as the average of the outcomes when repeatedly repeating the procedure (the average is an ...


2

In support of the Frechet mean argument, the mean of no numbers is defined as 0/0. This is the solution of the equation x * 0 = 0 (from the definition of division) which is solved by any x. So the mean is any number. Note that 0/0 is different from a/0 where a != 0, because x * 0 = a has no solutions. However there is no way to represent "any number" as a ...


2

This quantity always exists and is always equal to $1$. By Chebotarev's density theorem, the density of primes $p$ such that $f$ has $k$ roots $\bmod p$ is equal to the proportion of elements in the Galois group of $f$ which fix $k$ roots when acting on the roots. Since the Galois group acts transitively on the roots, the average number of fixed points is ...


1

Are you wanting straight average or weighted average? If straight average, I think that your formula on row 8 for each column should be =sum(c2:c5)/c6. That would give you the monthly average. Excel ignores blank cells in its built in functions. So you could just use =average(c2:c5) and it would give you 20 for the average number of employees in ...


1

So you want to find the average value of this function. In general, if you have an integrable function $f:[a, b] \rightarrow \mathbb{R}$, then the average value of that function is given by $\displaystyle \frac{1}{b-a}\int_a^b f(x) dx$. If you've been assigned this problem, you should be able to find this formula discussed somewhere in your textbook. ...


1

I was wondering that how did u rank the difficulty level of the question? I would suggest to make it an inverse function of average score for that question. for example :- if a question was answered by 4 test takers correctly(100%) and 3 others scored 50% and rest did not attempt then for 10 test taker its weigtage can be 10/(4 + 3*1.5 +0) if all answered ...


1

Even if you know the median, you would need to know what distribution it is (the shape of a graph of the data) because it could be skewed to one side. For things like age, it's likely to be a normal distribution. If that is the case, then you'd need to know more about the data (like the standard deviation or variance or total number of people etc.) to help ...



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