Questions tagged [average]

Averages summarise a set of values in one way or another. Without further qualification it most commonly refers to the arithmetic mean, but other averages such as root-mean-square exist.

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Expected value and average value problem

One has a very large batch of tablets. The weight (unit: gram) of a randomly selected tablet can be accurately considered as a normally distributed random variable X with mean μ and standard deviation ...
Need_MathHelp's user avatar
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What is the formula that connects the average distance to the nearest point and the average number of points per unit volume?

We have an infinite 3D space with points randomly disseminated inside it. Let $p$ be the average number of points per unit volume. Let $d$ be the average distance to the nearest point. What is the ...
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Average of multiple normally distributed Datasets

Heyo everyone :) My math education mostly stopped after school, so please bear with me while I try to describe my issue as best as I can: In a game I play there's a clear measure for your performance ...
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How to report the accuracy of the average of multiple sensors?

I am measuring the temperature of a metal plate using 4 thermal sensors. The sensor used is specified to have an accuracy of ±0.1°C. If I have for example those readings: The average of all sensors ...
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Splitting the barycenter of a curve

Definitions Consider a generic smooth closed curve $c(t):[0,1]\mapsto \mathbb{R}^2$ and define its barycenter $g$ as \begin{equation*} g \triangleq \frac{\int_0^1 c(t)\,\lVert \dot{c}(t)\rVert \text{ ...
matteogost's user avatar
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Generalization of Geometric Mean, Standard Deviation, etc.

The geometric mean can be thought of as the exponential of the arithmetic mean of the logarithms of your dataset $\{a_i\}_{i=1}^n$: $$GM = \exp\left(\dfrac{1}{n}\sum_{i=1}^{n}\ln(a_i)\right)$$ ...
Patch's user avatar
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Average item cost [closed]

So let's say I can open up crates that have balls in it, but the amount of balls received from each crate has different percentage probability. As shown bellow: ...
Jason C's user avatar
3 votes
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Stochastic average of a differential equation is not the same as average of its solutions

Assume a static (time-independent) random variable $r$ for which we know its probability distribution $P(r)$. Consider this to be a Gaussian distribution, such that $\langle r \rangle=0$ and $\langle ...
J.Agusti's user avatar
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combine multiple weighted averages

Suppose there is a dataset $X = [x_1, \cdots, x_n]$ where each element $x_i$ has a set of features (weights) $A_i = [a^1_i, a^2_i, \cdots , a^k_i]$. We then can have a multiple weighted average ...
goku7489's user avatar
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What type of mean satisfies "average rate of change is the mean of derivatives"?

From observations, for quadratic functions, the average rate of change in $[a,b]$ is the arithmetic average of the derivative evaluated at $a$ and $b$. $${\rm{Avg. ROC}_{[a,b]}}=\frac{f'(a)+f'(b)}{2}\...
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Folland chapter 3 exercise 22 Hardy-Littlewood maximal function and maximal theorem

Following is a question and it's solution from folland's real analysis chapter 3 exercise 22. Q22. If $f \in L^1\left(\mathbb{R}^n\right), f \neq 0$, there exist $C, R>0$ such that $H f(x) \geqq C||...
Siddharth Prakash's user avatar
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Birthday problem: how to show the scaling with $1/N^2$?

Suppose there is a sequence of$N$ numbers $x_1, x_2, x_3, ... x_N$. There are then gaps $|x_i - x_j|$, and the minimum gap: $\delta (N) = \text{min}_{i \ne j \le N} \{ | x_i -x_j | \}$. Let the mean ...
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What is the average of the squared ratio between the 2 norm and the max norm of a vector of iid gaussian variables?

I originally wanted to compute the following quantity $$ \langle v_{\max.}^2 \rangle_{S_d} = \left\langle \frac{\| x\|_{\infty}^2}{\|x\|_2^2} \right\rangle_{\mathcal N_n(0,1)} $$ which is the average ...
Piano Piano a Tratti Forte's user avatar
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What is the intuition behind the single-pass algorithm (Welford's method) for the corrected sum of squares?

The corrected sum of squares is the sum of squares of the deviations of a set of values about its mean. $$ S = \sum_{i=1}^k\space\space(x_i - \bar x)^2 $$ We can calculate the mean in a streaming ...
Foobar's user avatar
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Getting the right average rate of growth

I am trying to get the average rate of growth across a series of numbers that will lead to the same sum as the discrete rates of growth. For example, if I have 1 unit at $t=1$ that grows at 2%, then 3%...
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How do I go from an infinite amount of instantaneous rates to an average rate?

I have a function that gives me rate at a point (the derivative) and I have a value for average rate that was experimentally calculated. How would I go about creating an expression to solve for a ...
DaBest Gamer's user avatar
3 votes
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what more can I say using Sturm's comparison theorem?

Can I use Sturm's comparison theorem to say something about the average? Let $f(t)$ be a continuous function such that $\lim_{t \to \infty } f(t) = \infty$. Let us consider the following function $$ \...
alejandro's user avatar
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Mathematical Notation for mean of a variable in a class

I would ask for your help. How can I write down in mathematical notation the average for a specific feature that has a specific class. For example, in a dataset with two features: a continues HEIGHT, ...
GeorgeStu's user avatar
1 vote
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You're constantly drawing a random integer among {-1, 0, +1}, and adding the results. On averege, how many tries for a positive number result?

You have a randomizer that gives you either -1, 0, or +1. You start with 0, draw a random number, and add it to your total. On average, how many numbers you have to draw before you have a positive ...
Ellye F. S.'s user avatar
6 votes
2 answers
209 views

under which conditions $\lim_{t\to \infty} \frac{1}{t}\int_0^t f(s)ds$ implies $\lim_{t\to \infty} f(t) $

I am interested in knowing under which conditions $\lim_{t\to \infty} \frac{1}{t}\int_0^t f(s)ds$ implies that $\lim_{t\to \infty} f(t)$ exist. It is known that $\lim_{t\to \infty} \frac{1}{t}\int_0^t ...
alejandro's user avatar
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What is statistically more accurate - the average of a dataset of calculated concentrations or the sum of the total mass divided by the total volume? [closed]

Hopefully an easy question to answer. But if I have a series of five water samples, each sample is a different volume with a different number of plastic particles contained within. I can calculate the ...
E Wisniewski's user avatar
1 vote
1 answer
24 views

Weighted Average Over Time for Data Projection (aka Weighted Moving Average) - Mathematically Correct Way to Determine Weights?

Consider the following data set: Say graduation rates for a specific college's undergraduate program that are the following: Year # of Graduates Year 11 10,500 Year 12 10,750 Year 13 11,000 Year ...
SEStudent's user avatar
3 votes
0 answers
69 views

Conjecture about average cluster size in a prime-like sequence

Consider the sequence $u_n=n^a+p_n$, where $a$ is a real constant and $p_n$ is the $n$th prime. Write down the terms in $u_n$ increasing from left to right. From each term, draw a line segment ...
Dan's user avatar
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Derivation of the effective path length

I am trying to understand the derivation of the effective path length $d_{eff}$, which is the average length $L$ of the intersection between the rain cell of length $d_{0}$ and path $d$ shown below ...
radastro's user avatar
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Normalize data around a specific value

I am having trouble to solve a problem about normalization around a specific value. Let's say i have this data below: ValueA ValueB Division (A/B) Gap (Division - 0.625) 10 10 1 0.375 10 16 0.625 0 ...
romainL's user avatar
3 votes
2 answers
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If the average of the series is an integer, prove that there exists a number in the sequence that does not appear more than $n$ times.

We consider a sequence of $6n+5$ numbers $(n \in \mathbb{N})$ all belonging to the set {$4, 5, 6, 7$}. If the average of this series is an integer, prove that there exists a number in the sequence ...
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Arithmetic mean of proper-class-sized ('unsetly') list of real numbers

Suppose I have this list $1, 1, 1, 1, 1…$ with $\bf{Ord}$ (the proper class of ordinals) number of $1$'s. Then would the average (arithmetic mean) of the list be $1$ because there is literally no ...
3-1-4-One-Five's user avatar
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2 answers
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A tricky question on kinematics asked in a competitive exam

So I found this rather ambiguous question based on kinematics from a competitive examination, the question with options is - A car travels in such a way that its velocity becomes twice the previous ...
Nirmish Sarmalkar's user avatar
1 vote
1 answer
120 views

Determining the average rate of change of a product's price over time

I'm currently studying a model that describes the cost of 1 kg of a product $p$ in dollars over time, given by the equation $$\frac{d^3p}{dt^3} + a\frac{dp}{dt} = b,$$ where $a = 9[year^{ - 2}], b = ...
Ivn's user avatar
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Characterize a function given the average and the median

How to (1) obtain some examples for an $f$ function and/or (2) characterize the whole set of $f$ functions which has the following properties given $m$ ($\leftarrow$ the median) that $0 < m < ...
Bálint Sass's user avatar
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The relative difference of averages gives different result than the Average of relative differences

Say I have these values : Value 1 Value 2 Absolute difference (Value 1 - Value 2) Relative difference (Value 1 / Value 2 - 1) 0.80 0.30 0.50 167% 0.75 0.20 0.55 275% I'm interested in calculating ...
pollytatouin's user avatar
3 votes
4 answers
352 views

What is $\lim_{x\to\infty}\frac{\int_{0}^{x}\cos\{t-\cos t\}dt}{x}$?

I want to find a closed form for the average value of $\cos\{t-\cos t\}$ where $\{n\}$ denotes the fractional part of $n$. I do not have experience finding an average value over an infinite domain but ...
Dylan Levine's user avatar
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Average value of a function over some given interval

I was learning about simple harmonic motion from YouTube and in one of the lectures the teacher introduced to the formula for the average value of a function I know that formula for the average value ...
Anshu Gupta's user avatar
1 vote
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Help with formula to calculate "earnings"?

I'm looking to calculate earnings on a writing platform, based on a variety of metrics. Unfortunately I suck at math, and am unable to come up with the exact, proper formula to find an average. ...
FMB's user avatar
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Average of two discrete random variables

I have this graph B<-A->C and I have the joint probability distribution of A and B , P(A,B), and the joint probability distribution of A and C, P(A,C). I want to compute the average of B and C. ...
Moh's user avatar
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2 answers
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Inequality involving average value of a function

Consider the Integral $$\frac{1}{r}\int_{0}^{r}f(x)dx.$$ My observation, after testing several elementary functions, is that if $f(x)$ is a decreasing function, then $$f(r)\leq\frac{1}{r}\int_{0}^{r}f(...
mersa's user avatar
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Calculating weighted averages in ranking task

I have a set of eight principles and have been tasked with working out what the top five preferred principles are. In a series of interviews, participants were asked to rank their top five, with 1 ...
atem634's user avatar
4 votes
2 answers
143 views

Average Determinant

Let $S$ be the set of all $n \times n$ real matrices ($n$ > 1) with entries from a finite set $M$. Can we say that the average value of determinants of matrices in $S$ is always zero? See that $S$ ...
user_1729's user avatar
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Average of a function and center of mass

The geometric center of an object is the center of mass of an object with uniform density. The average height of a semicircular wire is $\pi R/4$ where R is the radius. But the actual y coordinate of ...
Hammock's user avatar
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How do I get a realistic average when having a highly-segregated data groups.

I have an instrument that collects travel_angle, work_angle and speed as the user moves this ...
Dimitar Veljanovski's user avatar
4 votes
0 answers
105 views

Does $f(x)=\frac1x\int_0^x\frac{1}{\sin t+\sin (\pi t)+2}dt$ have a supremum?

Does $f(x)=\frac1x\int_0^x\frac{1}{\sin t+\sin (\pi t)+2}dt$ have a supremum? (This question was inspired by What is the mean value of $|\sin x +\sin (\pi x)|$?.) My thoughts: $\frac{1}{\sin t+\sin (\...
Dan's user avatar
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Explicit formula for sequence where each term is the weighted average of adjacent terms

Consider the sequence $a_0, a_1, ..., a_{2n}$, where $a_0=0$, $a_{2n}=1$, and $a_i=wa_{i-1}+(1-w)a_{i+1}$ for all indexes $i\in [1, 2n-1]$. In other words, the first term of the sequence is 0, the ...
Random Person's user avatar
4 votes
2 answers
219 views

If the average sequence converges to zero, does a subsequence converge to zero?

Let $(a_n)_{n\in\mathbb{N}}$ be a positive sequence $0\leq a_n$, and suppose that the sequence of averages converges to zero: $$0\leq \left(\frac{1}{t}\sum_{i=0}^{t-1}a_n\right)_{t\geq 1}\to 0.$$ I ...
grouse's user avatar
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What is the intuition behind calculating the error of sum of squares and square of a shifted sum using moving averages?

Given $\mathbf{v} \in \mathbb{R}^d$ and $\beta \in \mathbb{R}$, then the following holds: $$ \begin{aligned} \sum_{i=1}^{d}v_i^2 -\frac{ \Big( (\sum_{i=1}^dv_i)-\beta \Big)^2 }{d} &= \beta(...
Saeed's user avatar
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-2 votes
1 answer
292 views

How to calculate the average stock price (level)

I know that we can calculate average stock price by using mean but this doesn't give a real picture of the portfolio specially if we have bought and sold the same stocks. How do we calculate average ...
Raymond's user avatar
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Design system debt calculation

Thanks in advance for your help. I am a designer working on a software platform. I am working on a calculation that captures the amount of redundancy/design debt in our design system on a component ...
SamCC's user avatar
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3 votes
1 answer
939 views

Average damage of critical hits

I am looking for a better understanding of the following formula. In games with Crit Rate and Crit Damage percentages (like Genshin impact or Honkai Star rail to quote a few) the expected average ...
SheeShee's user avatar
1 vote
0 answers
79 views

Replacing time-average for fast-varying function

Consider the second order differential equation as follows $$ \frac{d^2 f(t)}{dt^2} = \alpha(t) f(t), $$ where the variable $t$ is time. A result very often used in Physics is that, if $\alpha(t)$ ...
grav.field's user avatar
1 vote
1 answer
59 views

Average cost of an item from container with different chances [closed]

i need to find average cost for every item in container. It costs 50 currency to open one. I have following chances for items: lvl-1: 58% ...
eagle.s's user avatar
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1 vote
1 answer
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How to add new rows into a matrix $X$ without change its center?

Assume that you have a matrix $X \in \Re^{m \times n}$ and you have centered it column wise. $$\mu_j = \frac{1}{m}\sum_{i=1}^{m} X_{i,j}$$ $$X = X - \mu_j$$ MATLAB code: ...
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