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Questions tagged [median]

For questions about the numerical value separating the higher half of a data sample, a population, or a probability distribution, from the lower half.

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39 views

Writing median rather than mean

I have some code to calculate the mean bias between two arrays. $$mean_{bias}=\frac{\sum_{i=1}^{n}a_i-b_i}{n}$$ and was hoping to represent the median instead: $$median_{bias}=Median(a-b)$$ Does ...
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13 views

Convex hull and geometric median

Let $X\subseteq\mathbb R^d$ ($d\geq 1$) and let $\Delta_{(X)}$ be the collection of all families $(w_x)_{x\in X}$ of nonnegative numbers, such that $\{x\in X:w_x\neq 0\}$ is finite, and that sum to ...
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29 views

LP where I need to take medians of multiple sets of variables

I have an LP optimization problem where I need to impose a linear constraint on the result of a median filter applied to an image. The issue is that the image already has had possible transformation ...
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14 views

Distance to median vs average intra-distances

Consider $n$ points in a vector space, denoted $(a_1, \dotsc, a_n)$. I am wondering if the following inequality holds true: $$ \min_x \sum_{i=1}^n d(a_i, x) \leq \frac{1}{n-1} \sum_{i=1}^n \sum_{j \...
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1answer
47 views

Is there a way to find $\min\{m(|X-c|), \:c \in \mathbb{R}\}$?

Suppose X is a random variable, such that $F(t) := P(X < t) \in C(\mathbb{R})$. Is there a way to find $\min\{m(|X-c|), c \in \mathbb{R}\}$? Here $m$ stands for median. I know the solutions for ...
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1answer
82 views

Mean concentration implies median concentration

Exercise 2.14 in Wainwright, "High-Dimensional Statistics", states that if $X$ is such that $$P[|X-\mathbb{E}[X]|\geq t] \leq c_1 e^{-c_2t^2},$$ for $c_1, c_2$ positive constants, $t\geq 0$, then for ...
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26 views

Sample-quartile

I don't know : Is there a sample such that the mean does not lie between the lower and upper quartile? Is there a sample such that the median does not lie between the lower and upper quartile?
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28 views

How to show that the $\sum_{i=1}^{n}|A[i]-x|$ is minimal for $x=m$ with median $m$?

I am trying to solve this question, but have no idea how one can prove it: Let $m$ be the median of the array $A$ with $n$ real numbers. Show that $$\sum_{i=1}^{n}\bigg|A[i]-x\bigg|$$ is ...
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1answer
35 views

Property of median of probability distributiom

Suppose that a random variable $\mathbb{X}$ has density $f$ and a unique median $m$ . Suppose that $b$ is any real number. Show that $\mathbb{E(|X − b|) = E(|X − m|) + 2 \int_ b^ m (b − x)f(x)dx}$ , ...
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437 views

Find a number having minimum sum of distances between a set of numbers

Lets say we have a set of numbers $\{ 5, 7, 1, 2, 5, 100 \}$. I want to find a number $x$ such that the sum of distances of every number from the set to $x$ is minimal. My first thought was that $x$ ...
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26 views

Best Average to Represent Quantitative Data?

For a final assignment, my two group members and I have created a biased survey called Should Public Schools Implement a Uniform Policy? We have about 2-3 ...
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3answers
58 views

What Can Median Tell Me About my Data?

For a final assignment for math, I was required to take the "Yes!" side of 'Should Uniforms Be Implemented at School?' My job was to create a biased survey that would make people agree with our side ...
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1answer
25 views

Inequality of difference of medians of two datasets.

For $\{x_k\}_{k=1}^m$ and $\{y_k\}_{k=1}^m\in\mathbb{R}$ with $x_k-y_k=d_k$ define $x=\lim_{\delta\rightarrow 0}\left(\arg\min \frac{1}{m}\sum_{k=1}^m|x_k-x|+\delta x^2\right)$ and $y=\lim_{\delta\...
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36 views

Median Calculation on Grouped Interval Data

I was studying for my Probability & Statistics exam. I've encountered with an example which is about finding the median of a given grouped data. Here is the table of the data; So from the table, ...
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23 views

Let $s^2=\frac{1}{n}\sum_1^n( X_i-\bar{X})^2$ and $\widetilde{X}$ denote the sample median. Is $s^2\leq \widetilde{X}(1-\widetilde{X})$ true

$X_i\in[0,1]$ Let $s^2=\frac{1}{n}\sum_1^n( X_i-\bar{X})^2$ and $\widetilde{X}$ denote the sample median. Is the following true? $$s^2\leq \widetilde{X}(1-\widetilde{X})$$ I couldn't find any ...
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22 views

Find the density function of the median of a uniformly distributed random sample

I'm stuck on a question that asks me to find the density function of the median. I am not given anything aside from the information that $Y_1, Y_2,..., Y_{2n+1}$ is a random sample of uniformly ...
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1answer
37 views

Does $ \arg \min_{x_{2n}} \sum_{i = 1}^{N} \left( {s}_{i} - x_{2n} \right)^{2n} $, for $n>1$, have a name and/or application?

Given a list of $N$ real numbers $s_i$: the median $x_0$ is $$ \arg \min_{x_0} \sum_{i = 1}^{N} \left| {s}_{i} - x_0 \right| $$ the mean $x_2$ is $$ \arg \min_{x_2} \sum_{i = 1}^{N} \left( {s}_{i}...
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Is Geometric Median Affine Equivariance?

Does the geometric median have the natural property - Affine Equivariance? That is the depth of the geometric median and its relative location to other data points do not change under affine ...
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3answers
184 views

mean vs median geometric interpretation?

I'm looking at this picture on wikipedia, comparing the median and mean of an arbitrary distribution. But what does it mean exactly? From the figure, it looks like the mean is the center of mass of ...
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1answer
115 views

Concentration inequality applied for robust estimation of the mean

Problem: (Page 19 in "Vershynin, Roman (2018). High-Dimensional Probability. Cambridge University Press. ISBN 9781108415194.") Suppose we want to estimate the mean µ of a random variable $X$ from a ...
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2answers
60 views

Sum of areas of triangles formed from medians.

The lengths of medians of the given triangle are used to form a second triangle ,then the medians of that one are used to form a third triangle, and so on. Find the sum of the areas of all the ...
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1answer
41 views

Understanding Median Value in the BoxPlot for Profit/Loss Made in Stock Market?

I have multiple stocks and I took the PROFIT/LOSS for the stocks and plotted it using a box plot (using python seaborn library). Interestingly the median is seen as 0. So does this means that most of ...
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1answer
337 views

Medians which lie in sequence of even length.

Given a sequence of numbers say [1,2,2,2,4,3,3] from this sequence how many sub-sequences in order can be formed in which the median will lie in the sub-sequences ...
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1answer
33 views

Finding lengths of sides on triangles with 3 given medians and lengths

The medians of $△TUV$ are $\overline{TX}, \overline{UY},$ and $\overline{VW}$. They meet at a single point $Z$. In other words, $Z$ is the centroid of $△TUV$. Suppose $\overline{UY}=33$, $\overline{TZ}...
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35 views

Show that the constant m that minimizez $E{|X-m|}$ is the median of X; that is $F(m)=1/2$ [duplicate]

My question is from papoulis first edition problem 5-3: Show that the constant m that minimizez $E{|X-m|}$ is the median of X; that is $F(m)=1/2$
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1answer
25 views

Use simulations in R to numerically estimate medians and modes for discrete uniform variables.

The question I have been given is: Let X be Discrete Uniform on 1, 2, . . . , n. Please note that your answers to the questions below can depend on whether n is even or odd. (a) Use simulations in R ...
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54 views

Minimum of a sum proof.

The problem I am working on is: Let $ Y = \{{y_1,y_2,y_3,...y_n\}}$ and $c=median(Y)$. Prove that: $$ \text{min}\left[\sum_{i=1}^n \lvert y_{i}-c\rvert\right]=c $$ My question is: Is the $\text{...
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1answer
206 views

How to calculate Percentile of given value based on other values and percentile

I have a problem related to a formula and I need help from all of you. It may use some statistics formula to solve the problem. Your help would be appreciated. =========== We have following values (...
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30 views

Sample medain of the vector l2 norm from uniform distribtion in n-dimensional unit hypersphere

I want to solve next task. Suppose we have random variable $X$ with uniform distribution in $n$ dimensional hypersphere of radius $1$. We have $N$ independent samples. I want to find median of the ...
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22 views

Median of summing up indetically distributed independent random variables?

I have a real random variable $v$ taking 3 values only: $a<b<c$ (but maybe it is not important, just to make it easy I assume this). Suppose the mean of $v$ is the real number $m$ and the median ...
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42 views

Perpendicular line from a median in a triangle

In the right triangle ABC where ∠B = 90◦ , BC : AB = 1 : 2. Construct the median BD and let point E be on BD such that CE ⊥ BD. Determine BE : ED (Figure below) I already solved this problem, but the ...
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1answer
39 views

What's the least possible price of the most expensive widget?

Mary sold 15 widgets last week. The median sale price of all widgets was $130$ and the average sale price of all widgets was $150. What's the least possible price of the most expensive widget? The ...
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1answer
42 views

Restricting the median of marginals has implications for the joint distribution?

Consider a bivariate cumulative distribution function (cdf) $F: \bar{\mathbb{R}}^2\rightarrow [0,1]$ where $\bar{\mathbb{R}}^2\equiv \mathbb{R}\cup\{-\infty, \infty\}\times \mathbb{R}\cup\{-\infty, \...
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Expected value of the sample median from a folded normal distribution

Suppose $X_1, \ldots, X_n \sim N(0,\sigma^2)$ are iid. Find the expected value of $M$, the median of $\vert X_1 \vert, \ldots \vert X_n \vert$ What I have so far: The density of $\vert X_i \vert$ is ...
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1answer
42 views

Effect of scaling a probability distribution on median of distribution.

Say a probability distribution has density function $f(x),x\in[a,b]$ and cumulative distribution function as $F(x)$. Consider the scaled distribution $g(x)=\frac1\theta f(\theta x),x\in[a\theta,b\...
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109 views

Is There an $ {L}_{1} $ Norm Equivalent to Ordinary Least Squares?

The ordinary least squares (OLS) method is very useful. It gives you the solution to the problem $$ \arg \min_{x} {\left\| A x - b \right\|}_{2}^{2} $$ Now, if the problem is the same, but the $1$-...
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1answer
111 views

Median is twice the mean

I am stuck at solving the following problem (at what I believe is the last step): Determine which distributions on the non-negative reals, if any, with mean $\mu$ are such that $2\mu$ is a median. ...
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25 views

Comparison of errors of sample mean and sample median [closed]

I have a question considering the sample median of a mathematical sample. We have $n$ independent and identically $N(\mu,\sigma^2)$-distributed random variables $X_i$, $i=1,\dots,n$. Suppose that $\...
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1answer
149 views

Proof that every random variable has a median, proof check?

I already checked the question Does a median always exist?. But I am not convinced that the limits exist. Say I want to show $m:=\inf\{x\in\mathbb{R}~|\mathbb{P}(X\leq x)\geq \frac{1}{2}\}$. Then for ...
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89 views

Asymptotic variance of the median

The Question: You are given that if $U_1,\dots,U_n \sim \text{ Uniform}[0,1]$ are iid, and $M_U$ is the sample median, then $$\Bbb E[M_U]=\frac 12 \qquad \qquad \text{Var}\,(M_U) = \frac{1}{4(n+2)}$$...
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78 views

Statistics - Mean , Median, Mode , Standard deviation.

A vendor states that over the past year, the mean monthly sales were 10,000 with a standard deviation of 2,000. The sales in most months, however, were below 8,000 and the most frequent monthly sales ...
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225 views

Lower/Upper quartile problems

I have lots of problems with the idea of the lower quartile. Firstly, consider this example: Find the lower quartile of 1,2,3,4,5,6,7,8,9,10 On using the formula (n+1)/4 we achieve (10+1)/4=2.75 ...
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1answer
18 views

Table with scores and frequency is given. How to find the median that exceeds the mean of the marks.

[My approach to this question is: I have taken out mean which is 69.3 (Please verify) and then I took out median (68.178) and then I took difference but the answer is not falling in given options. I ...
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2answers
211 views

Convergence in probability implies convergence of the median?

Suppose that for every $\varepsilon>0$ the sequence of random variables $X_{n}$ satisfies: \begin{align*} P(|X_{n}|>\varepsilon) \to 0 \end{align*} as $n\to \infty$. Now let $M_{n}$ denote the ...
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1answer
45 views

median and average. which to use and how

game rules. it is me vs 1 opponent. we both have to pick 6 players from a list of 24. we add up the score of the six players and whoever has the highest total score wins. assuming opponent is expert ...
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1answer
1k views

What is a set of seven numbers that have a mean median mode and range of 10?

I have to answer this question for my homework and I have spent hours trying to research it but I haven't come across an answer. I thought it may be 10 but then I realised that it would mean that the ...
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158 views

How to find the Median of grouped data

So i have this frequency table and i want to find the median . I divide the total frequency , 20, by 2, which gives me 10. I pick up the very first number larger than 10 in the cumulative frequency to ...
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1answer
173 views

“Physical” proof that the medians are concurrent

Has this simple proof appeared in literature? This is essentially the same proof as one where you call the vertices $\vec{a},\vec{b},\vec{c}$, and observe that $\frac{\vec{a}+\vec{b}+\vec{c}}{3}$ lies ...
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1answer
30 views

Median of Random Variable

Let X be an integrable random variable. Show that the function $a \mapsto E|X −a|$ attains its minimum at $a = \mathrm{Med}(X)$. I think this means as $a$ approaches median of X. This makes intuitive ...
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1answer
37 views

Conditional expectation and conditional median

Suppose $(x_i,y_i)$ are random draws from the population and $\alpha, \beta$ are scalars. Let $u_i$ be unobserved effects(residuals). Consider the model $y_i=\alpha+\beta x_i +u_i$. By Law of Iterated ...