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.
0
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1answer
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 ...
0
votes
0answers
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 ...
1
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0answers
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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0answers
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:
$$
\min_x \sum_{i=1}^n d(a_i, x) \leq \frac{1}{n-1} \sum_{i=1}^n \sum_{j \...
3
votes
1answer
47 views
Is there a way to find $\min\{m(|X-c|), \:c \in \mathbb{R}\}$?
Suppose X is a random variable, such that $F(t) := P(X < t) \in C(\mathbb{R})$. Is there a way to find $\min\{m(|X-c|), c \in \mathbb{R}\}$? Here $m$ stands for median.
I know the solutions for ...
3
votes
1answer
82 views
Mean concentration implies median concentration
Exercise 2.14 in Wainwright, "High-Dimensional Statistics", states that if $X$ is such that $$P[|X-\mathbb{E}[X]|\geq t] \leq c_1 e^{-c_2t^2},$$ for $c_1, c_2$ positive constants, $t\geq 0$, then for ...
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1answer
26 views
Sample-quartile
I don't know : Is there a sample such that the mean does not lie between the lower and upper quartile?
Is there a sample such that the median does not lie between the lower and upper quartile?
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0answers
28 views
How to show that the $\sum_{i=1}^{n}|A[i]-x|$ is minimal for $x=m$ with median $m$?
I am trying to solve this question, but have no idea how one can prove it:
Let $m$ be the median of the array $A$ with $n$ real numbers. Show that
$$\sum_{i=1}^{n}\bigg|A[i]-x\bigg|$$
is ...
0
votes
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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votes
2answers
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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0answers
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 ...
1
vote
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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0answers
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, ...
2
votes
0answers
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})$$
I couldn't find any ...
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votes
0answers
22 views
Find the density function of the median of a uniformly distributed random sample
I'm stuck on a question that asks me to find the density function of the median. I am not given anything aside from the information that $Y_1, Y_2,..., Y_{2n+1}$ is a random sample of uniformly ...
3
votes
1answer
37 views
Does $ \arg \min_{x_{2n}} \sum_{i = 1}^{N} \left( {s}_{i} - x_{2n} \right)^{2n} $, for $n>1$, have a name and/or application?
Given a list of $N$ real numbers $s_i$:
the median $x_0$ is
$$ \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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0answers
23 views
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 ...
1
vote
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 ...
3
votes
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
...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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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0answers
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 $F(m)=1/2$
2
votes
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 ...
2
votes
0answers
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{...
0
votes
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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0answers
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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0answers
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 ...
0
votes
1answer
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 ...
0
votes
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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votes
0answers
82 views
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 ...
1
vote
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\...
4
votes
2answers
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 $1$-...
7
votes
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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0answers
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 $\...
2
votes
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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0answers
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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votes
2answers
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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0answers
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
...
0
votes
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 ...
-2
votes
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 ...
1
vote
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 ...
0
votes
0answers
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 ...
3
votes
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 ...
1
vote
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 ...
0
votes
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 ...
