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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18 views

Is the median of a discrete distribution with an infinite population a computable number?

For that matter, I ask for any quantile. I know that taking larger and larger sample sizes can get me closer to the true median, but does this satisfy the definition of Computable number?
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25 views

General Expectation Property for Gaussian

It is a well known property that $E[|X - p|]$ is minimized for $p$ such that $P(X \leq p) = \frac{1}{2}$ (i.e. $p$ is the median value for the random variable $X$). Now suppose $\mathbf{X} \sim \...
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35 views

Which of the statements are correct

I was trying to answer the question below. Given the sets S1 = {2,3,5,8,9} and S2 = {3,4,5,6,7} which of the following statements are correct? a. Median value and mean are 5 for both sets b. For S2 ...
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23 views

Median of a difference of random variables.

Let $X,Y$ two real random variables and let $Med(X)$ the median of $X$ and $Med(Y)$ the median of $Y$. We have that $\mathbb{P}(X\geq Med(X))\geq 1/2$ and $\mathbb{P}(Y\geq Med(Y))\geq 1/2$. My ...
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23 views

Convergence almost surely implies convergence of the medians.

Let $\{X_{n}\}_{n}$ be a sequence of real random variables and let $X$ be a random variable such that $\lim_{n}X_{n}(\omega) = X(\omega)$ for all $\omega\in\Omega$. It implies that $X_{n}$ converges ...
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1answer
16 views

Convergence of Random Variables and Median.

Let $\{X_{n}\}_{n}$ be a sequence of real random variables and let $X$ be a real random variable such that $\lim_{n} X_{n}(\omega) = X(\omega)$. Assume that $X_{n}(\omega)$ is a monotone decreasing ...
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15 views

Range of the median for a $Bin(n,p)$ distribution

I came across a question which went like this: Show that the median($m$) of any $Bin(n,p)$ distribution is such that $[np]\leq m \leq [np]+1$. I tried to prove this by the very basic notion of a ...
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14 views

Mean/median/etc of a zero-mean random variable with exponent

Suppose $\epsilon \sim N(0,\sigma^2)$, $a\in(0,1)$ and $x>0$ are constant. Is there any way of estimaing the following: $$(\frac{x+\epsilon}{x})^a$$ Although $\mathbb{E}[\frac{x+\epsilon}{x}]=1$, I ...
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1answer
50 views

Probability - median and mean

I'm having a difficult time with this one. Would be happy for some help. I'm kind of gets why its true but can't manage to prove it. Let $X$ be a random variable with median $\Bbb MX$ such that ...
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1answer
20 views

weighted median, but manually typed weights, not frequencies

Since theres some contoversy about the definition of the weighted median, I wonder if my doing is even possible: I have a large 2d matrix ...
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34 views

How to compute median without storing all the values?

I have a source of data that continuously gives me measurement results. At some point I compute the mean. Luckily to this end, I don't need to store all the values, I only store the sum and the number ...
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9 views

Change on individual medians affect which of the following measures

The median of 21 distinct median was increased by 1, which of the following measures is not effected by this change? (A)Mean. (B)Mean deviation about median. (C)Mid-range. (D)Standard deviation. The ...
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1answer
22 views

Infimum of the set of medians.

Let $\{F_{n}\}_{n}$ be a sequence of cumulative distribution functions such that converge to $F$, in the sense that $F_{n}(x)\rightarrow F(x)$ for all $x\in\mathbb{R}$. We define the function infimum ...
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1answer
39 views

Confidence Interval for the Median of Any Continuous Distribution

This is an example 4.4.7 from "Introduction to Mathematical Statistics" by Hogg, McKean, and Craig. Let $X$ be a random variable of the continuous type with cdf $F_{X}(x)$ and let $\xi_{1/2}$...
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1answer
25 views

Calculating the median from a sample

Say I have a sequence $a_1\leq a_2\leq\dots\leq a_n$ of $n$ numbers. Say I pick a subsequence of $k$ samples from this sequence. Can I approximate the median of the original sequence from the sample? ...
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12 views

How to calculate the sample variance if the data set is changed

I have a $15$ samples data-set in which $$\bar x= 15, s^2 = 3.2, 𝑄_0 = 11, 𝑄_1 = 14.0, 𝑄_2 = 𝑄_4 = 17$$ However, the exact number of each element is unknown. Now, I have added two new sample in ...
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Generating log normal distribution using median and 95th percentile

I have median and 95th percentile value of certain data which might follow log normal distribution, but I don't have that "data" ,I only have these two aforementioned aspects of data. The ...
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31 views

Limit of the Median of Random Variables

Let $X_{1},X_{2},\cdots$ be real random variables identically distributed. We consider the sequence $m_{n} := Med(X_{n})$, where $Med(\cdot)$ denotes the Median of a random variable. My question is ...
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2answers
45 views

Solve for the base of an isosceles triangle

The area of an isosceles triangle is $S$ and the angle between the medians to the legs, facing the base, is $\alpha$. Find the base of the triangle. Let $CH$ be the third median of the triangle. The ...
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31 views

Median instead of mean distance of an underdetermined system of linear equations

I am given a number $N$ of 2D point pairs $(x_i,y_i)$ with corresponding values $(v_i,w_i)$. My task is now to find a linear matrix s.t. $$ \begin{pmatrix} m_{x,1} & m_{y,1} & b_1 \\ m_{x,2} &...
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93 views

Find the ratio $\frac{AF}{FC}$

In the figure below, $AD$ is the median on $BC$. The point $E$ divides $A$ and $D$ in the ratio $1:2$. $BE$ produced meets $AC$ at $F$. Find the value of $AF:FC$ My try: I joined $E,C$.Let area of $\...
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1answer
17 views

Do CEVIANS in Triangles divides the emerging angle in the same ratio as they do to the opposite side .

Ok I know that suppose in a triangle ABC , there's a cevian AD intersecting BC at D and dividing it in the ratio 4:5 so which implies that it divides Area of Triangle also in ratio 4:5 by that cevian. ...
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1answer
71 views

Show that $|\operatorname{median}(X) - \operatorname{mean}(x)| \leq \sigma_X$ [duplicate]

Let $m$ denote median and $\bar{x}$ denote mean and $\sigma$ denote the standard deviation, I want to show that $|m - \bar{x}| \leq \sigma$. Since the LHS and RHS are both positive, we can prove $(m - ...
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20 views

Mean vs Median for Customer Valuation

This question is just as much of a math one as it is a valuation/business one, but here it goes: Say I have a heavily right skewed distribution (cumulative order amounts from customer over their 'life'...
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16 views

What seems more useful for this particular problem, Mean or Median and Why?

You are given an array of positive integers. You are told to make all the numbers equal by doing this operation, i.e. to increase/decrease the value of an array element. The cost of the operation will ...
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1answer
69 views

Find the area of a triangle $\triangle FGH$ [closed]

In triangle $\triangle FGH$, $GM$ is a median that lies on $4x-y=27$; height $HA$ lies on $x-y+3=0$; $F$ is $(4,5)$. Find the area of the triangle $\triangle FGH$. My attempt: F is not on any of the ...
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3answers
59 views

What is mean and median of $X$ if $P(X \geq \frac{1}{2}+x)=P(X \leq \frac{1}{2}-x)$?

Let $X$ be a random variable such that $E[X] < \infty$ and $P(X \geq \frac{1}{2}+x)=P(X \leq \frac{1}{2}-x), \forall x \in R$ then find $E(X)$ and median$(X)$. My attempt: $\int_{-\infty}^{\frac{...
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1answer
74 views

Variance of a sample median

Suppose $X_1,\ldots,X_n\sim\text{i.i.d.}\operatorname N(\mu,\sigma^2).$ I think I've only ever seen one way to prove that the sample mean of $X_1,\ldots,X_n$ has a smaller variance than does the ...
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1answer
107 views

median bisector?

The point $O$ is the incenter of the triangle $ABC$, $N$ is the centroid. Then it is possible to construct the point $M$ that can be loosely defined as the intersection point of the "median ...
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1answer
66 views

Bayes estimate under absolute loss function [closed]

Consider a Bayesian model where the prior distribution is $\lambda(\theta) = \frac{2}{(1+\theta)^3}$, $\theta > 0$ and $X\sim U(0,\theta).$ Find the Bayes estimate $\delta(x)$ if the loss function ...
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1answer
28 views

k-mean clustering minimize L1 distance

k-mean clustering minimize L1 norm In k-mean clustering, if I want to minimize the L1 distance from any point to cluster center, the error function and derivative is shown above. However, according to ...
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1answer
41 views

Extension of median minimizing the sum of absolute deviations (the $L_1$ norm)

This is an extension of the question asked in The Median Minimizes the Sum of Absolute Deviations (The $ {L}_{1} $ Norm) . Except with the extra constraint that $x \in S$. The solutions provided there ...
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1answer
50 views

Algorithm to compute “median”

Let $x_1,\dots,x_m\in\mathbb{R}^d$ be a finite set of points. I define the $d$-dimensional "median" $y\in\mathbb{R}^d$ to be the point minimizing the sum of distances to $x_j$, $$ y = \arg\...
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1answer
25 views

Median of a continuous random variable

Consider the cdf $F(x)=1-e^{-x}-xe^{-x}, 0\leq x <\infty$, zero elsewhere. Find the median of this distribution, The given CDF is a complicated and I am finding it difficult to find x for which P(X&...
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37 views

median of a discrete random variable

Find the median of the following distribution $p(x)=\frac{4!}{x!(4-x)!} {(\frac{1}{4})}^x {(\frac{3}{4})}^{4-x}$ , x=0,1,2,3,4, zero elsewhere. In the question, the median is defined as the value of x ...
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1answer
34 views

The method of moments estimator of median

In some textbook, it is written that the method of moment estimator can be applied even in estimating median. However, I can't come up with the idea. In many cases the median is estimated by $x_{n/2}$...
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97 views

Triangles and medians

Let G be the ABC barycenter. A line intersects the medians AD, BE and CF in X, Y and Z, respectively. Prove that $$\frac{XD}{XG}+\frac{YE}{YG}+\frac{ZF}{ZG}=3$$ By areas relations I found that $\frac{...
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26 views

Median (and consequently the mean) of an evenly-spaced list [closed]

Why is it the case you can find the median (and consequently the mean) of an evenly-spaced list by taking the mean of opposite terms? Where opposite refers to opposite positions (e.g. first and last). ...
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1answer
45 views

Find the density of the median of $2n$ i.i.d $U([0,1])$ random variables

As a part of a probability problem I found the PDF and CDF of the $i^{th}$ order statistic in a sample. When told that $$X_1,...,X_n{\sim}^{i.i.d}F$$ where F is countinuous, so I got to the conclusion ...
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1answer
38 views

Convention of finding Quartiles

Find the quartile deviation for the data $$ \begin{array}{|c|c|c|c|} \hline x& 2 & 3 & 4&5&6 \\ \hline f& 3 & 4 & 8&4&1\\ \hline\end{array} $$ My Attempt $$ \...
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1answer
115 views

Area of a triangle as a function of its bisectors.

Area of a triangle can be easily calculated using only its 3 medians is it also possible to find its Area as a function of its 3 bisectors? Lots of people tried to find the solution and oddly ...
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1answer
26 views

Find median of a combination of tables.

Values in List $P$ are distinct from values in List $Q$. List $R$ consists of all the values in list $P$ and all the values in list $Q$. Find the median of list $R$. My logic: Here the total number ...
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1answer
69 views

Mean deviation gives better results when deviations are taken from the median, why?

The mean deviation gives better resuls when deviations are taken from the median instead from the mean, because the sum of the deviations from the median is less than the sum of the deviations from ...
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1answer
37 views

Find global median from medians of subgroups

Supposte I have a list of numbers $\{x_1, \ldots, x_N\}$, not necessarily ordered, and I divide it in subsets $\{ \{x_1,\ldots,x_{d_1}\}, \{x_{d_1+1},\ldots,x_{d_2}\}, \ldots \}$, where $d_n$ is the ...
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27 views

How to derive formula of variance of sample median?

I wonder how to derive the formula of variance of median of sample. I searched it for a day but couldn't find, so I'm asking help here. You may just notice me the link of it. I saw the result form is ...
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47 views

The medians in a right triangle

$\triangle ABC$ is a right triangle $(\measuredangle ACB=90^\circ)$. The medians are $AA_1=4\sqrt{13}$ and $BB_1=2\sqrt{73}.$ Find the sides of $\triangle ABC$. By the Pythagorean theorem: $\triangle ...
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1answer
22 views

How to find the probability of the number of data

I have an exercise, i try to finish all of the question. I stack for the last one. Anyone can help me? I attach the question. I find the solution for (i) 2 (ii) 2.63 (iii) 0.055 And i need your help ...
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1answer
53 views

Median estimator; Theoretical statistics, Keener

I am reading the book Theoretical Statistics: Topics for a Core Course from Keener, 2010. In section 8.4 Medians and Percentiles includes: "Let $X_{1},...,X_{n}$ be random variables. These variables, ...
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20 views

Fitting lognormal distribution from D10, D50, and D90

I have a statistical sample where D10 = 8 (10% of population under the value 8), D50 = 11 (median), and D90 = 18 (90% of population under de value 18) Now, I need to fit the $\mu$ and the $\sigma^2$ ...
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1answer
31 views

Deriving median from minimum and range

Let's say you have an odd number of n distinct integer and you know the range of these integers. (E.g. your min integer is 10 and your max integer is 50, this implies that the range is 30) Is there a ...

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