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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The median minimizes the sum of absolute deviations (the $ {\ell}_{1} $ norm)

Suppose we have a set $S$ of real numbers. Show that $$\sum_{s\in S}|s-x| $$ is minimal if $x$ is equal to the median. This is a sample exam question of one of the exams that I need to take and I don't know how to proceed.
hattenn
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Why does the median minimize $E(|X-c|)$?

Suppose $X$ is a real-valued random variable and let $P_X$ denote the distribution of $X$. Then $$ E(|X-c|) = \int_\mathbb{R} |x-c| dP_X(x). $$ The medians of $X$ are defined as any number $m \in \mathbb{R}$ such that $P(X \leq m) \geq \frac{1}{2}$…
Tim
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Mean vs. Median: When to Use?

I know the difference between the mean and the median. The mean of a set of numbers is the sum of all the numbers divided by the cardinality. The median of a set of numbers is the middle number, when the set is organized in ascending or descending…
Mandelbrot
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For what values does the geothmetic meandian converge?

The geothmetic meandian, $G_{MDN}$ is defined in this XKCD as $$F(x_1, x_2, ..., x_n) = \left(\frac{x_1 +x_2+\cdots+x_n}{n}, \sqrt[n]{x_1 x_2 \cdots x_n}, x_{\frac{n+1}{2}} \right)$$ $$G_{MDN}(x_1, x_2, \ldots, x_n) = F(F(F(\ldots F(x_1, x_2,…
Pro Q
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Is $50$th percentile equal to median?

Consider we have the $100$ distinct integers between $1$ and $100$ inclusive. The median and fiftyth percentile can be calculated as below. Ordering: $1,2,3 ..... ,98, 99, 100$ The median is $(50+51)/2$ The $50$th percentile is $51$ ($51$ is…
Cardinal
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Derivation of formula for finding median for grouped data

I know the formula of formula for finding median for grouped data that is $$\mathrm{Median} = L_m + \left [ \frac { \frac{n}{2} - F_{m-1} }{f_m} \right ] \times c$$ and I know what all the letters stand for. But can anyone provide a derivation of…
Shivam Patel
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Prove that the sample median is an unbiased estimator

My book says that sample median of a normal distribution is an unbiased estimator of its mean, by virtue of the symmetry of normal distribution. Please advice how can this be proved.
preeti
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Is There Something Called a Weighted Median?

I was given some data that represents the number of lines in a document as well as the line count per hour (which is the lines in the document divided by the number of hours that the document was worked on). Considering the following data: lines…
Bilbert
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Calculating the median in the St. Petersburg paradox

I am studying a recreational probability problem (which from the comments here I discovered it has a name and long history). One way to address the paradox created by the problem is to study the median value instead of the expected value. I want to…
Thanassis
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Does a median always exist for a random variable?

Does a median always exist for a random variable? Note that a median of a random variable $X$ is defined as a number $m \in \mathbb{R}$ such that $P(X \leq m) \geq \frac{1}{2}$ and $P(X \geq m) \geq \frac{1}{2}$. Thanks and regards!
Tim
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Distance between mean and median

I want to solve the following problem in T.Tao's random matrix theory book. Let $X$ be a random variable with finite second momment. A median $M(X)$ of $X$ saisfies $\mathbb{P}(X>M(X)),\mathbb{P}(X
Lucien
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A continuous function defined on an interval can have a mean value. What about a median?

A function can have an average value $$\frac{1}{b-a}\int_{a}^{b} f(x)dx$$ Can a continuous function have a median? How would that be computed?
user2321
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How to win Matt Parker's jackpot - finding the median of the following distribution

In a recent video the legendary Matt Parker claimed he kept flipping a two-sided (fair) coin untill he scored a sequence of ten consecutive 'switch flips', i.e. letting $T$ denote a tail and $H$ a head, then a sequence of ten switch flips is defined…
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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 any median $m_X$ it holds that $$P[|X-m_X|\geq t]…
Lnct3
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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. My thoughts so far: Let's call the distribution…
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