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

learn more… | top users | synonyms

13
votes
6answers
7k views

The median minimizes the sum of absolute deviations

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 ...
10
votes
4answers
615 views

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?
9
votes
3answers
205 views

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 ...
7
votes
2answers
5k views

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.
7
votes
1answer
1k views

Median of the F-distribution

Is the median of the F-distribution with m and n degrees of freedom decreasing in n, for any m? From experiments it looks like it might be, but I have been unable to prove it.
5
votes
4answers
404 views

Median $\neq$ expectation

Do you have an example of real random variable such that its median is remarkably different from its expectation? I'd like an example where it is obvious that they are different.
5
votes
1answer
846 views

Asymptotic correlation between sample mean and sample median

Suppose $X_1,X_2,\cdots$ are i.i.d. $N(\mu,1)$. Show that the asymptotic correlation between sample mean and sample median (after suitably centering and renormalization) is $\sqrt{\frac{2}{\pi}}$.
5
votes
1answer
201 views

Why hasn't the median function made the mean (effectively) obsolete?

I've learned in my AP statistics class that means can really only be used on nearly-normal distributions, but I know that many studies, experiments, etc. don't always give that perfect normal curve ...
5
votes
0answers
198 views

How to show that this function respects the strict ordering of its input.

Suppose you have a vector $\pmb x=\{x_i\}_{i=1}^n$ where each entry is drawn from a continuous distribution and $n$ is even. Then, denote $i^*=\{1\leq j\neq i\leq n:|x_i-x_{j}|=\mbox{med}_j|x_i-x_j|\}...
4
votes
2answers
306 views

Why do Mean, Median, Mode, and Range present in school lessons?

I studied in East Europe and post Soviet mathematical education program have no Median, Mode, and Range terms. Mean (or average) on other hand was studied (with root mean square and sometimes with ...
4
votes
1answer
5k views

Why is the median the value that minimizes the sum of absolute deviations? [duplicate]

Apparently, the mean is the value that minimizes the sum of the squares of deviations, and this made sense to me because the sum of the squared differences can be represented as an equation: (a - x)^...
4
votes
2answers
1k views

Calculate average angle after crossing 360 degrees

For a piece of code I am writing to smooth out movements I need to calculate the average angle over the past 5 recorded angles given (used to give directionality to an object) This can be achieved ...
4
votes
2answers
117 views

The problem of congruent areas in a triangle.

A problem was posed in front of me and I couldn't solve it after multiple attempts-- Consider any triangle and 3 concurent cevians are drawn from each of its 3 points . Now the figure formed has 6 ...
4
votes
1answer
2k views

Median of medians algorithm

I am referring to the algorithm presented here used to find a good pivot: http://en.wikipedia.org/wiki/Selection_algorithm#Linear_general_selection_algorithm_-_Median_of_Medians_algorithm My ...
4
votes
2answers
52 views

Geometric Median Problem with a twist

Given two vectors $x, y$ in $\mathbb{R}^n$ and scalar $\alpha$, what is the value of $\alpha$ that minimizes $||\alpha x - y ||_1$? Give an algorithm to find the minimum. I've tried couple of ...
4
votes
2answers
44 views

Maximise difference between mean and median

I have a set of numbers and I want to maximise the difference between the mean and the median by removing a given number of elements. For instance I have the following set: ...
3
votes
2answers
2k views

Why add the number 1 to find the median number

Example 1: There are 45 numbers 45 plus 1 is 46, then divide by 2 and you get 23 So the median is the 23rd number in the sorted list. Example2: There are 66 numbers 66 plus 1 is 67, then divide ...
3
votes
2answers
44 views

Finding the median given the PDF $f(x) = cx^2$.

I'm new to stats and I facing problems in finding the median of a PDF. I have to find the median of this PDF $$ f(x) = \begin{cases} cx^2, & \text{if 0 $\le$ $x$ $\le$ 3} \\ 0, & \text{...
3
votes
1answer
77 views

Median of a multinomial variable

Let $k\in\mathbb N^+$ be a positive integer. Consider a set of i.i.d. random variables $X_1,X_2,\ldots, X_n$, each of which is distributed uniformly over $\{1,2,\ldots,2k+1\}$. For $i\in \{1,2,\...
3
votes
1answer
30 views

Median of Medians

Given a set A with median Am = 10 and set B with median Bm = 20 is it true that the median of the combined set C is $10 \le$ Cm$\le 20$ ? My first thought was that this wasn't true so I tried to find ...
3
votes
1answer
30 views

What is the third quartile when there are no data above the median?

I'm a programmer, so apologies if this is a bad question. I'm writing some code that needs to detect outliers. I am currently calculating first and third quartiles. In a sample set of data, I have ...
3
votes
1answer
79 views

How can I draw the skewed normal distribution curve

If its of real importance, I am trying to plot the data on gnuplot. I have the following of some experimental data, obtained by ...
3
votes
2answers
144 views

The minimum of the sum

I was trying to prove some theorem on my way. So the problem was: Theorem: Let $x_1, x_2, ..., x_n$ be the rising sorted sequence of numbers. The $ \sum_{k=1}^n |x_k-a|$ reaches it minimum if $...
3
votes
2answers
222 views

Functions minimized at the median of their arguments

I am doing research on problems of location of a public facility on a network which lead me to the following question. Is there an interesting way to characterize the class of functions $f : \mathbb{...
3
votes
1answer
184 views

Squence of order statistic converges to median?

Suppose that $(X_1,...,X_N)$ be $N$ iid samples from uniform distribution. Let $X_{(n)}$ be $n$-th order statistic. It sounds natural that $X_{N/2}$ converges to the median, $1/2$. (interpret $N/2$ as ...
3
votes
1answer
32 views

Median of waiting time for $k$-th ace from bridge cards

I can't figure out how to get median of a waiting time from the exercise 36 from W. Feller's book An Introduction to Probability Theory and Its Applications Vol.1 (bold in the quote): ...
3
votes
0answers
97 views

Generalizing the mode and mean like the quantile

The mode, median, and means of a series of number ($x_1,x_2,...,x_N$) can be roughly thought of as the points that minimize the $p$-norm of the sequence for $p\in \{0,1,2\}$. The median is $c=\min_c\...
2
votes
4answers
4k views

what's the name of the theorem:median of right-triangle hypotenuse is always half of it

This question is related to one of my previous questions. The answer to that question included a theorem: "The median on the hypotenuse of a right triangle equals one-half the hypotenuse". When I ...
2
votes
1answer
284 views

How to compute ideal investment leverage ratio to maximize median return? [closed]

If I had an investment that with 50% likelihood quadruples your investment on a given day and you lose it all also with 50% likelihood, what percent of your money should you invest each day to ...
2
votes
2answers
83 views

Proving that median of list $[x_1,x_2,…,x_n]$ minimises the sum $\sum_{i=1}^{i=n} |x_i-m|$ where $m$ is some number [duplicate]

The problem is in the title. Here is a detailed description: Let's say we have list $[x_i]_{i=1}^{i=n}$ where $x_i\in\Bbb{N}$. I want to pick such $m\in\Bbb{N}$ which minimises the sum $\sum_{i=1}^{i=...
2
votes
2answers
165 views

Prove that this segment bisects another

The circle touches the trapezoid $GFEC$ at the points $C$, $D$ and $E$. The point $A$ is the center of the circle. The rest of the information can be seen in the diagrams below. What we have to prove ...
2
votes
2answers
138 views

Median vs. Mean

Problem: Consider the following model: $y_i = \mu + \epsilon_i$, $i = 1,...,n$ Let the mean $\mu$ be estimated by minimizing the criterion $\sum|\mu - y_i|$ over $\mu$. Show that $m = $median($y_1,...
2
votes
2answers
60 views

Determine random variable such as Med[X]=a and E[X]=b.

Let $a<b$ (a,b real numbers). Determine a random variable $X$ such as $Med[X]=a$ and $E[X]=b$. Hmmm how does one find $E[X]$ and $Med[X]$ if we don't have the probability distribution? The only ...
2
votes
3answers
16k views

How to calculate the median of a continuous random variable

$X$ is a continuous random variable with probability density function $f(x)= \dfrac{2x}{15}$ where $1≤x≤4$. What is the median of X?
2
votes
1answer
32 views

Equivalent events in proof of Central Limit Theorem for Sample Median

Let $X_1,X_2,\ldots,X_n$ be i.i.d. random variables with cdf $F$ on $\mathbb{R}$ and $M_n$ be the sample median. The proof starts off with defining for any $a\in\mathbb{R}$, $S_n = \#\text{ of }X_i\...
2
votes
2answers
98 views

How can I calculate the median value?

$$f(a,b) = a^b$$ Where $0\le a \le1$ and $0\le b \le1$ and either $a\ne0$ or $b\ne0$ How can I calculate the median value of $f$ ? I can estimate it to be about 0.76536 by taking values along the ...
2
votes
2answers
3k views

Given a mean, median and sum how to find how many elements more than mean?

Given a mean, median and total sum, how can you calculate how many elements in the collection will be more than a mean value? Here is an example; I have a stack of poles. The total hight of all the ...
2
votes
2answers
152 views

Why can/should we use 5 instead of 10?

Problem: A pharmacy has a uniform annual demand for 200 bottles of a certain antibiotic. It costs \$10 per year for a storage place for one bottle, and $40 to place an order. How many times during ...
2
votes
2answers
649 views

Defining median for discrete distribution

In probability theory, a median of a probability distribution is a number $M$ such that the CDF of this distribution $F_\xi(x)$ satisfies $F_\xi(M)=\frac{1}{2} \tag1$ This works for continuous ...
2
votes
2answers
72 views

Proof of $\sigma^2\geq (\mu-m)^2$ without resorting to Jensen's or Chebychev's inequality.

I asked a group of undergrad students (engineering) to prove that $\sigma^2\geq (\mu-m)^2$, where $\sigma^2$, $\mu$ and $m$ are the variance, mean and median of a continuous random variable. For the ...
2
votes
1answer
42 views

Does three medians determine a triangle?

If given three medians, is there only one triangle that has these three medians? How do I prove that?
2
votes
2answers
27 views

Calculating new median if one of the observations from original calculation is removed

find new average if removing one element from current average Hey guys, found an old question that I would like to build on if possible, would appreciate your help. To piggyback on this old ...
2
votes
1answer
47 views

minimizing the sum of weighted absolute distance

Let $x_1, \ldots, x_n \in \mathbb{R}^d$ denote $n$ points in $d$-dimensional Euclidean space, and $w_1, \ldots, w_n \in \mathbb{R}_{\geq 0}$ any non-negative weights. $\arg\min_{\mu \in \mathbb{R}^d} ...
2
votes
1answer
46 views

Average weighted by inverse distance to median equal to median?

Problem Statement I have a set of $N$ ordered elements such that $x = \{x_1, x_2, ..., x_q, x_p, ..., x_N\}$ where $x_q \le x_m \le x_p$ and $x_m$ is the median of the set $x$. I define a particular ...
2
votes
4answers
297 views

Prove that the co-ordinates of the centroid of a triangle is an average of that of vertices

For a given triangle [ABC], how do I prove that the co-ordinates of the Centroid $O_{xy}$ (intersection of the medians) is the average of the individual vertices? $O_x = \left(\frac {A_x + B_x + C_x}...
2
votes
1answer
74 views

What is the median of the pdf $f(x) = e ^{-x}$?

I got its integral to get the CDF, which is equal to $-e^{-x}$, and I equate it to 0.5, but I'm stuck since I was getting an answer of ln(-0.5) which is a math error.
2
votes
2answers
74 views

Probability Question Random Variable Median

I have not idea where to start with this question can someone point me in the right direction...
2
votes
1answer
6k views

Compute number of comparisons in quicksort pivoting on median or third

With a few friends we read the Algorithm Design Manual from Skiena. One of his (chapter 4) exercises asks for the number of comparisons that the quicksort algorithm does (comparing an element to the ...
2
votes
2answers
64 views

Find factor of sample elements given the median

I have a sample $S(x)$ containing $n$ elements: $$S(x)=\{ s_1 x, s_2 x, \ldots, s_n x \},\qquad s_i \in \mathbb{R}, x\in \mathbb{R^{+}}$$ Every element in the sample is multiplied by $x$. Now ...
2
votes
0answers
31 views

Continuity of Median

There are three arbitrary random variables on $\mathbb{R}$: $X$, $Y$, $Z$. The median is defined as $\mathrm{Me}(X) = \mathrm{sup} \left\{t: F_X(t) \le \frac12\right\}$ (so it's hopefully always ...