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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0
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1answer
16k views

Finding the median value on a probability density function

Quick question here that I cannot find in my textbook or online. I have a probability density function as follows: $\begin{cases} 0.04x & 0 \le x < 5 \\ 0.4 - 0.04x & 5 \le x < 10 \\ ...
0
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0answers
23 views

Is the intuition correct?

Let $X$ be a random variable such that $E|X|<\infty$ and $$P\left(X\ge\frac{1}{2}+x\right)=P\left(X\le\frac{1}{2}-x\right)\ \forall x\in \mathbb{R}.$$ Then $E(X)=\frac{1}{2}$ and ...
3
votes
1answer
19 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 ...
1
vote
1answer
634 views

Given x is an exponential random variable, find median & probability

For the median, I believe that I should integrate the function, ∫x0λe−λtdt=1−e−λx Then I need 1−e−λm=.5 for m, which is equivalent to e−λm=.5. m=ln(2)/λ =>m=ln(2)/.2
1
vote
1answer
2k views

How do I find percentiles of data sets (Even vs odd)?

Given the following data set with an even number of values: $100, 100, 105, 113, 129, 132, 146, 152, 176, 200$ The value representing the 30th percentile, using the formula n(p/100) where n = sample ...
0
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1answer
26 views

calculation of median of grouped data

While calculating the median of grouped data of total frequency $N$, in order to find the median class which value should be taken into consideration to match against cumulative frequency : $\frac N2$ ...
0
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1answer
22 views

Ratios of median/mean and standard deviation/IQR in a normal distribution

I have some queries on the following question For a normal distribution, find the ratios of: (a) $\frac {\mbox{median}}{\mbox{mean}}$ (b) $\frac {\mbox{standard deviation}}{\mbox{interquartile ...
3
votes
1answer
30 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 ...
0
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0answers
9 views

median of bivariate truncated lognormal

$(X,Y)$ are bivariate lognormal random variables with mean $(\mu_1,\mu_2)$ and $\Sigma$ as the variance-covairance matrix, truncated to be in $[a_1, a_2]\times[a_1,a_2]$ where $a_1>0$ and ...
0
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0answers
27 views

In biweight midvariance, why would the median absolute deviation (MAD) be multiplied by the 0.75 standard normal quantile?

The biweight midvariance $\zeta^2$ is a measure of scale that is more robust to non-normal distributions. It is defined as follows by [1] in three steps. For observations $X_i$, $i = 1,2,\cdots,n$, ...
0
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0answers
17 views

Is there a generally unbiased estimator for the quantiles of a distribution?

1) Is there a generally unbiased estimator for the quantiles of a distribution? If not - I would be glad for an explanation (proof?) of why not. 2) Also (if not), is there a specific family of ...
7
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5answers
4k 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 ...
1
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1answer
30 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} ...
1
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0answers
26 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 ...
5
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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 ...
0
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2answers
44 views

High school Math: Finding the Median

I have a problem trying to understand the following two questions: In question 1, the median is found at the 50% of the total group. However, this strategy doesn't work for question 2. Why?
0
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1answer
21 views

How to find the median element when each element has three scores?

I know how to find the median with a list of one element but what to do with a list like this one: \begin{array}{|l|cr} & math & info & gestion\\ \hline Nelim & 0 & 4 & 0\\ ...
0
votes
1answer
22 views

Should I use interpolation when finding median, and quartiles?

I am a S1 maths (Edexcel) AS student in the UK. My question: Say we have a stem-and-leaf diagram with 26 values. We want to find the lower quartile. To get the marks for our specification, we need to ...
1
vote
1answer
50 views

Prove the median of a uniform distributions is $\frac{1}{2}(a+b)$

Let X~U(a,b) with a and b in the real line, such that b>a with X's pdf given by $$f_X(x)=\frac{1}{b-a}\mathbb{1}(a<x< b)$$ Show the median of X's distribution is given by $$m=\frac{1}{2}(b+a)$$ ...
4
votes
2answers
43 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 ...
0
votes
1answer
45 views

Show that $\mathbb{E}~|\xi - \mathrm{med}(\xi)| \leq \mathbb{E}~|\xi - \mathbb{E}(\xi)| $ [duplicate]

Assuming $\mu$ is the median of random variable $\xi$, prove, that, $$\mathbb{E}~|\xi - \mu| \leq \mathbb{E}~|\xi - \mathbb{E}\xi| $$ The case when $\mu = \mathbb{E}\xi $ is trivial, but I have ...
3
votes
2answers
123 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 ...
0
votes
2answers
378 views

question on five number summary & quantile.

i know that in five number summary : 25% of a data set lies between Min & 1st quartile. 50% of a data set lies between Min & 2nd quartile, that is, Median. 75% of a data set lies between ...
2
votes
1answer
32 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 ...
1
vote
2answers
37 views

Distribution with two (or more) medians

Is there any example with a distribution with two or more medians? I was reading about the median on wikipedia: https://en.wikipedia.org/wiki/Median and here it says that there may be more than one ...
0
votes
1answer
106 views

calculate minimum of max value of set of numbers

Suppose to have a set of numbers: $$S:=\{ 1,2,3,9,10,56,58,60\}$$ How can i group this number like this: group 1:$\{1,2,3,9,10\}$ group 2:$\{56,58,60\}$ and then take "56" as minimum value of ...
0
votes
2answers
420 views

In a distribution that is skewed by a few extreme outliers, what would be the best choice for a measure of central tendency?

According to my Professor, the answer to this question is the MEDIAN. However, wouldn't MODE be a better measure of central tendency? The mode is almost never affected by extreme outliers..please help ...
1
vote
2answers
36 views

Need help proving this geometry problem.

My friend asked me one question yesterday.It is as follows. Let there be two triangles ABD and ACD.D is a point on base BC such that BD=CD(given).Also,clearly side AD is common.Now we know median ...
3
votes
1answer
31 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): ...
0
votes
1answer
88 views

How to find median from a probability distribution?

Having trouble on something that should be really, really easy. I need to find the median of the following probability distribution...but according to the website I linked below...I'm doing it ...
0
votes
0answers
19 views

Median of n normal distributions

I have the following problem: I would like to know the distribution of $\mathrm{Med}[X1,\ldots,Xn]$, where $X1,\ldots,Xn$ are standard normal distributions $N(0,1)$. I know that one should consider ...
1
vote
1answer
104 views

How to understand the Repeated Median Estimator (Siegel 1982)?

I guess I did not understand the formula of the repeated median estimator introduced by Siegel (1982): This is the formula: If I got the following time series for instance: ...
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.
2
votes
4answers
168 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 + ...
1
vote
1answer
32 views

Non-Whole Median Numbers in Real Data [duplicate]

According to this CDC report, the median number of reported sexual partners for females aged 15-44 is 3.2, and for males 5.1. Tables on pages 19 and 20 report these statistics for a variety of ...
0
votes
0answers
9 views

What is the behavior of the spatial median in high-dimensional spaces?

I am a photographer who is investigating a technique known as image stacking, in which multiple images of the same subject are combined to reduce noise (by CLT). Commonly used techniques are mean and ...
1
vote
2answers
2k views

A set having the same mean, median, mode, and range

Is it possible to have a set with the same mean, median, mode, and range? If not, how can the following question be solved: Set $H$ contains five positive integers such that the mean, median, ...
0
votes
0answers
14 views

what's the best way to find a central tendency and filter outliners for this given problem?

I'm faced with the following problem: I have a group of Nodes which belong in the same network. They are constantly talking with one another providing services to one another. Suppose nodeA asks for ...
2
votes
1answer
274 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 ...
0
votes
1answer
57 views

The validity of dividing an average cost by an average duration?

Let's say you wanted to calculate, for example, the average (i.e. mean) cost per day for a hotel to accommodate a tourist. If only aggregate cost data is available, is dividing the average cumulative ...
5
votes
4answers
331 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.
0
votes
2answers
41 views

Is it possible to find Median of set contains string value as well as Null value?

I am doing one calculation on Median and AFSIK about Median, it can be calculated by a "set of Finite set of Numbers". Now if a set has following value as: A = {3,4.56, 2, 11, JAMES, , Joy, 54, 9} ...
2
votes
1answer
63 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.
1
vote
2answers
149 views

Where should I use median instead of average?

Is there a general law, or rule of thumb, or rationale, when to use median and when average? Although I know the difference and how they are computed, when I try to translate to simple English I ...
0
votes
1answer
35 views

Uniqueness of Spatial Median

https://projecteuclid.org/download/pdf_1/euclid.aos/1176350511 Can You help me understand why there is less-than sign in the proof? ...
2
votes
2answers
2k 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 ...
1
vote
0answers
39 views

Runtime of recursive algorithm - Master's Theorem

I wrote a computer program that solves a question, and I am interested in knowing what is the runtime. My aim is for $O(\log n)$, and I'd like someone more experienced (and smarter?) to review my ...
2
votes
1answer
30 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 ...
0
votes
1answer
83 views

Is it possible for the median to be left to the mean in the Gamma distribution?

Many textbooks teach a rule of thumb stating that the mean is right of the median under right skew, and left of the median under left skew but this is not always true. According to wikipedia, there ...
0
votes
1answer
40 views

Design an algorithm - Median, computer science

I was wondering if this question belongs here or on StackOverflow, but it is a question of mathematical nature so this seems more appropriate. We have an array $S$ of $n$ different numbers ...