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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8
votes
4answers
498 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?
7
votes
1answer
725 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
1answer
500 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 $\sqrt{\frac{2}{\pi}}$.
4
votes
2answers
207 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
2answers
3k 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.
3
votes
1answer
1k views

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

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 - ...
3
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 ...
3
votes
2answers
198 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 : ...
3
votes
1answer
48 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 ...
2
votes
4answers
2k 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
2answers
17 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

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 ...
2
votes
2answers
89 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 ...
2
votes
2answers
85 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 = ...
2
votes
2answers
1k 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
49 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
2answers
79 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
56 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
1answer
32 views

Does the median tend to the lower limit of the median class?

The median for a continous distribution is given by $M= L +\dfrac{(N/2-C)\cdot I}{f}$ where $M$ is median, $L$ is lower limit of median class, $N$ is the total frequency, $C$ is the cumulative ...
2
votes
1answer
35 views

Bounds on median of random walks

If $k$ random $n$-step $\pm 1$ walks start at 0, and the $i$th walk ends at position $X_i$, how big is $\text{median}_i \, |X_i|$? Is there a bound along the lines of $\text{P}(\text{median}_i \, ...
2
votes
1answer
38 views

Ordering of positive integers

Three positive integers: m. n and q, their average is 10, median is m+6. what is the greatest possible value of Q ? I understand how to find the solution which I am outlying below. My question is ...
2
votes
0answers
45 views

What are the properties of median-unbiased estimators?

On Wikipedia it says that " A median-unbiased estimator minimizes the risk with respect to the absolute-deviation loss function, as observed by Laplace." How to prove this? Note that I asked on Cross ...
2
votes
0answers
43 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 ...
1
vote
3answers
96 views

Median $\neq$ Expectation

Do you have an example of real random variable such that its median is very different from its expectation? I mean it is obvious that they are differents. Thank you
1
vote
3answers
6k 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?
1
vote
2answers
95 views

Formulation and computation of “the” unique median of an even-sized list

Consider an even-sized set of numbers $X = \{x_k\}$, such as $X = \{1, 2, 7, 10\}$. The median $m$ is defined as: $$m = \mathrm{arg \min_x} \sum_k \lvert x_k - x\rvert^1$$ Any $m \in [2, 7]$ is a ...
1
vote
1answer
218 views

Calculate Statistics (Check if the answers are correct)

Calculate the statistics below using the following data on a sample of the variable $X$: Data ($X$ sample) = $\{9, -1, 7, 0, -2, 5, 4, 9, 5 \}$ Using the sample data, calculate: Mean; Median; ...
1
vote
1answer
186 views

How to compute ideal investment leverage ratio based upon volatility? [closed]

I want to compute the ideal investment leverage ratio based upon an investment's volatility. For example, if I had an investment that with 50% likelihood quadruples your investment on a given day ...
1
vote
2answers
133 views

Find Weight for minimum Manhattan Distance

Let's say, I have three points $(1, 4)$, $(4, 3)$ and $(5, 2)$. I need to find weight $w_1$ and $w_2$ so that the point $(1, 4)$ be the centroid of the points in ...
1
vote
2answers
40 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 ...
1
vote
1answer
223 views

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 ...
1
vote
1answer
819 views

Algorithm to separate an array of numbers into “low”, “medium” and “high” ranges

this is my first post on Math-Exchange. I'm a programmer, and I work with object-oriented algorithms, but when things gets too "algebraic" I get stuck. So I thought I'd ask over here. I need to ...
1
vote
2answers
38 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 ...
1
vote
1answer
51 views

How to prove the property of the Lemoine point of a triangle?

From Wolfram MathWorld, I know there is a Lemoine point of triangle, also called symmedian point, the sum of squared distances of this point to all the three sides is algebraically minimum. How to ...
1
vote
1answer
425 views

Median of Medians in 2D Array/Matrix

This is a bit of a mathematics problem, and a MATLAB problem. In MATLAB, if I call median(M), where M is an ...
1
vote
2answers
107 views

What is the median in this gre question?

This is gre preparation question of data interpretation, Distribution of test score among students(Score range -> total % of students) 0-65 -> 16 65-69 -> 37 70-79 -> 25 80-89 -> 14 ...
1
vote
1answer
47 views

Median based on number of entries instead of values

I’m writing a computer program that provides some useful statistical information about files. Calculating the mean is trivial, and the mode at least has a simple definition, but the median is proving ...
1
vote
1answer
2k 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 ...
1
vote
2answers
522 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 ...
1
vote
0answers
18 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 ...
1
vote
0answers
66 views

A point minimizing total great circle distance to a given set of points on a hemisphere

If you have a set of points on a hemisphere, how do you find a point on that hemisphere that has the minimum total great circle distance to the points in the set.
1
vote
1answer
927 views

What is weighted median? [duplicate]

I heard median of any number of data. Suppose we have a data [13,23,23,23,54] The median of this data is is 23. How we calculate the weighted median of same data (as mentioned above). ** ...
1
vote
2answers
95 views

Median of i.i.d. uniform random variables on the interval $[-1/2, 1/2]$

Let $X_1, X_2, \ldots , X_{999}$ be independent and identically distributed random variables on the interval $[-1/2, 1/2]$. Let $X_{500}$ be the empirical median; that is, $X_{500} = X_k$ for some ...
1
vote
0answers
28 views

Error propagation of the median absolute deviation.

I wonder if there is a way to estimate error propagation of the mad (=median absolute deviation) for the difference between two median distributions, similar to how you estimate error propagation for ...
1
vote
2answers
59 views

Probability Question Random Variable Median

I have not idea where to start with this question can someone point me in the right direction...
1
vote
1answer
369 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
69 views

Means and Median of Probability

The question is: pennies come in rolls of 50. Out of the 200 rolls a bank is storing, 12 have all Canadian pennies: the remaining 188 rolls contain at least one American penny. In the 200 rolls, about ...
1
vote
1answer
116 views

Prove That The Composite Median Of Two Groups Lies Between Separate Medians of The Groups

Let $M_1$ be the median of $n_1$ observations and $M_2$ be the median of $n_2$ observations. Show that $M$, the median of$(n_1+n_2)$ observations combined, lies within $M_1$ and $M_2$.
0
votes
2answers
62 views

Why mean=median=mode of a bell curve?

I was reading this link - source - http://www.mathsisfun.com/data/standard-normal-distribution.html Here they say that for the bell curve - the mean = mode = median are identical. How mean i.e. ...
0
votes
1answer
76 views

Analysis of IQ scores given mean, median, sd, quartiles

The statistics below provides a summary of IQ scores of 100 children Mean: 100 Median: 102 Standard Deviation: 10 First Quartile: 84 Third Quartile: 110 About 50 of the children in this sample have ...
0
votes
1answer
101 views

Fractional part of Median always .5 or .0

If we find the mid value of two integer number,it's decimal part would always contain .5 or .0 exactly For Example: (5+10)/2=7 .5 (6+2)/2=4 .0 But,in some coding challenge they asked to calculate ...