For questions about the numerical value separating the higher half of a data sample, a population, or a probability distribution, from the lower half.
Questions tagged [median]
542 questions
188
votes
12 answers
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
- 2,287
67
votes
7 answers
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
- 49,162
32
votes
2 answers
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
- 691
22
votes
1 answer
Is $50$th percentile equal to median?
Consider we have the $100$ distinct integers between $1$ and $100$ inclusive. The median and fiftieth percentile can be calculated as follows:
Data set: $1,2,3 ..... ,98, 99, 100$
The median is $(50+51)/2 = 50.5$
The $50$th percentile is $51$…
Cardinal
- 890
21
votes
3 answers
For what values does the geothmetic meandian converge?
The geothmetic meandian, $G_{MDN}$ is defined in this XKCD as
$$F(x_1, x_2, ..., x_n) = \left(\frac{x_1 +x_2+\cdots+x_n}{n}, \sqrt[n]{x_1 x_2 \cdots x_n}, x_{\frac{n+1}{2}} \right)$$
$$G_{MDN}(x_1, x_2, \ldots, x_n) = F(F(F(\ldots F(x_1, x_2,…
Pro Q
- 943
21
votes
3 answers
Birthday-coverage problem
I heard an interesting question recently:
What is the minimum number of people required to make it more likely than not that all 365 possible birthdays are covered?
Monte Carlo simulation suggests 2287 ($\pm 1$, I think). More generally, with $p$…
Isaac
- 37,057
20
votes
2 answers
Derivation of formula for finding median for grouped data
I know the formula of formula for finding median for grouped data that is $$\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
- 4,139
16
votes
2 answers
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
- 1,543
16
votes
0 answers
Why Dottie$=\sqrt{1-\left(2\text I^{-1}_\frac12(\frac 12,\frac 32)-1\right)^2}=-\frac{2\sqrt3\text{InvT}(\frac14,3)}{\text{InvT}^2(\frac14,3)+3}$?
$\def\InvT{\operatorname{InvT}}$Introduction:
For some background information on the Dottie number D, see the great posts at:
What is the solution of $\cos(x)=x$?
Some definitions:
inverse beta regularized function $\text I^{-1}_x(a,b)$, beta…
Тyma Gaidash
- 13,576
14
votes
2 answers
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
- 1,447
- 1
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13
votes
1 answer
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
- 49,162
12
votes
3 answers
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
- 123
12
votes
5 answers
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
- 3,225
11
votes
5 answers
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
- 749
- 1
- 7
- 15
11
votes
3 answers
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…
Bib-lost
- 4,050
- 16
- 44