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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How does Excel calculate quartiles?

I have the data set [3,7,9,10]. Excel and Desmos both return the value 8 for the median, which agrees with my understanding of ...
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Median for continuous function explanation

So, I was wondering what the continous version of the median is (not a homework problem, just curious)? Can someone provide me with an explenation of the continous median, and maybe show how to derive ...
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Decision to use median or average in a skewed or uniform distribution [closed]

Goal: To determine whether to use the median or average (or a weighted combination of the ...
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Clarify answer for asymtotic of median smoothing

Calculating algorithmic complexity for median smoothing in Time Series I came up with the question same as this. Quote it here: A time series with T observations is given. Median smoothing with width ...
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Skewed data: choosing the average or median

Goal: The data below shows (in order) the date, median, average, ...
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69 views

Why are mean, median, and mode called central tendency?

Is there any difference between the two terms "Central tendency" and "representative values"? Why are mean, median, and mode called central tendency or representative values?
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Probability that the median roll of 21 rolls is 6

Let's say I roll a fair 6-sided dice 21 times. Assume 𝑀 tracks the median of the 21 numbers rolled. What is 𝑃(𝑀 = 6)? So far, I believe that this implies at least 11 of the rolls have to be 6, and ...
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A better definition of "median"?

Referring to the definition in Wikipedia, paraphrased: Formally, a median of a population is any value such that at most half of the population is less than the proposed median and at most half is ...
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How to get an "average" of a distribution with no average

Modelling something where I need some sort of average time, I reached Pareto Distribution, and the PDF is $$ R(t)=\begin{cases}\sqrt{\frac{1}{2\pi t^{3}}}, & \text{for } t > \frac{2}{\pi}\\ 0 &...
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Closed form expression for maximizer of $\mathbb{E}[\min\{X, c\}]$?

Let $X$ be a nonnegative random variable such that $\mathbb{E}[X] < \infty$. I am wondering if it possible to characterize the maximizer, $c > 0$ of the following function: $$ f(c) = \mathbb{E}\...
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minimizing the distance between values

Find all values of $a$ that minimize the expression $|a- x_1| + |a-x_2|+\cdots + |a-x_n|$, where $x_1\leq x_2\leq \cdots \leq x_n$. I know the minimum is achieved when $a=x_{\lfloor n/2\rfloor + 1}$ ...
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Is it true that the sum of squares of perpendicular medians in a triangle is equal to the square of the third median?

I am aware of the fact that two medians are perpendicular in a triangle if and only if the Sum of the squares of the lengths of the sides to which these medians are drawn is $5$ times greater than ...
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3 votes
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Solve a triangle given one angle of the median, and a side and an angle of its external triangle

This is a modified version of this question, and as such I'm using similar wording and visuals. Given: $α_2$, one of the two angles which the vertex $A$ is split by the median $m$; $\overline{AD}$, ...
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Median of function Z (Random Variable of Present Value of Benefit) of Multiple Decrement Model

I have a problem in determining median in life insurance context. Here we have $Z$ as random variable of present value of benefit of insurance like this: It will give benefits of $1000$ (at the moment ...
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1 answer
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Minimizing $E(|Y-f(X)|)$.

Assume that $X$ and $Y$ are random varaibles. The theory of conditional expectation says that the function f that minimizes $E[(Y-f(X))^2]$ is $f(x)=E[Y|X=x]$. My book says that the function that ...
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using linear programming to find median

I'm trying to figure out how to transform a median solving problem for a list of number x to linear programming and with standard form of $$\text{min }c^Tx \text{ s.t: } Ax\le b \text{ and }x \geq 0$$ ...
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How to calculate median, variance and correlation on two-dimensional random variable. [closed]

Two random variables, X and Y, have the joint density function: $$f(x, y) = \begin{cases} 2 & 0 < x \le y < 1 \\ 0 & ioc\end{cases}$$ Calculate the correlation coefficient between X and ...
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How close is the median of a uniformly chosen subsample to the median of the full set?

Take $N$ a set of n numbers, sample s numbers from $N$ uniformly and with replacement giving us the set $S$. What is the relationship between the median of $N$ and the median of $S$ ? I want a result ...
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1 vote
1 answer
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Why would you use a different calculation for finding median from a histogram rather than from a regular set of data?

I've always learned that you should use $\frac{n+1}{2}$ to find the median of a regular set of data. For example, the median of $$1, 2, 3, 4, 5, 6, 7$$ would be $4$ because of $\frac{n+1}{2}$. But ...
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Median for grouped/ungrouped data

If I am given a series of, let's say, 50 observations, of a continuous data, for example, the height of some group of people: 165.23 cm, 134.28 cm, and so on, and I want to find the box plot, I need ...
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2 votes
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54 views

Name and interpretation of the median of $xf(x).$

Is there a name for $\operatorname{median}(xf(x))$ where $f(x)$ is a probability density function on real numbers (or positive real numbers)? If yes, what are the properties of this number? This ...
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Mode and median of a discrete random variable - Exercise

I've the random variable $X=(0,25,25,50,100)$. Is it correct to say that: $$ mode = 25 $$ and $$ median = 25 $$ because $P(X \leq 25) = 2/5 = 0.4$ ?
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1 vote
1 answer
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Why is finding the median and quartiles different on a cumulative frequency graph? (Or is it?)

When finding the median we look for the (n+1)/2 value in the ordered data set. For the lower quartile it's the (n+1)/4 value and the upper quartile is 3/4(n+1). However, I've been helping my ...
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1 answer
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How to find the median of a p.d.f with unknown integral limits

I have been given the p.d.f: $$ f(x) = \frac{1}{b}\exp \left(\frac{-(x-a)}{b}\right)\exp\left(-\exp\left(\frac{-(x-a)}{b}\right)\right) $$ where $-∞ < a < ∞$ and $b > 0$. And I need to show ...
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Why do we get the median of an arithmetic series by getting the average?

I'm wondering why we get the median of an arithmetic series by getting the average. Example: $2 , 4 , 6 , 8$ The median is $5$⇒ $\frac{4+6}2$. Meanwhile, the average is also $5$⇒ $\frac{2+4+6+8}4$ A ...
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What's the median of $f(x) = 4xe^{-2x}$?

I'm trying to find the median of $f(x) = 4xe^{-2x}$. So far, I've tried solving for $q_{50}$ by plugging it into an integral and setting it equal to 0.5 like so: $\int_{0}^{q_{50}} 4xe^{-2x} dx = 0.5$....
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1 vote
1 answer
61 views

Is it possible to make a formula for median that works with both even or odd amounts of values?

Everyone here probably knows the median formula. It's the middle number of the set if the number of values in the set is odd, or the average of the two middle numbers in the set if the number of ...
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Absolute geometry and concurence of medians

I'm trying with a friend to find ....the simplest proof that in absolute(neutral) geometry the medians concur. In Bachmann's book (in German-Aufbau der Geoemtrie aus Spiegelungsbegriff) there is a ...
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3 votes
1 answer
215 views

how to get geometric median in 1D by solving an optimization problem?

I have a set of numbers $\{x_1,x_2,\cdots\}$. The median could be obtained by sorting, but I want to get via solving an optimization problem. I know that the media minimizes $$\sum_i \left| {{x_i} - \...
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2 votes
1 answer
82 views

What can we conclude from the property $P(X<Y)=P(X>Y)=0.5$ of two random variables $X$ and $Y$?

This is a similar question to Does the property $P(X<Y)=P(X>Y)=0.5$ imply equal expectations? I have a series of short questions related to: the medians of $X$ and $Y$, the means, and the fact ...
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How do I find an angle between two medians of triangle?

I have a hard time solving math task, which I have to do with vectors. I will post the photo of the triangle given in the task. Currently I am solving quite a lot of exercises with vectors, however I ...
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2 answers
71 views

Median of a uniform probability distribution

I am having a hard time with the solution to this problem. What I don't understand is why the $P(X\leq \frac{n+1}{2}) = \frac{n+1}{2}\frac{1}{n}$ What formula is being used here to calculate the ...
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What is the time complexity of the algorithm which extracts median of array using heap?

We have an array for which we have to extract the median of the array using a heap structure. Now consider the following algorithm: ...
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1 answer
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Relation between median and arithmetic mean

The median of the numbers in a list is $x$, and the arithmetic mean of the numbers in the list is $\frac{x+1}2$. Quantity $A = x$ and Quantity $B= \frac{x+1}2$ Quantity $A$ is greater. Quantity $B$ ...
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Non-Parametric Statistics: Probability of Median and Order Statistics

Let $X_1, ... , X_n$ be a random sample from a continuous distribution F. Let $\theta$ be the median of F. Using normal approximation, calculate the probability P($X_{(3)} \leq \theta \leq X_{(n-2)}$)....
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median of percentages?

I am looking to validate if my thoughts on answering the following question. The question is something I've made up / hypothetical / and may not reflect real life. Question: At a used car dealership 5%...
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sample median of the random sample from the exponential distribution

I have a question below but don't know how to do. Could anyone help? For part (a), I think that the exponential distribution is a decreasing function with negative slope, so it the sample median ...
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2 votes
0 answers
103 views

Is the coordinate-wise median equal to the geometric median if we apply a random rotation and average?

There are multiple ways to extend the median to higher dimensions. One natural way is to compute the coordinate-wise median, but this is not rotation-invariant. The coordinate-wise median is ...
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Get an estimate of Median and P90 percentiles across data

So, I have generated median and P90 percentiles for a group of data. I have multiple such groups of data with their own medians and P90 percentiles. Note that the group structure is critical I.e, I do ...
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1 vote
1 answer
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Median on 1-Sphere

Given a set of points $X \subset \mathbb{R}$, one can define the median $m^*$ using optimization. Namely $m^* = \textbf{argmin}_{m \in \mathbb{R}} \sum_{i \in X} |i - m|$. Here $|.|$ is absolute value....
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1 vote
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"median" of haversine distance

Suppose there is a set of points $V$ lives on surface of a 2-sphere $V \subset S^2$. One common distance between two points is the haversine distance or great circle distance $d: S^2\times S^2 \to \...
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1 vote
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Median factor of factorials

What is the order of $m(N)$, the median prime in the prime factorization of $N!$, as $N\to\infty$? For example, $m(6)=2$ because $6!=2^4\times3^2\times5$ and the median of $\{2,2,2,2,3,3,5\}$ is $2$. ...
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median of truncated Log-Cauchy distribution

Let $W$ be a standard Cauchy distributed random variable, i.e., $$ \mathbb{P}(W \leq w) = \frac{1}{2} + \frac{1}{\pi}\arctan w.$$ Let $X = x_0 e^{\mu + \sigma W }$. It can be seen that $\mathrm{median}...
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Statistical values about stock in different years

Analyst has collected the following data on the performance of the $X$ stock for $10$ different years. a) Calculate the arithmetic mean, the median, the mode, the standard deviation, the coefficient ...
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The range of a median function

This question is not about statistics. We have a median function with 3 parameters and we want to find the range (or the codomain) of the function as an interval. I discovered that it’s eventually the ...
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1 vote
2 answers
78 views

Why don't we use the mean of the middle pair to estimate the median in a cumulative frequency curve?

I just learned about cumulative frequency curve. The books says I could use this curve to estimate the median of the data. This is the picture that I cut from my book. As you can see, to estimate the ...
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Can we find sample quantiles in higher dimensions?

How can i extend the sample quantile to higher dimensions? For example, we can extend the median by $$ \text{Median} = \arg\min_c\sum_{j=1}^N\|x_j-c\|_p$$ Since the median is a sample quantile, I was ...
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How do I find the mode when my data points have decimals?

My data has numbers such as 70.35, 70.39, 70.1. None of the numbers are exactly the same. How do I find the mode?
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A rigorous bridge between Statistics and Set Theory. Deriving a median from a set

everyone! I'm trying to develop a new kind of database which is axiomatic and completely based on set theory. I use ZFC as a foundation and understand it on a moderate level, but still unable to see ...
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Choosing a median class interval when the median lies on a boundary (not estimating single median value)

I have a data set with a variable where income tax bands are coded as values. For example, 1 corresponds to the lowest income tax bracket. This means the codes are discrete integer values. I'm using ...
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