Mathematics Stack Exchange
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Questions tagged [central-tendency]

In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution. It may also be called a center or location of the distribution. Colloquially, measures of central tendency are often called averages.The tendency for the values of a random variable to cluster round its mean, mode, or median.

The most common measures of central tendency are the arithmetic mean, the median and the mode. A central tendency can be calculated for either a finite set of values or for a theoretical distribution, such as the normal distribution. Occasionally authors use central tendency to denote "the tendency of quantitative data to cluster around some central value."

The central tendency of a distribution is typically contrasted with its dispersion or variability; dispersion and central tendency are the often characterized properties of distributions. Analysts may judge whether data has a strong or a weak central tendency based on its dispersion.The term central tendency dates from the late 1920s.

22 questions
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Generalization of mean and median

It is well known that a median of a distribution $\mu$ can be defined as an $m$ such that $$m\in\operatorname*{arg\,min}_{c\in\mathbb{R}}\mathbb{E}_{X\sim\mu}[|X-c|].$$ Similarly, the mean of a distribution $\mu$ is defined as an $m$ such…
Tyler6
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Mid-range via minimax

Warning: crossposted at Statistics SE. Given vector ${\rm a} \in \Bbb R^n$, $$\begin{array}{ll} \displaystyle\arg\min_{x \in {\Bbb R}} & \left\| x {\Bbb 1}_n - {\rm a} \right\|_2^2\end{array} = \frac1n {\Bbb 1}_n^\top {\rm a} \tag{mean}$$ is the…
Rodrigo de Azevedo
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How do I find the expected return on a ticket?

A lottery has a grand prize of 400,000 dollars, five runner up prizes of 50,000 dollars each, nine third-place prizes of 10000 dollars each, and twenty-five consolation prizes of 1000 dollars each. If 2,000,000 tickets are sold for 1 dollar each and…
Kevin Diez
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Generalising "the mean of the means is the mean"

It's an interesting exercise to prove that the "mean of the means is the mean", by which I mean given a finite list $X=[x_1,\dots,x_n]$ we can define the arithmetic mean $$\mu(X)=\frac{1}{n}\sum_{i=1}^nx_i, \tag{1}$$ and we consider the list of…
Gentleman_Narwhal
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Why is it assumed that the mode, in a grouped frequency distribution, lies in the modal class(the class with maximum frequency)?

I am in 10th standard. We are given a formula to find the mode mode = l +[(f1-f0)/(2f1-f0-f2)]h Where l is the lower limit, F1 the modal class i.e the class with most frequencies, f0 the frequency of class preceding the modal class, F2 the frequency…
macaw_scrlt
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Where I am confusing about the concept of 'weightage' and 'Balance Point'?

By Balance point I understand equal weight on both sides of a lever. Various sources, including stackexchange threads use to treat 'mean' as 'balancing point'. We want to place a fulcrum on the line so that the line doesn't tip due to the…
user1072886
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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?
Sann
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No mode or all mode?

Let's say I have a data set:- $${1,2,3,4,5}$$ I decide to find the mode though it would not be an appropriate measure of central tendency. But still........ Now this is an exceptional case and for exceptional cases , I have the following…
Mohd Saad
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Does central tendency (Geometric Mean specifically) always lie in between the two numbers or can it lie outside the interval of the two numbers?

Does central tendency necessarily have to lie between the two numbers?For eg. I understand $2,4,8$ are in GP with $4$ being the GM of $2,4,8$. But what about $2,-4,8$ if we consider common ratio to be $-2$? $-4$ does not lie between $2,8$.Still is…
Soham
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What is the average of the following $\frac{n \text{ digits}}{(n-1) \text{ digits}}$?

My first assumptions were not correct. I am looking for an intuitive explanation for the average of the following formula as well as what that average is if one were to run every number combination through this. $$\frac{n+1 \text{ digits}}{n \text{…
Joe
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Central tendencies of grouped data.

In our curriculum we have various exercises on calculating the three elementary measures of central tendency - mean, median, and mode for grouped data. For the same, we have been taught the following formulae: Mean: $$\frac{\sum_i f_ix_i}{\sum_i…
Sahaj
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If the median of $X$ is greater than the median of $Y$ then $\Bbb P(X>Y) \geq \frac14$

The median value of a random variable $X$ is $m$ if $\Bbb P (X \geq m) \geq \frac12$ and $\Bbb P (X \leq m) \geq \frac12$. Let $X$ and $Y$ be independent real random variables. Show that if the median value of $X$ is larger than the median value of…
Ava Kate Lich
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Determine variance of item level scores within single test and sample of tests

I am decades out of school and need help determining the correct statistic / concept to use for my use case. My agency (human services) uses a specific functional test to determine how well clients are functioning in their daily lives. We are…
Patrick Wood
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Is there a non-mathematical definition of the geometric mean?

I am aware that the geometric mean is often used with the lognormal distribution, because then it directly relates to the arithmetic mean with the normal distribution. But I was trying to think of an intuitive defintion of the geometric mean. For…
Peter_Pan
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Why use mean when you can use median?

I know that if I calculate the median, I am going to get a number that represents central tendency even if there are outliers that would cause the distribution to be skewed right or left. According to the Aerd Statistics website: When you have a…
Darien Springer
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