The sum of several numbers divided by their count nicely summarizes the overall size of the numbers involved in a single value.

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28
votes
2answers
2k views

What's the mean of all real numbers?

At first, I had thought the average must be zero, since for every positive number there's an equal magnitude negative number to cancel out the positive number's effect on the average, leaving only ...
23
votes
11answers
4k views

Prove that if two miles are run in 7:59 then one mile MUST be run under 4:00.

I'm in an argument with someone who claims that a two mile in 7:59 does not imply that one mile (at some point within the two miles) was covered in under 4:00. This is obviously wrong, but I'm not ...
23
votes
4answers
1k views

What is to geometric mean as integration is to arithmetic mean?

The arithmetic mean of $y_i ... y_n$ is: $$\frac{1}{n}\sum_{i=1}^n~y_i $$ For a smooth function $f(x)$, we can find the arithmetic mean of $f(x)$ from $x_0$ to $x_1$ by taking $n$ samples and using ...
16
votes
8answers
3k views

Unorthodox way of getting the average of two numbers

I can't believe the alternative method I just saw to calculate the average of two numbers: ...
12
votes
3answers
3k views

Incremental averageing

Is there a way to incrementally calculate (or estimate) the average of a vector (a set of numbers) without knowing their count in advance? For example you have ...
12
votes
2answers
17k views

Why is an average of an average usually incorrect?

Can someone explain why taking an average of an average usually results in a wrong answer? Is there ever a case where the average of the average can be used correctly? As an example, let's say ...
8
votes
5answers
355 views

Puzzle in Percentages

Okay, this is a real-time problem. The following is a picture of Customer satisfaction rating, which was displayed next to an item in an online shopping website. Satisfied customers click the ...
8
votes
4answers
2k views

How do I figure out what kind of distribution this is?

i've sampled a real world process, network ping times. The "round-trip-time" is measured in milliseconds. Results are plotted in a histogram: Ping times have a minimum value, but a long upper tail. ...
7
votes
2answers
163 views

What does the $L^p$ norm tend to as $p\to 0$?

This is something I was thinking about, so I'm going to post it as a question and post my own answer. I hope that anyone who wants will comment, correct me if I'm wrong, and add their own knowledge ...
7
votes
3answers
264 views

Asymptotic difference between a function and its binomial average

The origin of this question is the identity $$\sum_{k=0}^n \binom{n}{k} H_k = 2^n \left(H_n - \sum_{k=1}^n \frac{1}{k 2^k}\right),$$ where $H_n$ is the $n$th harmonic number. Dividing by $2^n$, we ...
6
votes
4answers
1k views

How can I calculate “most popular” more accurately?

I'm developing a website at the moment. The website allows users to "rate" a post from 0 to 5. Posts can then be displayed in order of popularity. At the moment, my method of calculation is pretty ...
6
votes
4answers
974 views

Average bus waiting time

My friends and I were "thinking" yesterday in the pub about the following: if a person is standing on a bus stop that is served by a single bus which comes every p minutes, we would expect the average ...
6
votes
1answer
184 views

Why is the $0$th power mean defined to be the geometric mean?

Mentioned in the wikipedia article, the $0$th power mean is defined to be the geometric mean. Why is this? I understand that a convenient consequence is that the means are ordered by their exponent. ...
6
votes
4answers
114 views

Is there a way to get an average that weights each item inversely based on distance from the mean?

I'm not sure how to phrase this, or even if such a thing exists. Sorry! I have a bunch of data points, which are mostly pretty tight. Each group should hone in on a specific point in space. ...
6
votes
1answer
103 views

How to compute the mean average exponent of the naturals? What is the limit for large numbers?

With a friend I was trying to get an understanding for why the expected gap between primes is logarithmic. With that motivation I tried to express the average exponent of numbers. By average ...
6
votes
1answer
6k views

Average percent increase not equal to total percent increase?

I tried searching around for this but it was difficult to boil down the search terms. Plus nothing seemed to be showing up anyway. What's an easy way to show that the average percentage increase of n ...
6
votes
2answers
62 views

Intuition on why the average minimizes the euclidean distance

I saw that there was a question with a very similar (if not identical) flavour to my question, but the answer was derived from derivative, the method that I already knew solved this problem. I feel ...
6
votes
0answers
193 views

What's the average distance between two discs in the plane?

Consider two discs in the plane of radius $r$ and $s$, with centers separated by a distance $l$. If we choose a point uniformly at random from each disc, what is the expected distance between the two ...
5
votes
4answers
151 views

$x$ is equal to at least $51$ of $a_1,\frac{a_1+a_2}{2},\ldots,\frac{a_1+a_2+\ldots+a_{100}}{100}$. Prove that $2$ of $a_1,\ldots,a_{100}$ are equal.

If $x$ is equal to at least $51$ number of the array $a_1, \frac{a_1+a_2}{2},\ldots,\frac{a_1+a_2+\ldots+a_{100}}{100}$, prove that $2$ numbers of the array $a_1,a_2\ldots,a_{100}$ are equal. ...
5
votes
7answers
439 views

3 trams are coming every 10, 15 and 15 minutes. On average, how long do I have to wait for any tram to come?

3 trams are coming to the stop every 10, 15 and 15 minutes. On average, how long do I have to wait for any tram to come? It's a practical problem, not some kind of a riddle for which I have a ...
5
votes
6answers
620 views

Confusion with the definition of mean value

For some reason the formula for mean started to trouble me: $$\mu = \frac{1}{b-a}\int_a^b f(x)\:dx$$ The reason this confuses me a bit is because when I read this formula I read it as: ...
5
votes
2answers
1k views

Why is median age a better statistic than mean age?

If you look at Wolfram Alpha or this Wikipedia page List of countries by median age Clearly median seems to be the statistic of choice when it comes to ages. I am not able to explain to myself ...
5
votes
6answers
19k views

Sum of averages vs average of sums

I have essentially a table of numbers -- a time series of measurements. Each row in the table has 5 values for the 5 different categories, and a sum row for the total of all categories. If I take ...
5
votes
1answer
119 views

continuously averaging a stream of numbers

Ok, so I'd like a little help with the mathematical theory behind something I'm doing in a program. I've got a stream of numeric inputs, and I need to find the average of all the numbers in this ...
5
votes
3answers
2k views

Which average to use? (RMS vs. AM vs. GM vs. HM)

The generalized mean (power mean) with exponent $p$ of $n$ numbers $x_1, x_2, \ldots, x_n$ is defined as $$ \bar x = \left(\frac{1}{n} \sum x_i^p\right)^{1/p}. $$ This is equivalent to the harmonic ...
5
votes
1answer
916 views

Calculate average latitude / longitude

I have the array of geographical coordinates (latitude & longitude). What is the best way to calculate average latitude and longitude? Thanks!
5
votes
1answer
477 views

Harmonic mean and logarithmic mean

The harmonic mean of a finite set of positive real numbers $\{x_1, x_2, \ldots, x_n\}$ is defined to be $$H(\{x_1, x_2, \ldots, x_n\}) = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + ...
5
votes
2answers
498 views

How many decimal places are needed for incremental average calculation?

If using the following formula to incrementally calculate average value: $$m_n = m_{n-1} + \frac{a_{n}-m_{n-1}}{n}$$ And the number of decimal places to where you can save $m_n$ is limited to $d$ ...
4
votes
2answers
1k views

What is the average of rolling two dice and only taking the value of the higher dice roll?

What is the average result of rolling two dice, and only taking the value of the higher dice roll? To make sure the situation I am asking about is clear, here is an example: I roll two dice and one ...
4
votes
1answer
214 views

Prove that if 2000 m is run in 4:50, then one mile is run under 4:00

Prove that if 2000 m is run in 4:50, then one continuous mile is run under 4:00. I was thinking -- yeah, this follows easily from the other question, where I asked that if 2 miles are run in ...
4
votes
1answer
3k views

5 star ratings. Bayesian or Weighted average?

I am building a website which has 5 star ratings to rate items. These ratings unlike the normal ones are of unequal weights. i.e. ...
4
votes
2answers
146 views

Basic question about natural density

Suppose that we have a sequence of finite sets $A_1, A_2, \ldots$, which partition $\mathbb{N}$. I am making no other assumptions on the $A_n$ - i.e. there could be any amount of interleaving between ...
4
votes
5answers
196 views

what is the argument for not using the average of an average?

I want to disprove someone's calculation of percentage of cash sales for a year by taking summing percentage of cash sales by month and dividing by 12. I sense the correct way is to take total cash ...
4
votes
1answer
650 views

Regular average calculated accumulatively

is it possible to calculate the regular average of a sequence of numbers when i dont know everything of the sequence, but just everytime i get a new number i know the total count of numbers and the ...
4
votes
2answers
2k views

Find the average of a collection of points (in 2D space)

I'm a bit rusty on my math, so please forgive me if my terminology is wrong or I'm overlooking extending a simple formula to solve the problem. I have a collection of points in 2D space (x, y ...
4
votes
1answer
1k views

How to calculate an overall 5 star rating based on other weighted 5 star ratings

I have these 3 ratings. I want to combine them to produce an overall rating out of 5. The overall rating can be a float, so no rounding is necessary, but it cannot be negative. I'm just not sure ...
4
votes
1answer
352 views

Average over angles

I feel quite dumb for asking this, but it's getting a bit frustrating. I have a function which depends on $\theta$ and $\phi$ in spherical coordinates: $$ f(\theta, \phi) = \sin^2 \theta ...
4
votes
2answers
130 views

Is there a name (and use) for an average based on the unique values of a set of data?

Consider the following data points: 1, 1, 2, 3, 4 I understand that... the average is the total of the numbers divided by the count of numbers in the set, the median is the central value based on ...
4
votes
1answer
178 views

Power Mean Random Distribution

I'm trying to find a the distribution for the power mean of $n$ random variables on $[0,1]$. I've got the arithmetic mean: $\frac{n}{(n-1)!}\sum_{k=0}^{\lfloor ...
4
votes
2answers
2k views

Real world examples for Mode in Statistics

I am looking for some real world examples for mode in Statistics involving topics which students like say Football or Social networks. Also they need to clearly identify differences in the usefulness ...
4
votes
1answer
118 views

Probability value in a skewed data

A question goes like this: Given that there 2 questions out of 3 questions to pass an exam and follow are details of number of people who attended the questions correctly, what is the maximum number ...
4
votes
0answers
277 views

Average transformation matrix?

I have several estimates of the transformation matrix between two planes and some values that give some indication of the error involved in the estimate. How can I use this information to gain the ...
3
votes
4answers
177 views

If $\sum\limits_{k=1}^n y_k\geq n$ and $\sum\limits_{k=1}^n \frac{1}{y_k}\geq n$, then $\prod\limits_{k=1}^n y_k\geq 1$?

Let $y_1,\ldots y_n$ be positive real numbers satisfying $y_1+\cdots+y_n\geq n$ and $\displaystyle{\frac{1}{y_1}+\cdots+\frac{1}{y_n}\geq n}$. Is it true that $y_1y_2\cdots y_n\geq 1$?
3
votes
3answers
213 views

Erased number?

A set of consecutive positive integers starting with 1 is written on the board. A student came along and erased one number. Average of remaining numbers is 61 15/20 . What was the number erased
3
votes
3answers
226 views

Definition of random

Suppose that you has to guess given a set of numbers If they are random. The mathematical expectation Is there a definition of randomness that allow this prove/test? Is even possible? if so: How ...
3
votes
1answer
2k views

How to add and subtract values from an average?

Say I have 100 numbers that are averaged: number of values = 100 total sum of values = 2000 mean = 2000 / 100 => 20 If I want to add a value and find out the ...
3
votes
1answer
7k views

Average of multiple vectors

I have more than two vectors (these are 2D vectors) and I want to calculate the mean vector. What is the correct way to do it? All my vectors share their origins at (0,0).
3
votes
2answers
3k views

Calculate average wind direction

What is the best way to average wind direction? I have found many conflicting suggestions elsewhere. Best one I saw is to average SINs of all angles in radians and take inverse SIN of the result. ...
3
votes
2answers
125 views

How to find the average value of $y = e^x$ between $x = e$ and $x = 2e$? [closed]

What approach would be ideal in finding the average value of $y = e^x$ between $x = e$ and $x = 2e$?
3
votes
2answers
183 views

If I put $M$ balls into $N$ boxes at random, what is the average number of balls in the boxes that are not empty?

I have a very brief question: if I put $M$ balls into $N$ boxes at random, what is the average number of balls in the boxes that are not empty?