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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43
votes
11answers
8k views

What is the average of no numbers?

I have two programs that both behave nearly identically: they both take in any numbers you give them and can tell you the average and how many numbers were given. However, when you don't give them any ...
43
votes
3answers
1k views

Expected number of people to not get shot?

Suppose $n$ gangsters are randomly positioned in a square room such that the positions of any three gangsters do not form an isosceles triangle. At midnight, each gangster shoots the person that is ...
42
votes
3answers
67k 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 ...
30
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 ...
28
votes
4answers
2k views

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

The arithmetic mean of $y_i \ldots y_n$ 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 ...
25
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
3answers
10k 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 ...
19
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: ...
16
votes
2answers
360 views

Geometric mean of prime gaps?

The arithmetic mean of prime gaps around $x$ is $\ln(x)$. What is the geometric mean of prime gaps around $x$ ? Does that strongly depend on the conjectures about the smallest and largest gap such as ...
13
votes
3answers
384 views

If my average speed is 50 mph, is it better to go 10% faster or 5 mph faster? (These are not the same)

BACKGROUND Suppose we have a journey between A and B where different sections of roads have different speed limits and our average speed is 50 mph. If when I repeat the journey I travel 10% faster ...
10
votes
3answers
8k 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 ...
9
votes
4answers
3k 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. ...
9
votes
1answer
16k 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).
8
votes
5answers
408 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
1answer
367 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. ...
8
votes
4answers
3k 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 ...
8
votes
1answer
15k 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 ...
7
votes
2answers
352 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
3k views

Average distance between two points in a circular disk

How can I find an average distance between two points lying inside a circular disk of a certain radius? I wonder if there is any other way except of using a Monte Carlo method?
7
votes
3answers
293 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 ...
7
votes
2answers
165 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 ...
6
votes
4answers
3k 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
3answers
14k views

what is the difference between average and expected value?

I have been going through the definition of expected value in Wikipedia (http://en.wikipedia.org/wiki/Expected_value) beneath all that jargon it seems that the expected value of a distribution is the ...
6
votes
1answer
5k 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. ...
6
votes
1answer
2k 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 ...
6
votes
3answers
10k views

Simple question regarding ratio and average

I have 2 lists of numbers (with an equal number of numbers in each). Each number is then divided by the number of which it is paired with (by index), and a ratio is received. I then want to calculate ...
6
votes
4answers
142 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
578 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 ...
6
votes
1answer
3k 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 ...
6
votes
7answers
45k 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 ...
6
votes
1answer
250 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
2answers
747 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
251 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
4k 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 ...
5
votes
4answers
178 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
6answers
2k 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
7answers
2k 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
2answers
3k 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
2answers
4k 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 ...
5
votes
2answers
303 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?
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
700 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 ...
5
votes
1answer
1k 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
774 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
941 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 $d$ ...
5
votes
1answer
353 views

Average sine of an angle between two rays in a cone

I'm looking for an average value of sine of an angle between two rays, lying within a cone with a certain angle. Given a cone with an aperture of ${2\chi}$ and two rays lying within the cone. The ...
5
votes
3answers
11k views

Standard deviation of the weighted mean [duplicate]

How do you find the standard deviation of the weighted mean? The weighted mean is defined: $\bar{x}_w = \frac{\sum{wx}}{\sum{w}}$ The weighted standard deviation (since it is not specified, I take ...
5
votes
1answer
572 views

Find center of mass of triangle with varying density

My objective: Find the center of mass of a thin triangular plate bounded by the y-axis and the lines $y= 7x+3$ and $y= 36-4x$. Assume that the density is given by $\delta(x,y) = 7x+2y+2$. In ...
4
votes
1answer
676 views

Sum of averages to averages of sum

I have the following values $\overline{b}$, and $\overline{u}$ which is : $\overline{b} = \dfrac{b_1+b_2+b_3+b_4}{7}$, $\overline{u} = \dfrac{u_1+u_2+u_3+u_4}{7}$ is there any way I can get the ...
4
votes
1answer
228 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 ...