# What is Arithmetic mean: Definition and 21 Discussions

In mathematics and statistics, the arithmetic mean ( , stress on first and third syllables of "arithmetic"), or simply the mean or the average (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results of an experiment or an observational study, or frequently a set of results from a survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics, because it helps distinguish it from other means, such as the geometric mean and the harmonic mean.
In addition to mathematics and statistics, the arithmetic mean is used frequently in many diverse fields such as economics, anthropology and history, and it is used in almost every academic field to some extent. For example, per capita income is the arithmetic average income of a nation's population.
While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influenced by outliers (values that are very much larger or smaller than most of the values). For skewed distributions, such as the distribution of income for which a few people's incomes are substantially greater than most people's, the arithmetic mean may not coincide with one's notion of "middle", and robust statistics, such as the median, may provide better description of central tendency.

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1. ### B Arithmetic mean of an infinite number of points?

So I was thinking about arithmetic, geometric and harmonic means when I had a thought. Let's say we have a curve y = x^2. We want to find the AM of the points on the curve between x=1 and x=2 i.e. y = 1 and y = 4. To make thing easier, we'll start with just the endpoints and keep adding...
2. ### MHB Calculation of probability with arithmetic mean of the sum of random variables

Calculation of probability with arithmetic mean of random variables There are 4 people, each of whom has one deck of cards with 500 cards that are numbered from 1 to 500 with no duplicates. Each person draws a card from his deck and I would like to calculate the probability of the event that...
3. ### I Geometric mean versus arithmetic mean

The Beer-Lambert law gives the intensity of monochromatic light as a function of depth ##z## in the form of an exponential attenuation: $$I(z)=I_{0}e^{-\gamma z},$$ where ##\gamma## is the wavelength-dependent attenuation coefficient. However, if two different wavelengths are present...
4. ### Spivak's "Calculus": AM-GM inequality problem.

Homework Statement The problem is stated as follows: "The result in Problem 1-7 has an important generalization: If ##a_1,...,a_n≥0##, then the "arithmetic mean" ##A_n=\frac {a_1+...+a_n} {n}## and "geometric mean" ##G_n=\sqrt[n] {a_1...a_n}## Satisfy ##G_n≤A_n## Suppose that ##a_1\lt A_n##...
5. ### MHB Can the geometric and arithmetic means be applied to algebraic expressions?

Given two positive numbers a and b, we define the geometric mean and the arithmetic mean as follows G. M. = sqrt{ab} A. M. = (a + b)/2 If a = 1 and b = 2, which is larger, G. M. or A. M. ? G. M. = sqrt{1•2} G. M. = sqrt{2} A. M. = (1 + 2)/2 A. M = 3/2 Conclusion: G. M. > A. M. Correct...
6. ### I Do I use instrument error or arithmetic mean error?

Hi. Let's say I have data which I have measured. For example I measured a length of an object and the measurment was repeated 5 times. An instrument which I used to measure has an error, value of which I know. My options are to either to just go with the instrument error (probably not, right?)...
7. ### Arithmetic mean Fermi Dirac & Bose Einstein

Hi everybody, I was doing one asignment form class, I was tasked to prove that in one system, the arimetic mean of FD and BE distributions is equal to MB's distribution for undishtingable particles. After doing the numbers I found out that it actually was, but I don't know why this happens, can...
8. ### Prove that for a,b,c > 0, geometric mean <= arithmetic mean

Homework Statement Let ## a,b,c \in \mathbb{R}^{+} ##. Prove that $$\sqrt[3]{abc} \leq \frac{a+b+c}{3}.$$ Note: ## a,b,c ## can be expressed as ## a = r^3, b = s^3, c = t^3 ## for ## r,s,t > 0##. Homework Equations ## P(a,b,c): a,b,c \in \mathbb{R}^{+} ## ## Q(a,b,c): \sqrt[3]{abc} \leq...
9. ### Geometric Mean vs. Arithmetic Mean in Bandpass Filters

Why is the geometric mean used to define the center frequency of a bandpass filter instead of the arithmetic mean? I read in this book that 1. All the lowpass elements yield LC pairs that resonate at ω = 1. 2. Any point of the lowpass response is transformed into a pair of points of the...
10. ### Arithmetic mean and standard deviation

Homework Statement Showing all your working, calculate the arithmetic mean and standard deviation of the number of days lost. Table shows man days lost to sickness.. Days lost: 1-3 4-6 7-9 10-12 13-15 Frequency 8 7 10 9 6 Homework Equations...
11. ### Arithmetic mean complex numbers

Can the arithmetic mean of a data set of complex numbers be calculated? if so, can the method be demonstrated?
12. ### Find the least value of a

Homework Statement If four distinct points on the curve y=2x^4+7x^3+3x-5 are collinear, then find the arithmetic mean of x-coordinates of the aforesaid points. Homework Equations The Attempt at a Solution I think that the four points mentioned must be the roots of the equation.
13. ### Pigeonhole Principle and the Arithmetic Mean

My discrete mathematics book gives the following definition for the pigeonhole principle: If m objects are distributed into k containers where m > k, then one container must have more than \lfloor\frac{m-1}{k}\rfloor objects. It then states as a corollary that the arithmetic mean of a set...
14. ### Sequence and series - Arithmetic mean question have been breaking my head

Homework Statement Two consecutive numbers from 1,2,3...n are removed A.M of remaining numbers is 105/4. Find n and those numbers removed .Homework Equations Answer n = 50 those numbers are 7 and 8 The Attempt at a Solution I solved this question like a few weeks ago but now it escaped my...

PROVE mean (X bar) of a continuous distribution is given by: ∫x.f(x)dx {'a' is the lower limit of integration and 'b' is the upper limit}
16. ### What is the Limit of the Arithmetic Mean as n Approaches Infinity?

Homework Statement prove: lim x_n = L. Then \lim_{n\to\infty}\frac{x_1+\cdots+x_n}{n}=L Homework Equations The Attempt at a Solution i don't know abolutely. i tried definition \left|\frac{x_1+\cdots+x_n}{n}-L\right|=\frac{1}{n}\left|(x_1-L)+\cdots+(x_n-L)\right|
17. ### Prove this inequality : Geometric Mean and Arithmetic Mean

Homework Statement let r_{1}, r_{2}, ... , r_{n} be strictly positive numbers. Suppose an investment of one dollar at the beginning of the year k grows to 1+r_{k} at the end of year k (so that r_{k} is the "return on investment" in year k). Then the value of an investment of one dollar at...
18. ### Is the Calculation of Arithmetic Means Between Two Numbers Always Intuitive?

Homework Statement A question gives the problem find the two arithmetic means between 4 and 19. The answer is 9 and 14. Homework Equations (a1+a2+a3+an)/n The Attempt at a Solution Logic would dictate that the arithmetic mean would be adding 4 and 19 then dividing by two. Leaving...
19. ### Arithmetic mean of two unknowns in a system of equations

Homework Statement When finding the arithmetic mean in a system of equations is there any reason why the method that I am using is wrong? Find the arithmetic mean of x and y in the following set of equations Homework Equations 3x + 5y = 65 and 7x + 14y = 175 The Attempt at...
20. ### How to Bound the Limit of Arithmetic Mean from Below

Homework Statement Prove if that if the limit of a_n = c as n approaches infinity, then the limit of o_n = c as n approaches infinity, where o_n is the arithmetic mean (a_1 + ... + a_n)/n Homework Equations I can't figure out how to bound it from below. The Attempt at a Solution...
21. ### Arithmetic mean always greater than geometric mean

Hey, (sin A + sin B + sin C)/3 >= \sqrt[3]{}(sin A*sin B*sin C) I know this is true by Arithmetic mean always greater than geometric mean... but is there any other way of proving this?