Arithmetic Mean
The arithmetic mean is the sum of all the numbers in a data set divided by the quantity of numbers in that set. More precisely,
The arithmetic mean \(\overline{x}\) of a collection of \(n\) numbers (from \(a_1\) through \(a_n\)) is given by the formula
\[\overline{x}=\displaystyle \frac{1}{n}\sum_{i=1}^n a_i = \frac{a_1+a_2+a_3+\dots + a_n}{n}.\ _\square\]
Note that this definition refers to the arithmetic mean, as distinct from other types of means like geometric mean or harmonic mean.
The arithmetic mean is commonly referred to as the average, because it is a common measure of central tendency among a data set. However, there are other ways of measuring an average, including median and mode, so the term should be clarified if there is any uncertainty as to which average a person is using.
Contents
Visualizing the Mean
The arithmetic mean can be visualized as a balancing point on a scale. Half the numerical "mass" of the data set will land above the value of the mean, while the other half will land below. The mean may or may not be one of the numbers that appears in the number set.
Examples
If Clara scores 100 in calculus, 90 in literature, and 95 in physics, what is her average score? Our instinct tells us that she scored an average of 95 points, since 95 is exactly in the middle of 90 and 100. A more mathematical approach would be: \(\frac{100+90+95}{3}=95.\)
What is the arithmetic mean of 3, -14, 25, 103 and 48?
We have \[\frac{3-14+25+103+48}{5}=\frac{165}{5}=33.\ _\square\]
What is the arithmetic mean of all the positive integers in the interval \([1, 10]?\)
We have \[\frac{\displaystyle{\sum_{k=1}^{10}} k}{10}=\frac{55}{10}=\frac{11}{2}.\ _\square \]
If the arithmetic mean of five numbers \(2, 3, 9, 15\) and \(a\) is \(4,\) what is \(a?\)
We have \[\begin{align} \frac{2+3+9+15+a}{5}=\frac{29+a}{5} &=4 \\ 29+a &=20 \\ a &=-9. \ _\square \end{align}\]
\(a,b,\) and \(c\) follow an arithmetic progression. Denote \(x\) as the geometric mean of \(a\) and \(b\), and \(y\) the geometric mean of \(b\) and \(c.\)
Find the arithmetic mean of \(x^2 \) and \(y^2\) in terms of \(a,b,\) and/or \(c\).
What is the arithmetic mean of the first 100 positive integers?
The average of 999 numbers is 999.
From these numbers I choose 729 of them, and coincidentally, their average is 729.
Find the average of the remaining numbers.
See Also
References
- Medina, G. Example: Arithmetic mean as center of mass. Retrieved from http://www.texample.net/tikz/examples/balance/