Mean
The mean is the arithmetic average of a dataset, found by adding all values and dividing by the count.
Formula
\text{Mean} = \dfrac{\text{sum of all values}}{\text{number of values}}
Definition
The mean (average) is the number you get by adding all values in a set and dividing by how many there are, representing a "fair share" value. Formally, the arithmetic mean is $\bar{x} = \frac{1}{n}\sum x_i$; it is the most common measure of center but is sensitive to outliers because extreme values pull it toward them. It is also the maximum likelihood estimator of the population mean $\mu$ when data are normally distributed, with sampling distribution $\bar{x} \sim N(\mu, \sigma^2/n)$ exactly for normal data and approximately by the CLT for large $n$ from any distribution with finite variance.
Example
Five students who scored $80$, $85$, $90$, $75$, and $70$ on a test have mean $= (80+85+90+75+70)/5 = 400/5 = 80$. Incomes (in thousands) of $30$, $35$, $40$, $42$, $38$, and one person earning $200$ give mean $= (30+35+40+42+38+200)/6 = 385/6 = 64.2$, which does not represent most people well; the median ($39$) is more appropriate here. Beyond the arithmetic mean, the geometric mean $GM = \left(\prod x_i\right)^{1/n}$ is appropriate for growth rates and ratios, the harmonic mean $HM = n / \sum (1/x_i)$ is appropriate for rates (e.g., average speed), and the inequality $HM \le GM \le AM$ holds for positive data.
Key Insight
The mean is like leveling out all the values so everyone has the same amount: if you poured all the water from cups of different sizes into equal cups, each cup's level is the mean. It minimizes the sum of squared deviations, $\sum (x_i - c)^2$ is minimized when $c = \bar{x}$, a property that makes the mean the foundation of least-squares regression. Jensen's inequality generalizes this: for a convex function $\phi$, $\phi(E[X]) \le E[\phi(X)]$, underpinning results across probability, information theory, and economics.