Mean
Statistics & ProbabilityThe 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. It represents a "fair share" value.
Example
Five students scored $80$, $85$, $90$, $75$, and $70$ on a test. Mean $= (80+85+90+75+70)/5 = 400/5 = 80$.
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.
Definition
The arithmetic mean is the sum of all data values divided by the number of values: $\bar{x} = \frac{1}{n}\sum x_i$. It is the most common measure of center but is sensitive to outliers because extreme values pull it toward them.
Example
Incomes (in thousands): $30$, $35$, $40$, $42$, $38$, and one person earning $200$. Mean $= (30+35+40+42+38+200)/6 = 385/6 = 64.2$. The mean of $64.2$ does not represent most people well; the median ($39$) is more appropriate here.
Key Insight
The mean minimizes the sum of squared deviations: $\sum (x_i - c)^2$ is minimized when $c = \bar{x}$. This property makes the mean the foundation of least-squares regression.
Definition
The arithmetic mean $\bar{x} = \frac{1}{n}\sum x_i$ is the maximum likelihood estimator of the population mean $\mu$ when data are normally distributed. Its sampling distribution satisfies $\bar{x} \sim N(\mu, \sigma^2/n)$ exactly for normal data and approximately by the CLT for large $n$ from any distribution with finite variance.
Example
Beyond 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). The arithmetic-geometric-harmonic inequality: $HM \le GM \le AM$ holds for positive data.
Key Insight
Jensen's inequality generalizes this: for a convex function $\phi$, $\phi(E[X]) \le E[\phi(X)]$. This underpins results across probability, information theory, and economics (e.g., risk aversion means preference for $E[W]$ over the lottery).