The mean is the average value of a set of numbers. In economics, it is used to summarize data (average income, average inflation) and to connect empirical averages to theoretical expectations.
Arithmetic mean (the usual definition)
For observations (x_1,\dots,x_n), the arithmetic mean is:
[ \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i. ]
- (\bar{x}) is a sample mean (computed from data).
- (\mu) is often used for the population mean (the mean of the underlying distribution).
Weighted mean
When observations carry different importance (population shares, expenditure weights, portfolio weights), economists use a weighted mean:
[ \bar{x}w = \sum{i=1}^{n} w_i x_i \quad \text{with} \quad w_i \ge 0,; \sum_{i=1}^{n} w_i = 1. ]
Examples include inflation measurement (price indices) and average wages computed across groups with different employment shares.
Mean as an expected value
In probability and econometrics, the mean is the expected value of a random variable (X):
[ E[X] = \sum_x x,P(X=x) \quad \text{(discrete)}, \qquad E[X] = \int x f(x),dx \quad \text{(continuous)}. ]
Under standard conditions, the sample mean (\bar{x}) converges to (E[X]) as the sample gets large (law of large numbers). This is why averages are so central to empirical work.
Why the mean can mislead
The mean uses all values, so it is sensitive to outliers and skewed distributions.
Example: incomes (30, 30, 40, 200) (in thousands) have:
- mean (= (30+30+40+200)/4 = 75)
- median (= 35)
If the distribution has a long right tail (as income often does), the mean can be much higher than what a “typical” person experiences.