Meðalgildi

Meðalgildi, meaning mean value, is a mathematical concept used to describe the central tendency or average of a quantity. It appears in statistics, calculus, and signal processing as a way to summarize or characterize data, functions, or signals.

In statistics, the most common form is the arithmetic mean of a finite set of numbers x1, x2, ..., xn: x̄ = (1/n) ∑_{i=1}^n x_i. This value minimizes the sum of squared deviations from the data and serves as a typical summary of the data. In probability theory, the mean of a random variable is its expected value, representing the long-run average outcome of repeated trials.

In calculus, the mean value of a function f on an interval [a, b] is the average value f̄ = (1/(b − a)) ∫_a^b f(x) dx, provided f is continuous. This notion is related to, but distinct from, the Mean Value Theorem, which guarantees a point c with f′(c) = (f(b) − f(a))/(b − a).

In applied contexts such as signal processing, the mean value may refer to the average level of a signal over time or over a period, computed by integrating or summing over the period and dividing by its length.

Examples are straightforward: for the numbers 3, 7, 10, the mean is (3 + 7 + 10) / 3 = 6.67. Meðalgildi is a broad concept with related forms such as the geometric and harmonic means, which capture different notions of central tendency.