Mean (mathematics)

A mathematical mean assigns a representative value to several quantities. A real mean of $x_1,\dots ,x_n$ normally lies between their minimum and maximum and returns $x$ when every input equals $x$. Different means preserve different operations: the arithmetic mean averages sums, while the geometric mean averages positive multiplicative factors.

For $2$ and $8$, the arithmetic mean is $5$ and the geometric mean is $4$. If equal distances are traveled at speeds $2$ and $8$, the average speed is instead their harmonic mean, $2/(1/2+1/8)=16/5$. The useful choice depends on which quantity is being added: distances, times, or logarithms of growth factors.

For distinct positive real numbers $a,b$, their logarithmic mean isusing the natural logarithm. Set $L(a,a)=a$ by continuity: the difference quotient of the natural logarithm tends to its derivative $1/a$, so the reciprocal tends to $a$. For example, $L(1,4)=3/\log 4\approx 2.164$.

An integral representation explains its name and its relation to multiplicative interpolation:For $a\ne b$, integrate $a\exp (t\log (b/a))$ directly; for $a=b$, the integrand is constant. Every interpolated value lies between $a$ and $b$, so the logarithmic mean does too. It is symmetric in $a,b$ and homogeneous: $L(ca,cb)=cL(a,b)$ for $c>0$.