The Fréchet mean is the point, x, that minimizes the Fréchet function, in cases where such a unique minimizer exists. The value at a point p, of the Fréchet function associated to a random point X on a complete metric space (M, d) is the expected squared distance from p to X. In particular, the Fréchet mean of a set of discrete random points x_i is the minimizer m of the weighted sum of squared distances from an arbitrary point to each point of positive probability (weight), assuming this minimizer is unique. In symbols:.

The name Karcher mean is sometimes used instead of "Fréchet mean", where this refers to Hermann Karcher.

Arithmetic mean

For real numbers, the arithmetic mean

Arithmetic mean

In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

 is a Fréchet mean, using as distance function the usual Euclidean distance.

Geometric mean

On the positive real numbers, the (hyperbolic) distance function can be defined. The geometric mean

Geometric mean

The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...

 is the corresponding Fréchet mean.Indeed is then an isometry from the euclidean space to this "hyperbolic" space and must respect the Fréchet mean: the Fréchet mean of the is the image by of the Fréchet mean (in the euclidean sense) of the , i.e. it must be:
.

Harmonic mean

On the positive real numbers, the metric

Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

 (distance function) can be defined. The harmonic mean

Harmonic mean

In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....

 is the corresponding Fréchet mean.

Power means

Given a non-zero real number , the power mean can be obtained as a Fréchet mean by introducing the metric.

f-mean

Given an invertible function , the f-mean can be defined as the Fréchet mean obtained by using the metric
. This is sometimes called the Generalised f-mean or Quasi-arithmetic mean

Quasi-arithmetic mean

In mathematics and statistics, the quasi-arithmetic mean or generalised f-mean is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function f...

.

Weighted means

The general definition of the Fréchet mean that includes the possibility of weighting observations can be used to derive weighted versions for all of the above types of means.