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The mean (L 2 center) and midrange (L ∞ center) are unique (when they exist), while the median (L 1 center) and mode (L 0 center) are not in general unique. This can be understood in terms of convexity of the associated functions (coercive functions).
The median of a normal distribution with mean μ and variance σ 2 is μ. In fact, for a normal distribution, mean = median = mode. The median of a uniform distribution in the interval [a, b] is (a + b) / 2, which is also the mean. The median of a Cauchy distribution with location parameter x 0 and scale parameter y is x 0, the location parameter.
Taking the mean μ of X to be 0, the median of Y will be 1, independent of the standard deviation σ of X. This is so because X has a symmetric distribution, so its median is also 0. The transformation from X to Y is monotonic, and so we find the median e 0 = 1 for Y. When X has standard deviation σ = 0.25, the distribution of Y is weakly skewed.
Splitting the observations either side of the median gives two groups of four observations. The median of the first group is the lower or first quartile, and is equal to (0 + 1)/2 = 0.5. The median of the second group is the upper or third quartile, and is equal to (27 + 61)/2 = 44. The smallest and largest observations are 0 and 63.
The degenerate distribution at x 0, where X is certain to take the value x 0. This does not look random, but it satisfies the definition of random variable. This is useful because it puts deterministic variables and random variables in the same formalism. The discrete uniform distribution, where all elements of a finite set are equally likely ...
The mean of a set of observations is the arithmetic average of the values; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely value (mode). For example, mean income is typically skewed upwards by a small number of people with very large incomes, so that the majority have an ...
The shape of a distribution will fall somewhere in a continuum where a flat distribution might be considered central and where types of departure from this include: mounded (or unimodal), U-shaped, J-shaped, reverse-J shaped and multi-modal. [1] A bimodal distribution would have two high points rather than one. The shape of a distribution is ...
For n = 1 or 2, the midrange and the mean are equal (and coincide with the median), and are most efficient for all distributions. For n = 3, the modified mean is the median, and instead the mean is the most efficient measure of central tendency for values of γ 2 from 2.0 to 6.0 as well as from −0.8 to 2.0.