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Like the statistical mean and median, the mode is a way of expressing, in a (usually) single number, important information about a random variable or a population. The numerical value of the mode is the same as that of the mean and median in a normal distribution, and it may be very different in highly skewed distributions.
The median of a symmetric unimodal distribution coincides with the mode. The median of a symmetric distribution which possesses a mean μ also takes the value μ. 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 normal distribution with density () (mean and variance >) has the following properties: It is symmetric around the point =, which is at the same time the mode, the median and the mean of the distribution. [22]
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 ...
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the ...
the weighted arithmetic mean of the median and two quartiles. Winsorized mean an arithmetic mean in which extreme values are replaced by values closer to the median. Any of the above may be applied to each dimension of multi-dimensional data, but the results may not be invariant to rotations of the multi-dimensional space. Geometric median
Average of chords. In ordinary language, an average is a single number or value that best represents a set of data. The type of average taken as most typically representative of a list of numbers is the arithmetic mean – the sum of the numbers divided by how many numbers are in the list. For example, the mean or average of the numbers 2, 3, 4 ...
This distribution for a = 0, b = 1 and c = 0.5—the mode (i.e., the peak) is exactly in the middle of the interval—corresponds to the distribution of the mean of two standard uniform variables, that is, the distribution of X = (X 1 + X 2) / 2, where X 1, X 2 are two independent random variables with standard uniform distribution in [0, 1]. [1]
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