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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.
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For a symmetric distribution (where the median equals the midhinge, the average of the first and third quartiles), half the IQR equals the median absolute deviation (MAD). The median is the corresponding measure of central tendency. The IQR can be used to identify outliers (see below). The IQR also may indicate the skewness of the dataset. [1]
In statistics, the mode is the value that appears most often in a set of data values. [1] If X is a discrete random variable, the mode is the value x at which the probability mass function takes its maximum value (i.e., x=argmax x i P(X = x i)).
Comparison of mean, median and mode of two log-normal distributions with different skewness. The mode is the point of global maximum of the probability density function. In particular, by solving the equation ( ln f ) ′ = 0 {\displaystyle (\ln f)'=0} , we get that:
the point minimizing the sum of distances to a set of sample points. This is the same as the median when applied to one-dimensional data, but it is not the same as taking the median of each dimension independently. It is not invariant to different rescaling of the different dimensions. Quadratic mean (often known as the root mean square)
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The standard Gumbel distribution is the case where = and = with cumulative distribution function = ()and probability density function = (+).In this case the mode is 0, the median is ( ()), the mean is (the Euler–Mascheroni constant), and the standard deviation is /