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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]
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:
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]
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 absolute deviation is a measure of statistical dispersion. Moreover, the MAD is a robust statistic, being more resilient to outliers in a data set than the standard deviation. In the standard deviation, the distances from the mean are squared, so large deviations are weighted more heavily, and thus outliers can heavily influence it ...