among those some are extremely rich. However, many are rather poor. A well-known class of distributions that can be arbitrarily skewed is given by the log-normal distribution. It is obtained by transforming a random variable having a normal distribution into random variable . Then the logarithm of random variable is normally distributed, hence the name. Taking the mean μ of to be 0, the median of will be 1, independent of the standard deviation σ of . This is so because has a symmetric distribution, so its median is also 0. The transformation from to is monotonic, and so we find the median for . When has standard deviation σ = 0.25, the distribution of is weakly skewed. Using formulas for the log-normal distribution, we find: Indeed, the median is about one third on the way from mean to mode. When has a larger standard deviation, , the distribution of is strongly skewed. Now Here, Pearson's rule of thumb fails. Van Zwet condition Van Zwet derived an inequality which provides sufficient conditions for this inequality to hold. The inequality Mode ≤ Median ≤ Mean holds if F( Median - ) + F( Median + ) ≥ 1 for all where F() is the cumulative distribution function of the distribution. Unimodal distributions It can be shown for a unimodal distribution that the median and the mean lie within (3/5)1/2 ≈ 0.7746 standard deviations of each other. In symbols, where is the absolute value. A similar relation holds between the median and the mode: they lie within 31/2 ≈ 1.732 standard deviations of each other: History The term mode originates with Karl Pearson in 1895. Pearson uses the term mode interchangeably with maximum-ordinate. In a footnote he says, "I have found it convenient to use the term mode for the abscissa corresponding to the ordinate of maximum frequency." See also Arg max Central tendency Descriptive statistics Moment (mathematics) Summary statistics Unimodal function References External links A Guide to Understanding & Calculating the Mode Mean, Median and Mode short beginner video from Khan Academy Means Summary statistics Articles with example MATLAB/Octave code