Editing Mode (statistics) (section)

===Example for a skewed distribution===

An example of a [[Skewness|skewed]] distribution is [[Distribution of wealth|personal wealth]]: Few people are very rich, but among those some are extremely rich. However, many are rather poor.

[[Image:Comparison mean median mode.svg|thumb|300px|Comparison of [[mean]], [[median]] and mode of two [[log-normal distribution]]s with different [[skewness]].]]
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 {{mvar|X}} having a normal distribution into random variable {{math|''Y'' {{=}} ''e''<sup>''X''</sup>}}. Then the logarithm of random variable {{mvar|Y}} is normally distributed, hence the name.

Taking the mean μ of {{mvar|X}} to be 0, the median of {{mvar|Y}} will be 1, independent of the [[standard deviation]] σ of {{mvar|X}}. This is so because {{mvar|X}} has a symmetric distribution, so its median is also 0. The transformation from {{mvar|X}} to {{mvar|Y}} is monotonic, and so we find the median {{math|''e''<sup>0</sup> {{=}} 1}} for {{mvar|Y}}.

When {{mvar|X}} has standard deviation σ = 0.25, the distribution of {{mvar|Y}} is weakly skewed. Using formulas for the [[log-normal distribution]], we find:
:<math>\begin{array}{rlll}
\text{mean}   & = e^{\mu + \sigma^2 / 2} & = e^{0 + 0.25^2 / 2} & \approx 1.032 \\
\text{mode}   & = e^{\mu - \sigma^2}     & = e^{0 - 0.25^2}     & \approx 0.939 \\
\text{median} & = e^\mu                  & = e^0                & = 1
\end{array}</math>
Indeed, the median is about one third on the way from mean to mode.

When {{mvar|X}} has a larger standard deviation, {{math|σ {{=}} 1}}, the distribution of {{mvar|Y}} is strongly skewed. Now
:<math>\begin{array}{rlll}
\text{mean}   & = e^{\mu + \sigma^2 / 2} & = e^{0 + 1^2 / 2} & \approx 1.649 \\
\text{mode}   & = e^{\mu - \sigma^2}     & = e^{0 - 1^2}     & \approx 0.368 \\
\text{median} & = e^\mu                  & = e^0             & = 1
\end{array}</math>
Here, [[Skewness#Pearson's skewness coefficients|Pearson's rule of thumb]] fails.