Use Unlike mean and median, the concept of mode also makes sense for "
nominal data" (i.e., not consisting of
numerical values in the case of mean, or even of ordered values in the case of median). For example, taking a sample of
Korean family names, one might find that "
Kim" occurs more often than any other name. Then "Kim" would be the mode of the sample. In any voting system where a plurality determines victory, a single modal value determines the victor, while a multi-modal outcome would require some tie-breaking procedure to take place. Unlike median, the concept of mode makes sense for any random variable assuming values from a
vector space, including the
real numbers (a one-
dimensional vector space) and the
integers (which can be considered embedded in the reals). For example, a distribution of points in the
plane will typically have a mean and a mode, but the concept of median does not apply. The median makes sense when there is a
linear order on the possible values. Generalizations of the concept of median to higher-dimensional spaces are the
geometric median and the
centerpoint.
Uniqueness and definedness For some
probability distributions, the expected value may be infinite or undefined, but if defined, it is unique. The mean of a (finite) sample is always defined. The median is the value such that the fractions not exceeding it and not falling below it are each at least 1/2. It is not necessarily unique, but never infinite or totally undefined. For a data sample it is the "halfway" value when the list of values is ordered in increasing value, where usually for a list of even length the numerical average is taken of the two values closest to "halfway". Finally, as said before, the mode is not necessarily unique. Certain
pathological distributions (for example, the
Cantor distribution) have no defined mode at all. For a finite data sample, the mode is one (or more) of the values in the sample.
Properties Assuming definedness, and for simplicity uniqueness, the following are some of the most interesting properties. • All three measures have the following property: If the random variable (or each value from the sample) is subjected to the linear or
affine transformation, which replaces by , so are the mean, median and mode. • Except for extremely small samples, the mode is insensitive to "
outliers" (such as occasional, rare, false experimental readings). The median is also very robust in the presence of outliers, while the mean is rather sensitive. • In continuous
unimodal distributions the median often lies between the mean and the mode, about one third of the way going from mean to mode. In a formula, median ≈ (2 × mean + mode)/3. This rule, due to
Karl Pearson, often applies to slightly non-symmetric distributions that resemble a normal distribution, but it is not always true and in general the three statistics can appear in any order. • For unimodal distributions, the mode is within standard deviations of the mean, and the root mean square deviation about the mode is between the standard deviation and twice the standard deviation.
Example for a skewed distribution An example of a
skewed distribution is
personal wealth: Few people are very rich, but among those some are extremely rich. However, many are rather poor. ,
median and mode of two
log-normal distributions 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 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: \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} 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 \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} 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 \text{mode} \leq \text{median} \leq \text{mean} holds if F(\text{median} - x) + F(\text{median} + x) \geq 1 for all where is the
cumulative distribution function of the distribution. ==Unimodal distributions==