Like the statistical mean and the median, the mode is a way of capturing important information about a random variable or a population in a single quantity. The mode is in general different from mean and median, and may be very different for strongly skewed distributions.
The mode is not necessarily unique, since the same maximum frequency may be attained at different values. The most ambiguous case occurs in uniform distributions, wherein all values are equally likely.
The mode of a discrete probability distribution is the value x at which its probability mass function takes its maximum value. In other words, it is the value that is most likely to be sampled. The mode of a continuous probability distribution is the value x at which its probability density function attains its maximum value, so, informally speaking, the mode is at the peak.
As noted above, the mode is not necessarily unique, since the probability mass function or probability density function may achieve its maximum value at several points x1, x2, etc.
When a probability density function has multiple local maxima, it is common to refer to all of the local maxima as modes of the distribution (even though the above definition implies that only global maxima are modes). Such a continuous distribution is called multimodal (as opposed to unimodal).
In symmetric unimodal distributions, such as the normal (or Gaussian) distribution (the distribution whose density function, when graphed, gives the famous "bell curve"), the mean (if defined), median and mode all coincide. For samples, if it is known that they are drawn from a symmetric distribution, the sample mean can be used as an estimate of the population mode.
The mode of a data sample is the element that occurs most often in the collection. For example, the mode of the sample [1, 3, 6, 6, 6, 6, 7, 7, 12, 12, 17] is 6. Given the list of data [1, 1, 2, 4, 4] the mode is not unique, unlike the arithmetic mean.
For a sample from a continuous distribution, such as [0.935..., 1.211..., 2.430..., 3.668..., 3.874...], the concept is unusable in its raw form, since each value will occur precisely once. The usual practice is to discretize the data by assigning the values to equidistant intervals, as for making a histogram, effectively replacing the values by the midpoints of the intervals they are assigned to. The mode is then the value where the histogram reaches its peak. For small or middle-sized samples the outcome of this procedure is sensitive to the choice of interval width if chosen too narrow or too wide; typically one should have a sizable fraction of the data concentrated in a relatively small number of intervals (5 to 10), while the fraction of the data falling outside these intervals is also sizable. An alternate approach is kernel density estimation, which essentially blurs point samples to produce a continuous estimate of the probability density function which can provide an estimate of the mode.
Unlike median, the concept of mean 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.
For some probability distributions, the expected value may be infinite or undefined, but if defined, it is unique. The average 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 both 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.
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 X having a normal distribution into random variable Y = exp(X). Then the logarithm of random variable Y is normally distributed, hence the name. Taking the mean μ of X to be 0, the median of Y will be 1, independent of the standard deviation σ of X. This is so because X has a symmetric distribution, so its median is also 0. The transformation from X to Y is monotonic, and so we find the median exp(0) = 1 for Y.
When X has standard deviation σ = 0.2, the distribution of Y is not very skewed. We find (see under Log-normal distribution), with values rounded to four digits:
Indeed, the median is about one third on the way from mean to mode.
When X has a much larger standard deviation, σ = 5, the distribution of Y is strongly skewed. Now
Here, Pearson's rule of thumb fails.