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Encyclopedia > Mode (statistics)

In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. The term is applied both to probability distributions and to collections of experimental data. In some fields, notably education, sample data are often called scores, and the sample mode is known as the modal score.[1] A graph of a normal bell curve showing statistics used in educational assessment and comparing various grading methods. ... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... Sampling is that part of statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern, especially for the purposes of statistical inference. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... Data produced by an experimental or quasi-experimental design. ...


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. In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ... In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ... In descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate as much as possible as simply as possible. ...


The mode is not necessarily unique, since the same maximum frequency may be attained at different values. The worst case is given by so-called uniform distributions, in which all values are equally likely. In mathematics, the uniform distributions are simple probability distributions. ...

Contents

Mode of a probability distribution

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. In mathematics, a probability distribution is called discrete, if it is fully characterized by a probability mass function. ... In probability theory, a probability mass function (abbreviated pmf) gives the probability that a discrete random variable is exactly equal to some value. ...


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. By one convention, a probability distribution is called continuous if its cumulative distribution function is continuous. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...


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). A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some &#949; > 0 such that f(x*) &#8805; f(x) for all x with |x-x*| < &#949;. Stated less formally, a local maximum... Figure 1. ... In mathematics, a function f(x) between two ordered sets is unimodal if for some value m (the mode), it is monotonically increasing for x ≤ m and monotonically decreasing for x ≥ m. ...


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. Figures with the axes of symmetry drawn in. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ...


Mode of a sample

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. In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ...


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. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... personal space, proxemics. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... Example of a histogram of 100 normally distributed random values. ...


Comparison of mean, median and mode

See also: mean and median

For a probability distribution, the mean is also called the expected value of the random variable. For a data sample, the mean is also called the average. In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ... In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In mathematics, an average or central tendency of a set (list) of data refers to a measure of the middle of the data set. ...


When do these measures make sense?

Unlike mean and median, the concept of mode also makes sense for "nominal data" (i.e., not consisting of numerical values). For example, taking a sample of Korean family names, one might find that "Kim" occurs more often than any other name. Then "Kim" might be called the mode of the sample. However, this use is not common. The level of measurement of a variable in mathematics and statistics describes how much information the numbers associated with the variable contain. ... This article discusses the use of the word Number in Mathematics. ... The Korean name Hong Gildong (a common anonymous name, like John Doe in American English). ... Kim is the most common family name in Korea. ...


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. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... 2-dimensional renderings (ie. ... The integers are commonly denoted by the above symbol. ... Two intersecting planes in three-dimensional space In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. ... In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...


Uniqueness and definedness

For the remainder, the assumption is that we have (a sample of) a real-valued random variable.


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. Furthermore, like the mean, the mode of a probability distribution can be (plus or minus) infinity, but unlike the mean it cannot be just undefined. For a finite data sample, the mode is one (or more) of the values in the sample and is itself then finite.


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 X by aX+b, so are the mean, median and mode.
  • However, if there is an arbitrary monotonic transformation, only the median follows; for example, if X is replaced by exp(X), the median changes from m to exp(m) but the mean and mode won't.
  • Except for extremely small samples, the median is totally insensitive to "outliers" (such as occasional, rare, false experimental readings). The mode is also very robust in the presence of outliers, while the mean is rather sensitive.
  • In continuous unimodal distributions the median lies, as a rule of thumb, 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, is however not always true and the three statistics can appear in any order.[2] It often applies to slightly non-symmetric distributions that resemble a normal distribution.

In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces (strictly speaking, two affine spaces) consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... In mathematics, a function f(x) between two ordered sets is unimodal if for some value m (the mode), it is monotonically increasing for x ≤ m and monotonically decreasing for x ≥ m. ... Karl Pearson FRS (March 27, 1857 – April 27, 1936) established the discipline of mathematical statistics. ...

Example for a skewed distribution

A well-known 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. Example of the experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... Differences in national income equality around the world as measured by the national Gini coefficient. ...


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, whence the name. In probability and statistics, the log-normal distribution is the probability distribution of any random variable whose logarithm is normally distributed. ...


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. In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ...


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: In probability and statistics, the log-normal distribution is the probability distribution of any random variable whose logarithm is normally distributed. ...

  • mean = 1.0202
  • mode = 0.9608

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

  • mean = 7.3891
  • mode = 0.0183

Here, Pearson's rule of thumb fails, though for this distribution it is true for the logarithms of the mean, median and mode.


See also

In mathematics, a function f(x) between two ordered sets is unimodal if for some value m (the mode), it is monotonically increasing for x &#8804; m and monotonically decreasing for x &#8805; m. ... In descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate as much as possible as simply as possible. ... Descriptive statistics are used to describe the basic features of the data in a study. ... In statistics, central tendency is an average of a set of measurements, the word average being variously construed as mean, median, or other measure of location, depending on the context. ... In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ... In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ... In mathematics, arg max (or argmax) stands for the argument of the maximum, that is to say, the value of the given argument for which the value of the given expression attains its maximum value: This is well-defined only if the maximum is reached at a single value. ... -1...

External links

  • A problem involving the mean, the median, and the mode

References

  • Paul T. von Hippel. Mean, Median, and Skew: Correcting a Textbook Rule. J. of Statistics Education 13:2 (2005)

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