The **mode** is the value that appears most often in a set of data values.^{ [1] } If **X** is a discrete random variable, the mode is the value x (i.e, * X* =

- Mode of a sample
- Comparison of mean, median and mode
- Use
- Uniqueness and definedness
- Properties
- Example for a skewed distribution
- Van Zwet condition
- Unimodal distributions
- History
- See also
- References
- External links

Like the statistical mean and median, the mode is a way of expressing, in a (usually) single number, important information about a random variable or a population. The numerical value of the mode is the same as that of the mean and median in a normal distribution, and it may be very different in highly skewed distributions.

The mode is not necessarily unique to a given discrete distribution, since the probability mass function may take the same maximum value at several points *x*_{1}, *x*_{2}, etc. The most extreme case occurs in uniform distributions, where all values occur equally frequently.

When the probability density function of a continuous distribution has multiple local maxima it is common to refer to all of the local maxima as modes of the distribution. Such a continuous distribution is called multimodal (as opposed to unimodal). A mode of a continuous probability distribution is often considered to be any value x at which its probability density function has a locally maximum value, so any peak is a mode.^{ [2] }

In symmetric unimodal distributions, such as the normal distribution, the mean (if defined), median and mode all coincide. For samples, if it is known that they are drawn from a symmetric unimodal distribution, the sample mean can be used as an estimate of the population mode.

The mode of a 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] its mode is not unique. A dataset, in such a case, is said to be bimodal, while a set with more than two modes may be described as multimodal.

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 no two values will be exactly the same, so each value will occur precisely once. In order to estimate the mode of the underlying distribution, the usual practice is to discretize the data by assigning frequency values to intervals of equal distance, 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.

The following MATLAB (or Octave) code example computes the mode of a sample:

`X=sort(x);% x is a column vector datasetindices=find(diff([X;realmax])>0);% indices where repeated values change[modeL,i]=max(diff([0;indices]));% longest persistence length of repeated valuesmode=X(indices(i));`

The algorithm requires as a first step to sort the sample in ascending order. It then computes the discrete derivative of the sorted list, and finds the indices where this derivative is positive. Next it computes the discrete derivative of this set of indices, locating the maximum of this derivative of indices, and finally evaluates the sorted sample at the point where that maximum occurs, which corresponds to the last member of the stretch of repeated values.

Type | Description | Example | Result |
---|---|---|---|

Arithmetic mean | Sum of values of a data set divided by number of values | (1+2+2+3+4+7+9) / 7 | 4 |

Median | Middle value separating the greater and lesser halves of a data set | 1, 2, 2, 3, 4, 7, 9 | 3 |

Mode | Most frequent value in a data set | 1, 2, 2, 3, 4, 7, 9 | 2 |

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.

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.^{[ citation needed ]} For a finite data sample, the mode is one (or more) of the values in the sample.

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. - 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.
^{ [4] }^{ [5] } - For unimodal distributions, the mode is within √3 standard deviations of the mean, and the root mean square deviation about the mode is between the standard deviation and twice the standard deviation.
^{ [6] }

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.

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* = *e*^{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 *e*^{0} = 1 for Y.

When X has standard deviation σ = 0.25, the distribution of Y 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 X has a larger standard deviation, σ = 1, the distribution of Y is strongly skewed. Now

Here, Pearson's rule of thumb fails.

Van Zwet derived an inequality which provides sufficient conditions for this inequality to hold.^{ [7] } The inequality

- Mode ≤ Median ≤ Mean

holds if

- F( Median - x ) + F( Median + x ) ≥ 1

for all x where F() is the cumulative distribution function of the distribution.

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.^{ [8] } In symbols,

where is the absolute value.

A similar relation holds between the median and the mode: they lie within 3^{1/2} ≈ 1.732 standard deviations of each other:

The term mode originates with Karl Pearson in 1895.^{ [9] }

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."

In statistics, a **central tendency** is a central or typical value for a probability distribution.

In probability theory and statistics, **kurtosis** is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosis describes the shape of a probability distribution and there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population. Different measures of kurtosis may have different interpretations.

In statistics and probability theory, the **median** is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic feature of the median in describing data compared to the mean is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of a "typical" value. Median income, for example, may be a better way to suggest what a "typical" income is, because income distribution can be very skewed. The median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median is not an arbitrarily large or small result.

In statistics, a **normal distribution** is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is

In probability theory and statistics, **skewness** is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.

In probability theory and statistics, **variance** is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by , , , , or .

In probability theory, a **log-normal distribution** is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then *Y* = ln(*X*) has a normal distribution. Equivalently, if Y has a normal distribution, then the exponential function of Y, *X* = exp(*Y*), has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and engineering sciences, as well as medicine, economics and other topics.

In probability and statistics, **Student's t-distribution** is any member of a family of continuous probability distributions that arise when estimating the mean of a normally distributed population in situations where the sample size is small and the population's standard deviation is unknown. It was developed by English statistician William Sealy Gosset under the pseudonym "Student".

In probability theory, **Chebyshev's inequality** guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean. Specifically, no more than 1/*k*^{2} of the distribution's values can be *k* or more standard deviations away from the mean. The rule is often called Chebyshev's theorem, about the range of standard deviations around the mean, in statistics. The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers.

In probability theory and statistics, the **Gumbel distribution ** is used to model the distribution of the maximum of a number of samples of various distributions.

In mathematics, the **moments** of a function are quantitative measures related to the shape of the function's graph. If the function represents mass, then the first moment is the center of the mass, and the second moment is the rotational inertia. If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics.

In probability theory, the ** Vysochanskij–Petunin inequality ** gives a lower bound for the probability that a random variable with finite variance lies within a certain number of standard deviations of the variable's mean, or equivalently an upper bound for the probability that it lies further away. The sole restrictions on the distribution are that it be unimodal and have finite variance. The theorem applies even to heavily skewed distributions and puts bounds on how much of the data is, or is not, "in the middle."

In statistics, a **sampling distribution** or **finite-sample distribution** is the probability distribution of a given random-sample-based statistic. If an arbitrarily large number of samples, each involving multiple observations, were separately used in order to compute one value of a statistic for each sample, then the sampling distribution is the probability distribution of the values that the statistic takes on. In many contexts, only one sample is observed, but the sampling distribution can be found theoretically.

In probability and statistics, a **mixture distribution** is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized. The underlying random variables may be random real numbers, or they may be random vectors, in which case the mixture distribution is a multivariate distribution.

In statistics, a **bimodal****distribution** is a probability distribution with two different modes, which may also be referred to as a bimodal distribution. These appear as distinct peaks in the probability density function, as shown in Figures 1 and 2. Categorical, continuous, and discrete data can all form bimodal distributions.

In mathematics, **unimodality** means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object.

This **glossary of statistics and probability** is a list of definitions of terms and concepts used in the mathematical sciences of statistics and probability, their sub-disciplines, and related fields. For additional related terms, see Glossary of mathematics.

In statistics, the **68–95–99.7 rule**, also known as the **empirical rule**, is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution: 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean, respectively.

In probability theory, an **exponentially modified Gaussian distribution** describes the sum of independent normal and exponential random variables. An exGaussian random variable *Z* may be expressed as *Z* = *X* + *Y*, where *X* and *Y* are independent, *X* is Gaussian with mean *μ* and variance *σ*^{2}, and *Y* is exponential of rate *λ*. It has a characteristic positive skew from the exponential component.

In statistics and probability theory, the **nonparametric skew** is a statistic occasionally used with random variables that take real values. It is a measure of the skewness of a random variable's distribution—that is, the distribution's tendency to "lean" to one side or the other of the mean. Its calculation does not require any knowledge of the form of the underlying distribution—hence the name nonparametric. It has some desirable properties: it is zero for any symmetric distribution; it is unaffected by a scale shift; and it reveals either left- or right-skewness equally well. In some statistical samples it has been shown to be less powerful than the usual measures of skewness in detecting departures of the population from normality.

- ↑ Damodar N. Gujarati.
*Essentials of Econometrics*. McGraw-Hill Irwin. 3rd edition, 2006: p. 110. - ↑ Zhang, C; Mapes, BE; Soden, BJ (2003). "Bimodality in tropical water vapour".
*Q. J. R. Meteorol. Soc*.**129**(594): 2847–2866. doi:10.1256/qj.02.166. - ↑ "AP Statistics Review - Density Curves and the Normal Distributions". Archived from the original on 2 April 2015. Retrieved 16 March 2015.
- ↑ "Relationship between the mean, median, mode, and standard deviation in a unimodal distribution".
- ↑ Hippel, Paul T. von (2005). "Mean, Median, and Skew: Correcting a Textbook Rule".
*Journal of Statistics Education*.**13**(2). doi: 10.1080/10691898.2005.11910556 . - ↑ Bottomley, H. (2004). "Maximum distance between the mode and the mean of a unimodal distribution" (PDF).
*Unpublished Preprint*. - ↑ van Zwet, WR (1979). "Mean, median, mode II".
*Statistica Neerlandica*.**33**(1): 1–5. doi:10.1111/j.1467-9574.1979.tb00657.x. - ↑ Basu, Sanjib; Dasgupta, Anirban (1997). "The mean, median, and mode of unimodal distributions: a characterization".
*Theory of Probability & Its Applications*.**41**(2): 210–223. doi:10.1137/S0040585X97975447. - ↑ Pearson, Karl (1895). "Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material".
*Philosophical Transactions of the Royal Society of London A*.**186**: 343–414. doi: 10.1098/rsta.1895.0010 .

- "Mode",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - A Guide to Understanding & Calculating the Mode
- Weisstein, Eric W. "Mode".
*MathWorld*. - Mean, Median and Mode short beginner video from Khan Academy

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