Mode (statistics)

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The mode of a set of data values ​​is the most common value. This is the value x in which its probability mass function takes its maximum value. In other words, it is the most likely value to be sampled.

Like statistical and median means, mode is a way of expressing, in (usually) a single number, important information about random or population variables. The numerical value of this mode is equal to the mean and median in the normal distribution, and it may be very different in the highly skewed distribution.

This mode is not always unique for certain discrete distributions, since the probability mass function can take the same maximum value at some point x ₁ , x ₂ , etc. The most extreme case occurs in a uniform distribution, where all values ​​occur just as often.

When the probability density function of a continuous distribution has some local maxima it is common to refer to all local maxima as the distribution mode. This sustainable distribution is called multimodal (the opposite of unimodal). A continuous probability distribution mode is often regarded as any value x in which the probability density function has a local maximum value, so each peak is mode.

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

Video Mode (statistics)

Mode sampel

Sample mode is the most common element in the collection. For example, the sample modes [1, 3, 6, 6, 6, 6, 7, 7, 12, 12, 17] are 6. Based on the data list [1, 1, 2, 4, 4] the modes are not unique - the dataset can said to be bimodal, while a set with more than two modes can be described as multimodal.

For a sample of a continuous distribution, such as [0.935..., 1,211..., 2,430..., 3,668..., 3,874...], this concept can not be used in raw form, since no two values ​​will be exactly so each value will happen exactly once. To estimate the underlying distribution mode, the usual practice is to discriminate data by assigning frequency values ​​to the same distance interval, such as to create a histogram, effectively replacing the value with the midpoint of the specified interval. The mode is the value at which the histogram reaches its peak. For small or medium samples, the result of this procedure is sensitive to the choice of interval widths if selected too narrow or too wide; usually a person must have most of the data concentrated in a small number of intervals (5 to 10), while the fraction of data falling outside this interval is also quite large. An alternative approach is the kernel density estimation, which essentially obscures the point example to produce a continuous estimate of the probability density function which can provide fashion estimates.

The following sample MATLAB (or Octave) code calculates the sample mode:

The algorithm requires as a first step to sort the sample in ascending order. It then calculates the discrete instance of the sorted list, and finds the index where the derivative is positive. Next he calculates the discrete derivatives of this set of indices, finds the maximum derivative of this index, and ultimately evaluates the ordered sample at the point where the maximum occurs, corresponding to the last member of the repetitive stretch values.

Maps Mode (statistics)

Comparison of mean, median and mode

Use

Unlike averages and medians, the concept of mode also makes sense for "nominal data" (ie, does not consist of numerical values ​​in the average case, or even values ​​ordered in the median case). For example, taking samples of Korean surnames, one might find that "Kim" more often appears than any other name. Then "Kim" will be the sample mode. In a voting system in which plurality determines victory, the value of a single capital determines the winner, while multi-modal results will require some breaking procedures to take place.

Unlike the median, the mode concept makes sense for any random variable with the assumption of the value of the vector space, including the real number (one-dimensional vector space) and the integer (which can be considered embedded in real). For example, the distribution of points on the plane will usually have mean and mode, but the median concept does not apply. The median makes sense when there is a linear sequence of possible values. The generalization of the median concept to a higher dimensional space is the geometric median and the center point.

The uniqueness and limits

For some probability distributions, the expected value may be unlimited or undefined, but if defined, it is unique. The mean of the sample (until) is always defined. The median is a value such that the fraction does not exceed it and does not fall below it at least 1/2 each. It is not always unique, but it is never indefinitely or completely undefined. For a sample of data, this is the "middle" value when the list of values ​​is ordered in an increased value, where usually for a numerical average length list taken from two values ​​closest to "half". Finally, as said before, this mode is not necessarily unique. Certain pathological distributions (eg, Cantor distribution) do not have a specified mode at all. For a limited sample of data, the mode is one (or more) of the values ​​in the sample.

Properties

Assuming the definition, and for the uniqueness of simplicity, here are some of the most interesting properties.

• The three sizes have the following properties: If the random variable (or any value of the sample) is subjected to a linear or affine transform that replaces X by aX b , as well as mean, median, and mode.
• Except for very small samples, the modes are not sensitive to "outliers" (such as occasional, rare, false experiment readings). The median is also very strong with the outlier, while the mean is rather sensitive.
• In median continuous unimodal distributions often lie between mean and mode, about one-third of the path from mean to mode. In the formula, median? (2 ÃÆ'â € "mean mode)/3. This rule, because Karl Pearson, often applies to a slightly non-symmetrical distribution resembling a normal distribution, but not always true and in general all three statistics can appear in any order.
• For unimodal distributions, the modes are in ${\ displaystyle {\ sqrt {3}}}$

Example for oblique distribution

The example of oblique distribution is personal wealth: Few people are very rich, but among them there are very rich. However, many are rather poor.

A well-known distribution class that can be arbitrarily provided by a log-normal distribution. This is obtained by changing the random variable X having the normal distribution to be a random variable Y = e ^X . Then the logarithm of the random variable Y is normally distributed, hence its name.

Taking the meaning? from X to 0, median Y will be 1, regardless of standard deviation? from X . This is so because X has a symmetrical distribution, so the medium is also 0. The transformation from X to Y is monotonic, and so we find the median e ⁰ = 1 for Y .

Indeed, the median is about a third on the way from mean to mode.

Here, Pearson's rule of thumb failed.

Van Zwet Condition

Van Zwet lowers the inequality that provides sufficient conditions for this inequality. Inequality

Mode <= Median <= Mean

berlaku jika

F (Median - x ) F (Median x )> = 1

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

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Unimodal distribution

Hal ini dapat ditunjukkan untuk distribusi unimodal yang median ${\ displaystyle {\ tilde {X}}}  ÂÂ$ dan rerata ${\ displaystyle {\ bar {X}}}  ÂÂ$ terletak di dalam (3/5) ^1/2 ? 0,7746 standar deviasi satu sama lain. Dalam simbol,

${\ displaystyle {\ frac {\ left | {\ tilde {X}} - {\ bar {X}} \ right |} {\ sigma}} \ leq (3/5) ^ {1/2}}  ÂÂ$

di mana ${\ displaystyle | \ cdot |}  ÂÂ$ adalah nilai absolutnya.

Hubungan serupa berlaku antara median dan mode: mereka berada dalam 3 ^1/2 ? 1.732 standar deviasi satu sama lain:

${\ displaystyle {\ frac {\ left | {\ tilde {X}} - \ mathrm {mode} \ right |} {\ sigma}} \ leq 3 ^ {1/2}.}  ÂÂ$

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Histori

The term mode originated with Karl Pearson in 1895.

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See also

• Arg max
• Bimodal distribution
• Central tendency
  • Means
  • Median
• Descriptive statistics
• Moments (math)
• Summary statistics
• Unimodal function

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References

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External links

• Hazewinkel, Michiel, ed. (2001) [1994], "Fashion", Encyclopedia of Mathematics , Springer Science Business Media BV/Kluwer Academic Publishers, ISBN 978-1-55608-010-4
• Guides for Understanding & amp; Calculating Mode
• Weisstein, Eric W. "Mode". MathWorld .
• Means, Median and Short video modes from Khan Academy

Source of the article : Wikipedia