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The "average" number; found by adding all data points and dividing by the number of data points. The middle number; found by ordering all data points and picking out the one in the middle or if there are two middle numbers, taking the mean of those two numbers. The most frequent number—that is, the number that occurs the highest number of times. Want to learn more about mean, median, and mode? Check out the more in-depth examples below, or check out this video explanation.

There are many different types of mean, but usually when people say mean, they are talking about the arithmetic mean. The arithmetic mean is the sum of all of the data points divided by the number of data points. Here's the same formula written more formally:. Find the mean of this data: Start by adding the data: There are 4 4 4 data points. The mean is 3 3 3. What is the arithmetic mean of the following numbers? Want to practice more of these? Check out this exercise on calculating the mean.

How to find the mean, median, mode and range

The median is the middle point in a dataset—half of the data points are smaller than the median and half of the data points are larger. Let's look at some more examples. The Jameson family drove through 7 states on their summer vacation. Gasoline prices varied from state to state. What is the median gasoline price? There were 3 states with higher gasoline prices and 3 with lower prices.

During the first marking period, Nicole's math quiz scores were 90, 92, 93, 88, 95, 88, 97, 87, and What was the median quiz score?

Measuring center in quantitative data

The median quiz score was Four quiz scores were higher than 92 and four were lower. In each of the examples above, there is an odd number of items in each data set. In Example 1, there are 7 numbers in the data set; in Example 2 there are 9 numbers. Let's look at some examples in which there is an even number of items in the data set. A marathon race was completed by 4 participants.

Previously, this article discussed the univariate median, when the sample or population had one-dimension. When the dimension is two or higher, there are multiple concepts that extend the definition of the univariate median; each such multivariate median agrees with the univariate median when the dimension is exactly one. Medoid is defined as. The medoid is often used in clustering using the k-medoid algorithm.

Two Numbers in the Middle

The marginal median is defined for vectors defined with respect to a fixed set of coordinates. A marginal median is defined to be the vector whose components are univariate medians. The marginal median is easy to compute, and its properties were studied by Puri and Sen. For N vectors in a normed vector space , a spatial median minimizes the average distance.


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The spatial median is unique when the data-set's dimension is two or more and the norm is the Euclidean norm or another strictly convex norm. Other names are used especially for finite sets of points: The spatial median is a robust and highly efficient estimator of a central tendency of a population. An alternative generalization of the spatial median in higher dimensions that does not relate to a particular metric is the centerpoint.

When dealing with a discrete variable, it is sometimes useful to regard the observed values as being midpoints of underlying continuous intervals. An example of this is a Likert scale, on which opinions or preferences are expressed on a scale with a set number of possible responses. If the scale consists of the positive integers, an observation of 3 might be regarded as representing the interval from 2.

It is possible to estimate the median of the underlying variable. But the interpolated median is somewhere between 2. In other words, we split up the interval width pro rata to the numbers of observations. For univariate distributions that are symmetric about one median, the Hodges—Lehmann estimator is a robust and highly efficient estimator of the population median; for non-symmetric distributions, the Hodges—Lehmann estimator is a robust and highly efficient estimator of the population pseudo-median , which is the median of a symmetrized distribution and which is close to the population median.

Find the Median (the Middle Number)!

The Theil—Sen estimator is a method for robust linear regression based on finding medians of slopes. In the context of image processing of monochrome raster images there is a type of noise, known as the salt and pepper noise , when each pixel independently becomes black with some small probability or white with some small probability , and is unchanged otherwise with the probability close to 1. In cluster analysis , the k-medians clustering algorithm provides a way of defining clusters, in which the criterion of maximising the distance between cluster-means that is used in k-means clustering , is replaced by maximising the distance between cluster-medians.

This is a method of robust regression. The line could then be adjusted to fit the majority of the points in the data set. Nair and Shrivastava in suggested a similar idea but instead advocated dividing the sample into three equal parts before calculating the means of the subsamples. Any mean -unbiased estimator minimizes the risk expected loss with respect to the squared-error loss function , as observed by Gauss. A median -unbiased estimator minimizes the risk with respect to the absolute-deviation loss function, as observed by Laplace.

Other loss functions are used in statistical theory , particularly in robust statistics. The theory of median-unbiased estimators was revived by George W. This requirement seems for most purposes to accomplish as much as the mean-unbiased requirement and has the additional property that it is invariant under one-to-one transformation. Further properties of median-unbiased estimators have been reported. There are methods of construction median-unbiased estimators that are optimal in a sense analogous to minimum-variance property considered for mean-unbiased estimators.

Such constructions exist for probability distributions having monotone likelihood-functions. The procedure holds for a smaller class of probability distributions than does the Rao—Blackwell procedure but for a larger class of loss functions. The idea of the median appeared in the 13th century in the Talmud [54] [55] further [ citation needed ] for possible older mentions.

The idea of the median also appeared later in Edward Wright 's book on navigation Certaine Errors in Navigation in in a section concerning the determination of location with a compass. Wright felt that this value was the most likely to be the correct value in a series of observations. In , Roger Joseph Boscovich developed a regression method based on the L1 norm and therefore implicitly on the median. In , Laplace suggested the median be used as the standard estimator of the value of a posterior pdf.

Gustav Theodor Fechner used the median Centralwerth in sociological and psychological phenomena. Gustav Fechner popularized the median into the formal analysis of data, although it had been used previously by Laplace. Francis Galton used the English term median in , [60] having earlier used the terms middle-most value in , and the medium in From Wikipedia, the free encyclopedia.


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This article is about the statistical concept. For other uses, see Median disambiguation. Sheskin 27 August Handbook of Parametric and Nonparametric Statistical Procedures: Retrieved 25 February Statistical Methods for Spc and Tqm. Experiment, Design and Statistics in Psychology. Retrieved 16 March Random Vectors and Random Sequences.

Statistical data analysis based on the L1-norm and related methods". Laplace, Fisher and the Discovery of the Concept of Sufficiency". Kendall's Advanced Theory of Statistics.

BBC Bitesize - How to find the mean, median, mode and range

The Jackknife, the Bootstrap and other Resampling Plans. Another Look at the Jackknife". Probab Theory Related Fields. Robust nonparametric statistical methods.