|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
|
| |
|
| In [[statistics]], a '''central tendency''' (or, more commonly, a '''measure of central tendency''') is a central value or a typical value for a [[probability distribution]].<ref name=Weisberg>Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications in the Social Sciences, ISBN 0-8039-4007-6 p.2 </ref> It is occasionally called an [[average]] or just the '''center''' of the distribution. The most common measures of central tendency are the [[arithmetic mean]], the [[median]] and the [[Mode (statistics)|mode]]. A central tendency can be calculated for either a finite set of values or for a theoretical distribution, such as the [[normal distribution]]. Occasionally authors use central tendency (or '''centrality'''), to mean "the tendency of quantitative data to cluster around some central value,". <ref name=Dodge>Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP for [[International Statistical Institute]]. ISBN 0-19-920613-9 (entry for "central tendency") </ref><ref name=Upton>Upton, G.; Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP ISBN 978-0-19-954145-4 (entry for "central tendency") </ref> This meaning might be expected from the usual dictionary definitions of the words [[wiktionary:tendency|tendency]] and [[wiktionary:centrality|centrality]]. Those authors may judge whether data has a strong or a weak central tendency based on the [[statistical dispersion]], as measured by the standard deviation or something similar.
| |
|
| |
|
| The term "central tendency" dates from the late 1920s.<ref name=Upton/>
| | Many sites present exactly the same boring templates or skins for that format. While the theme does not affect readers who appreciate your blog via FEED (unless they click-through to the blog), but it surely affects some other blog customer who says the blog from the web browser. [http://www.51q51q.com/selecting-the-top-web-design-on-your-website/ My Link] contains more concerning why to do it. The more unique looking a blog style is, the easier it is to build brand recognition and a great name that gets you into viral marketing. Your personal blog will not stick out in the group, if your blog looks like many other blogs and you will lose out on visitors. In the past I have been using free Wordpress templates myself. I applied slight improvements, but it never really separated my website from others enough to be really special. <br><br>The issue with a lot of websites is to find out what"s their primary specialty. What"s the primary subject a writer is authoring? Often there"s no-way to tell what a blog is all about - particularly when you enter a from an older publishing through a search-engine listing. The homepage might in the course of time have a tiny snippet of text describing the main topics a weblog, but aged articles don"t show this normally. To discover more, consider checking out: [http://www.dh-xd.com/today-could-be-the-right-time-for-a-vocation-in-graphic-design/ find here]. O-r the blog owner doesn"t even bother at all to share with what his main intention for your blog is. With a custom blog design this can easily be integral and helps to make the blog look different from someone else. <br><br>By using a custom header as a good example it can help to get the unique search you are seeking. To start with if it is unique it makes my blog stand out more. Second it can help you to thus get visitors to keep coming back and show what your blog is all about. Make your Blog look unique + make sure your readers know what your blog is all about. Visiting [http://www.sodahead.com//user/profile/3923291/cheaptylerlock150/?editMode=true tyler collins pastor] likely provides lessons you can use with your aunt. Simply because they look for a (one) publishing of their search term that is matched by your blog, it generally does not mean that your blog is truly of interest to them. Nevertheless you may be able to catch their interest by pointing out what your blog is really about. If you find their attention you may achieve a fresh audience that a) comes back to see more from your blog and b) ultimately tells others about your great blog..<br><br>If you have any type of questions concerning where and ways to utilize cost of health insurance ([http://www.fizzlive.com/magnificentwoma27 http://www.fizzlive.com]), you can contact us at our own web-site. |
| | |
| ==Measures of central tendency==
| |
| | |
| The following may be applied to one-dimensional data. Depending on the circumstances, it may be appropriate to transform the data before calculating a central tendency. Examples are squaring the values or taking logarithms. Whether a transformation is appropriate and what it should be depend heavily on the data being analyzed.
| |
| | |
| *[[Arithmetic mean]] (or simply, mean) – the sum of all measurements divided by the number of observations in the data set
| |
| *[[Median]] – the middle value that separates the higher half from the lower half of the data set. The median and the mode are the only measures of central tendency that can be used for [[Level of measurement#Ordinal scale|ordinal data]], in which values are ranked relative to each other but are not measured absolutely.
| |
| *[[Mode (statistics)|Mode]] – the most frequent value in the data set. This is the only central tendency measure that can be used with [[Level of measurement#Nominal scale|nominal data]], which have purely qualitative category assignments.
| |
| *[[Geometric mean]] – the [[Nth root|''n''th root]] of the product of the data values, where there are ''n'' of these. This measure is valid only for data that are measured absolutely on a strictly positive scale.
| |
| *[[Harmonic mean]] – the [[Multiplicative inverse|reciprocal]] of the arithmetic mean of the reciprocals of the data values. This measure too is valid only for data that are measured absolutely on a strictly positive scale.
| |
| *[[Weighted mean]] – an arithmetic mean that incorporates weighting to certain data elements
| |
| *[[Truncated mean]] – the arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded.
| |
| **[[Interquartile mean]] (a type of truncated mean)
| |
| *[[Midrange]] – the arithmetic mean of the maximum and minimum values of a data set.
| |
| *[[Midhinge]] – the arithmetic mean of the two [[quartile]]s.
| |
| *[[Trimean]] – the weighted arithmetic mean of the median and two quartiles.
| |
| *[[Winsorized mean]] – an arithmetic mean in which extreme values are replaced by values closer to the median.
| |
| | |
| Any of the above may be applied to each dimension of multi-dimensional data, but the results may not be invariant to rotations of the multi-dimensional space. In addition, there is the
| |
| | |
| *[[Geometric median]] - which minimizes the sum of distances to the data points. This is the same as the median when applied to one-dimensional data, but it is not the same as taking the median of each dimension independently. It is not invariant to different rescaling of the different dimensions.
| |
| | |
| ==Solutions to variational problems==
| |
| Several measures of central tendency can be characterized as solving a variational problem, in the sense of the [[calculus of variations]], namely minimizing variation from the center. That is, given a measure of [[statistical dispersion]], one asks for a measure of central tendency that minimizes variation: such that variation from the center is minimal among all choices of center. In a quip, "dispersion precedes location". In the sense of [[Lp space|''L''<sup>''p''</sup> spaces]], the correspondence is:
| |
| {| class="wikitable"
| |
| ! ''L''<sup>''p''</sup> !! dispersion !! central tendency
| |
| |-
| |
| ! ''L''<sup>1</sup>
| |
| | [[average absolute deviation]]
| |
| | [[median]]
| |
| |-
| |
| ! ''L''<sup>2</sup>
| |
| | [[standard deviation]]
| |
| | [[mean]]
| |
| |-
| |
| ! ''L''<sup>∞</sup>
| |
| | [[maximum deviation]]
| |
| | [[midrange]]
| |
| |}
| |
| | |
| Thus standard deviation about the mean is lower than standard deviation about any other point, and the maximum deviation about the midrange is lower than the maximum deviation about any other point. The uniqueness of this characterization of mean follows from [[convex optimization]]. Indeed, for a given (fixed) data set ''x'', the function
| |
| | |
| :<math>f_2(c) = \|x-c\|_2</math>
| |
| | |
| represents the dispersion about a constant value ''c'' relative to the ''L''<sup>2</sup> norm. Because the function ''ƒ''<sub>2</sub> is a strictly [[convex function|convex]] [[coercive function]], the minimizer exists and is unique.
| |
| | |
| Note that the median in this sense is not in general unique, and in fact any point between the two central points of a discrete distribution minimizes average absolute deviation. The dispersion in the ''L''<sup>1</sup> norm, given by
| |
| :<math>f_1(c) = \|x-c\|_1</math>
| |
| is not ''strictly'' convex, whereas strict convexity is needed to ensure uniqueness of the minimizer. In spite of this, the minimizer is unique for the ''L''<sup>∞</sup> norm.
| |
| | |
| | |
| ==Relationships between the mean, median and mode== | |
| {{Main|Nonparametric skew#Relationships between the mean, median and mode}}
| |
| | |
| For [[unimodal distribution]]s the following bounds are known and are sharp:<ref name=Johnson1951>Johnson NL, Rogers CA (1951) "The moment problem for unimodal distributions". ''Annals of Mathematical Statistics'', 22 (3) 433–439</ref>
| |
| | |
| : <math> \frac{| \theta - \mu |}{ \sigma } \le \sqrt{ 3 } ,</math>
| |
| | |
| : <math> \frac{| \nu - \mu |}{ \sigma } \le \sqrt{ 0.6 } ,</math>
| |
| | |
| : <math> \frac{| \theta - \nu |}{ \sigma } \le \sqrt{ 3 } ,</math>
| |
| | |
| where ''μ'' is the mean, ''ν'' is the median, ''θ'' is the mode, and ''σ'' is the standard deviation.
| |
| | |
| For every distribution,<ref name=Hotelling1932>Hotelling H, Solomons LM (1932) The limits of a measure of skewness. Annals Math Stat 3, 141–114</ref><ref name=Garver1932>Garver (1932) Concerning the limits of a mesuare of skewness. Ann Math Stats 3(4) 141–142</ref>
| |
| | |
| : <math> \frac{| \nu - \mu |}{ \sigma } \le 1.</math>
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| {{Statistics|descriptive}}
| |
| | |
| {{DEFAULTSORT:Central Tendency}}
| |
| [[Category:Summary statistics]]
| |
| [[Category:Multivariate statistics]]
| |
| [[Category:Statistical terminology]]
| |
| | |
| [[de:Lagemaß]]
| |
| [[eu:Zentro-neurri]]
| |
| [[fi:Keskiluku]]
| |
Many sites present exactly the same boring templates or skins for that format. While the theme does not affect readers who appreciate your blog via FEED (unless they click-through to the blog), but it surely affects some other blog customer who says the blog from the web browser. My Link contains more concerning why to do it. The more unique looking a blog style is, the easier it is to build brand recognition and a great name that gets you into viral marketing. Your personal blog will not stick out in the group, if your blog looks like many other blogs and you will lose out on visitors. In the past I have been using free Wordpress templates myself. I applied slight improvements, but it never really separated my website from others enough to be really special.
The issue with a lot of websites is to find out what"s their primary specialty. What"s the primary subject a writer is authoring? Often there"s no-way to tell what a blog is all about - particularly when you enter a from an older publishing through a search-engine listing. The homepage might in the course of time have a tiny snippet of text describing the main topics a weblog, but aged articles don"t show this normally. To discover more, consider checking out: find here. O-r the blog owner doesn"t even bother at all to share with what his main intention for your blog is. With a custom blog design this can easily be integral and helps to make the blog look different from someone else.
By using a custom header as a good example it can help to get the unique search you are seeking. To start with if it is unique it makes my blog stand out more. Second it can help you to thus get visitors to keep coming back and show what your blog is all about. Make your Blog look unique + make sure your readers know what your blog is all about. Visiting tyler collins pastor likely provides lessons you can use with your aunt. Simply because they look for a (one) publishing of their search term that is matched by your blog, it generally does not mean that your blog is truly of interest to them. Nevertheless you may be able to catch their interest by pointing out what your blog is really about. If you find their attention you may achieve a fresh audience that a) comes back to see more from your blog and b) ultimately tells others about your great blog..
If you have any type of questions concerning where and ways to utilize cost of health insurance (http://www.fizzlive.com), you can contact us at our own web-site.