View source for Univariate (statistics)

'''Univariate''' is a term commonly used in statistics to describe a type of data which consists of observations on only a single characteristic or attribute. A simple example of univariate data would be the salaries of workers in industry.<ref>{{cite book|last1=Kachigan|first1=Sam Kash|title=Statistical analysis : an interdisciplinary introduction to univariate & multivariate methods|date=1986|publisher=Radius Press|location=New York|isbn=0-942154-99-1}}</ref> Like all the other data, univariate data can be visualized using graphs, images or other analysis tools after the data is measured, collected, reported, and analyzed.<ref name="Introductory statistics">{{cite book|last1=Lacke|first1=Prem S. Mann; with the help of Christopher Jay|title=Introductory statistics.|date=2010|publisher=John Wiley & Sons|location=Hoboken, NJ|isbn=978-0-470-44466-5|edition=7th}}</ref>
== Univariate data types ==
Some univariate data consists of numbers (such as the height of 65 inches or the weight of 100 pounds), while others are nonnumerical (such as eye colors of brown or blue). Generally, the terms categorical univariate data and numerical univariate data are used to distinguish between these types.

=== Categorical univariate data ===
Categorical univariate data consists of non-numerical [[Observation|observation]]s that may be placed in categories. It includes labels or names used to identify an attribute of each element. Categorical univariate data usually use either [[Nominal number|nominal]] or [[Ordinal number|ordinal]] scale of measurement.<ref name="Cengage Learning">{{cite book|last1=Anderson|first1=David R.|last2=Sweeney|first2=Dennis J.|last3=Williams|first3=Thomas A.|title=Statistics For Business & Economics|publisher=Cengage Learning|isbn=978-0-324-80926-8|pages=1018|edition=Tenth}}</ref>

=== Numerical univariate data ===
Numerical univariate data consists of observations that are numbers. They are obtained using either [[Interval (mathematics)|interval]] or [[Ratio|ratio]] scale of measurement. This type of univariate data can be classified even further into two subcategories: discrete and continuous.<ref name="Introductory statistics"/> A numerical univariate data is discrete if the set of all possible values is [[Finite set|finite]] or countably [[Infinite set|infinite]]. Discrete univariate data are usually associated with counting (such as the number of books read by a person). A numerical univariate data is continuous if the set of all possible values is an interval of numbers. Continuous univariate data are usually associated with measuring (such as the weights of people).

== Data analysis and applications {{anchor|Analysis}}==

Univariate analysis is the simplest form of analyzing data. ''Uni'' means "one", so the data has only one variable (''[[Univariate|univariate]]'').<ref name=stathow>{{cite web|url=http://www.statisticshowto.com/univariate/|title=Univariate analysis|website=stathow}}</ref> Univariate data requires to analyze each [[Variable (mathematics)|variable]] separately. Data is gathered for the purpose of answering a question, or more specifically, a research question. Univariate data does not answer research questions about relationships between variables, but rather it is used to describe one characteristic or attribute that varies from observation to observation.<ref name=study.com>{{cite web|url=http://study.com/academy/lesson/univariate-data-definition-analysis-examples.html|title=Univariate Data|website=study.com}}</ref> Usually there are two purposes that a researcher can look for. The first one is to answer a research question with descriptive study and the second one is to get knowledge about how [[Social:Variable and attribute (research)|attribute]] varies with individual effect of a variable in [[Regression analysis]]. There are some ways to describe patterns found in univariate data which include graphical methods, measures of central tendency and measures of variability.<ref>{{cite web|last1=Trochim|first1=William|title=Descriptive Statistics|url=http://www.socialresearchmethods.net/kb/statdesc.php|website=Web Center for Social Research Methods|access-date=15 February 2017}}</ref>

Like other forms of statistics, it can be inferential or [[Descriptive statistics|descriptive]]. The key fact is that only one variable is involved.

Univariate analysis can yield misleading results in cases in which [[Multivariate analysis|multivariate analysis]] is more appropriate.

=== Measures of central tendency ===
Central tendency is one of the most common numerical descriptive measures. It's used to estimate the central location of the univariate data by the calculation of [[Mean|mean]], [[Median|median]] and [[Mode (statistics)|mode]].<ref>{{cite book|last1=Stepanski|first1=Norm O'Rourke, Larry Hatcher, Edward J.|title=A step-by-step approach to using SAS for univariate & multivariate statistics|date=2005|publisher=Wiley-Interscience|location=New York|isbn=1-59047-417-1|edition=2nd}}</ref> Each of these calculations has its own advantages and limitations. The mean has the advantage that its calculation includes each value of the data set, but it is particularly susceptible to the influence of [[Outlier|outlier]]s. The median is a better measure when the data set contains [[Outlier|outlier]]s. The mode is simple to locate. The important thing is that it's not restricted to using only one of these measure of central tendency. If the data being analyzed is categorical, then the only measure of central tendency that can be used is the mode. However, if the data is numerical in nature ([[Ordinal number|ordinal]] or [[Interval (mathematics)|interval]]/[[Ratio|ratio]]) then the mode, median, or mean can all be used to describe the data. Using more than one of these measures provides a more accurate descriptive summary of central tendency for the univariate.<ref>{{cite book|last1=Longnecker|first1=R. Lyman Ott, Michael|title=An introduction to statistical methods and data analysis|date=2009|publisher=Brooks/Cole|location=Pacific Grove, Calif.|isbn=978-0-495-10914-3|edition=6th ed., International}}</ref>

=== Measures of variability ===
A measure of variability or [[Statistical dispersion|dispersion]] (deviation from the mean) of a univariate data set can reveal the shape of a univariate data distribution more sufficiently. It will provide some information about the variation among data values. The measures of variability together with the measures of central tendency give a better picture of the data than the measures of central tendency alone.<ref>{{cite book|title=Statistical Data Analysis A Practical Guide.|last2=Militky|first2=Jirí|date=2011|publisher=Woodhead Pub Ltd|isbn=978-0-85709-109-3|location=New Delhi|last1=Meloun|first1=Milan}}</ref> The three most frequently used measures of variability are [[Range (statistics)|range]], [[Variance|variance]] and [[Standard deviation|standard deviation]].<ref>{{cite book|title=Statistics|date=2007|publisher=Norton|isbn=978-0-393-92972-0|edition=4.|location=New York [u.a.]|last1=Purves|first1=David Freedman; Robert Pisani; Roger}}</ref> The appropriateness of each measure would depend on the type of data, the shape of the distribution of data and which measure of central tendency are being used. If the data is categorical, then there is no measure of variability to report. For data that is numerical, all three measures are possible. If the distribution of data is symmetrical, then the measures of variability are usually the variance and standard deviation. However, if the data are [[Skewness|skewed]], then the measure of variability that would be appropriate for that data set is the range.<ref name="Cengage Learning"/>

=== Descriptive methods ===
Descriptive statistics describe a sample or population. They can be part of [[Exploratory data analysis|exploratory data analysis]].<ref name = "Everitt">{{cite book | last = Everitt | first = Brian | title = The Cambridge Dictionary of Statistics | publisher = Cambridge University Press | location = Cambridge, UK New York | year = 1998 | isbn = 0521593468 | url-access = registration | url = https://archive.org/details/cambridgediction00ever_0 }}</ref>

The appropriate statistic depends on the [[Level of measurement|level of measurement]]. For nominal variables, a frequency table and a listing of the [[Mode (statistics)|mode(s)]] is sufficient. For ordinal variables the [[Median|median]] can be calculated as a measure of [[Central tendency|central tendency]] and the [[Range (statistics)|range]] (and variations of it) as a measure of dispersion. For interval level variables, the [[Arithmetic mean|arithmetic mean]] (average) and [[Standard deviation|standard deviation]] are added to the toolbox and, for ratio level variables, we add the [[Geometric mean|geometric mean]] and [[Harmonic mean|harmonic mean]] as measures of central tendency and the [[Coefficient of variation|coefficient of variation]] as a measure of dispersion.

For interval and ratio level data, further descriptors include the variable's [[Skewness|skewness]] and [[Kurtosis|kurtosis]].

=== Inferential methods ===
Inferential methods allow us to infer from a sample to a population.<ref name = "Everitt" /> For a nominal variable a one-way chi-square (goodness of fit) test can help determine if our sample matches that of some population.<ref>{{Cite web|url=http://www.vassarstats.net/csfit.html|title = One-Way Chi-Square}}</ref> For interval and ratio level data, a one-sample t-test can let us infer whether the mean in our sample matches some proposed number (typically 0). Other available tests of location include the one-sample [[Sign test|sign test]] and Wilcoxon signed rank test.

== Graphical methods==
The most frequently used graphical illustrations for univariate data are:

=== Frequency distribution tables  ===
Frequency is how many times a number occurs. The frequency of an observation in statistics tells us the number of times the observation occurs in the data. For example, in the following list of numbers {''1, 2, 3, 4, 6, 9, 9, 8, 5, 1, 1, 9, 9, 0, 6, 9''}, the frequency of the number 9 is 5 (because it occurs 5 times in this data set).

=== Bar charts ===
[[File:Barplot.jpg|thumb|This is an example of barplot.]]
Bar chart is a [[Graph (discrete mathematics)|graph]] consisting of rectangular bars. These bars actually represents [[Number|number]] or percentage of observations of existing categories in a variable. The [[Physics:Length|length]] or [[Height|height]] of bars gives a visual representation of the proportional differences among categories.

=== Histograms ===
[[File:Histarman2.jpg|thumb|histogram]]
Histograms are used to estimate distribution of the data, with the frequency of values assigned to a value range called a [[Data binning|bin]].<ref>{{cite book|last1=Diez|first1=David M.|last2=Barr|first2=Christopher D.|last3=Çetinkaya-Rundel|first3=Mine|title=OpenIntro Statistics|date=2015|publisher=OpenIntro, Inc.|isbn=978-1-9434-5003-9|page=30|edition=3rd}}</ref>

=== Pie charts ===
Pie chart is a circle divided into portions that represent the relative frequencies or percentages of a population or a sample belonging to different categories.
[[File:Figure 2023-11-12 031825.png|thumb|This is a pie chart of the most common fairy tale animals]]

== Univariate distributions ==
[[Univariate distribution]] is a dispersal type of a single random variable described either with a [[Probability mass function|probability mass function]] (pmf) for discrete probability distribution, or [[Physics:Probability density function|probability density function]] (pdf) for continuous probability distribution.<ref>{{cite book|last1=Samaniego|first1=Francisco J.|title=Stochastic modeling and mathematical statistics : a text for statisticians and quantitative scientists|date=2014|publisher=CRC Press|location=Boca Raton|isbn=978-1-4665-6046-8|page=167}}</ref> It is not to be confused with multivariate distribution.

===Common discrete distributions===
Uniform distribution (discrete) <br>
[[Bernoulli distribution]] <br>
[[Binomial distribution]] <br>
[[Geometric distribution]] <br>
[[Negative binomial distribution]] <br>
[[Poisson distribution]] <br>
[[Hypergeometric distribution]] <br>
[[Zeta distribution]]

===Common continuous distributions===
[[Uniform distribution (continuous)]] <br>
[[Normal distribution]] <br>
[[Gamma distribution]] <br>
[[Exponential distribution]] <br>
[[Weibull distribution]] <br>
[[Cauchy distribution]] <br>
[[Beta distribution]]

==See also==
* [[Univariate]] 
* [[Univariate distribution]]
* [[Bivariate analysis]]
* [[Multivariate analysis]]
* [[List of probability distributions]]

==References==
<references />

[[Category:Mathematical terminology]]

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