A descriptive statistic (in the

count noun In linguistics, a count noun (also countable noun) is a noun that can be modified by a quantity and that occurs in both singular and plural forms, and that can co-occur with quantificational determiners like ''every'', ''each'', ''several'', ...

sense) is a

summary statistic In descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate the largest amount of information as simply as possible. Statisticians commonly try to describe the observations in * a measure of ...

that quantitatively describes or summarizes features from a collection of

information Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random ...

, while descriptive statistics (in the mass noun sense) is the process of using and analysing those statistics. Descriptive statistic is distinguished from

inferential statistics Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers propertie ...

(or inductive statistics) by its aim to summarize a

sample Sample or samples may refer to: Base meaning * Sample (statistics), a subset of a population – complete data set * Sample (signal), a digital discrete sample of a continuous analog signal * Sample (material), a specimen or small quantity of s ...

, rather than use the data to learn about the

population Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a ...

that the sample of data is thought to represent. This generally means that descriptive statistics, unlike inferential statistics, is not developed on the basis of

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

, and are frequently

nonparametric statistics Nonparametric statistics is the branch of statistics that is not based solely on parametrized families of probability distributions (common examples of parameters are the mean and variance). Nonparametric statistics is based on either being dist ...

. Even when a data analysis draws its main conclusions using inferential statistics, descriptive statistics are generally also presented. For example, in papers reporting on human subjects, typically a table is included giving the overall

sample size Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a populatio ...

, sample sizes in important subgroups (e.g., for each treatment or exposure group), and demographic or clinical characteristics such as the

average In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7 ...

age, the proportion of subjects of each sex, the proportion of subjects with related

co-morbidities In medicine, comorbidity - from Latin morbus ("sickness"), co ("together"), -ity (as if - several sicknesses together) - is the presence of one or more additional conditions often co-occurring (that is, concomitant or concurrent) with a primary c ...

, etc. Some measures that are commonly used to describe a data set are measures of

central tendency In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications in ...

and measures of variability or

dispersion Dispersion may refer to: Economics and finance * Dispersion (finance), a measure for the statistical distribution of portfolio returns * Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variat ...

. Measures of central tendency include the

mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the '' ari ...

, median and

mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Arts and entertainment * '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine * ''Mode'' magazine, a fictional fashion magazine which is ...

, while measures of variability include the standard deviation (or

variance 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 numbe ...

), the minimum and maximum values of the variables,

kurtosis In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurt ...

and

skewness 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. For a unimodal ...

.Investopedia
Descriptive Statistics Terms
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Use in statistical analysis

Descriptive statistics provide simple summaries about the sample and about the observations that have been made. Such summaries may be either

quantitative Quantitative may refer to: * Quantitative research, scientific investigation of quantitative properties * Quantitative analysis (disambiguation) * Quantitative verse, a metrical system in poetry * Statistics, also known as quantitative analysis ...

, i.e.

summary statistics In descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate the largest amount of information as simply as possible. Statisticians commonly try to describe the observations in * a measure of ...

, or visual, i.e. simple-to-understand graphs. These summaries may either form the basis of the initial description of the data as part of a more extensive statistical analysis, or they may be sufficient in and of themselves for a particular investigation. For example, the shooting percentage in

basketball Basketball is a team sport in which two teams, most commonly of five players each, opposing one another on a rectangular court, compete with the primary objective of shooting a basketball (approximately in diameter) through the defender's h ...

is a descriptive statistic that summarizes the performance of a player or a team. This number is the number of shots made divided by the number of shots taken. For example, a player who shoots 33% is making approximately one shot in every three. The percentage summarizes or describes multiple discrete events. Consider also the grade point average. This single number describes the general performance of a student across the range of their course experiences. The use of descriptive and summary statistics has an extensive history and, indeed, the simple tabulation of populations and of economic data was the first way the topic of statistics appeared. More recently, a collection of summarisation techniques has been formulated under the heading of

exploratory data analysis In statistics, exploratory data analysis (EDA) is an approach of analyzing data sets to summarize their main characteristics, often using statistical graphics and other data visualization methods. A statistical model can be used or not, but pri ...

: an example of such a technique is the

box plot In descriptive statistics, a box plot or boxplot is a method for graphically demonstrating the locality, spread and skewness groups of numerical data through their quartiles. In addition to the box on a box plot, there can be lines (which are ca ...

. In the business world, descriptive statistics provides a useful summary of many types of data. For example, investors and brokers may use a historical account of return behaviour by performing empirical and analytical analyses on their investments in order to make better investing decisions in the future.

Univariate analysis

Univariate analysis Univariate analysis is perhaps the simplest form of statistical analysis. Like other forms of statistics, it can be inferential or descriptive. The key fact is that only one variable is involved. Univariate analysis can yield misleading results i ...

involves describing the

distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations * Probability distribution, the probability of a particular value or value range of a vari ...

of a single variable, including its central tendency (including the

, median, and

) and dispersion (including the

range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...

and

quartiles In statistics, a quartile is a type of quantile which divides the number of data points into four parts, or ''quarters'', of more-or-less equal size. The data must be ordered from smallest to largest to compute quartiles; as such, quartiles are ...

of the data-set, and measures of spread such as the

and standard deviation). The shape of the distribution may also be described via indices such as

and

. Characteristics of a variable's distribution may also be depicted in graphical or tabular format, including

histograms A histogram is an approximate representation of the distribution of numerical data. The term was first introduced by Karl Pearson. To construct a histogram, the first step is to " bin" (or "bucket") the range of values—that is, divide the en ...

and

stem-and-leaf display A stem-and-leaf display or stem-and-leaf plot is a device for presenting quantitative data in a graphical format, similar to a histogram, to assist in visualizing the shape of a distribution. They evolved from Arthur Bowley's work in the early ...

Bivariate and multivariate analysis

When a sample consists of more than one variable, descriptive statistics may be used to describe the relationship between pairs of variables. In this case, descriptive statistics include: * Cross-tabulations and

contingency tables In statistics, a contingency table (also known as a cross tabulation or crosstab) is a type of table in a matrix format that displays the (multivariate) frequency distribution of the variables. They are heavily used in survey research, business ...

* Graphical representation via

scatterplot A scatter plot (also called a scatterplot, scatter graph, scatter chart, scattergram, or scatter diagram) is a type of plot or mathematical diagram using Cartesian coordinates to display values for typically two variables for a set of data ...

s * Quantitative measures of dependence * Descriptions of

conditional distribution In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the ...

s The main reason for differentiating univariate and bivariate analysis is that bivariate analysis is not only a simple descriptive analysis, but also it describes the relationship between two different variables. Quantitative measures of dependence include correlation (such as

Pearson's r In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficient ...

when both variables are continuous, or

Spearman's rho In statistics, Spearman's rank correlation coefficient or Spearman's ''ρ'', named after Charles Spearman and often denoted by the Greek letter \rho (rho) or as r_s, is a nonparametric measure of rank correlation (statistical dependence between ...

if one or both are not) and

covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the ...

(which reflects the scale variables are measured on). The slope, in regression analysis, also reflects the relationship between variables. The unstandardised slope indicates the unit change in the criterion variable for a one unit change in the predictor. The standardised slope indicates this change in standardised (

z-score In statistics, the standard score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores above the mean ...

) units. Highly skewed data are often transformed by taking logarithms. The use of logarithms makes graphs more symmetrical and look more similar to the

normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...

, making them easier to interpret intuitively.

References

External links

* Descriptive Statistics Lecture: University of Pittsburgh Supercourse: http://www.pitt.edu/~super1/lecture/lec0421/index.htm {{Authority control