Statistics Definitions > Coefficient Definition

Contents:

1. Coefficient Definition in Statistics
2. Coefficient Definition in General Mathematics

Coefficient Definition: Statistics

A coefficient measures a certain property or characteristic of a data set, phenomenon, or process, given specified conditions. You’ll come across many different coefficient definitions, each of which is specific to a test or procedure:

Correlation coefficients tell us whether two sets of data are connected.

• The Pearson’s correlation coefficient(r) tells us the degree of correlation between two variables. It is probably the most widely used correlation coefficient.
• The Spearman rank correlation coefficient is the nonparametric version of the Pearson correlation coefficient.
• The point biserial correlation coefficient is another special case of Pearson’s correlation coefficient. It measures the relationship between one continuous variable and one naturally binary variable.
• The validity coefficient tells you how strong or weak your experiment results are.
• Moran’s I measures how one object is similar to others surrounding it.

Coefficients are also used as measures of reliability:

• The coefficient alpha (Cronbach’s alpha) is a way to measure reliability, or internal consistency of a psychometric instrument.
• The intraclass correlation coefficient measures the reliability of ratings or measurements for clusters — data that has been collected as groups or sorted into groups.
• Test-Retest reliability coefficients measure test consistency — the reliability of a test measured over time.

Coefficients that measure agreement (e.g. two judges agreeing on a certain ranking) include:

• The polychoric correlation coefficient measures agreement between multiple raters for ordinal variables.
• The tetrachoric correlation coefficient is used to measure agreement for binary variables.
• The coefficient of concordance is used to assess agreement between different raters.

Other types of coefficients:

• The coefficient of variation tells us how data points are dispersed around the mean.
• The gamma coefficient tells us how closely two pairs of data match.
• Pearson’s coefficient of skewness tells us how much and in what direction data is skewed.
• The Jaccard similarity coefficient compares members for two sets to see which members are shared and which are distinct.
• The Durbin Watson coefficient is a measure of autocorrelation (also called serial correlation) in residuals from regression analysis.
• The coefficient of determination is used to analyze how differences in one variable can be explained by a difference in a second variable.
• The standardized beta coefficient compares the strength of the effect of each individual independent variable to the dependent variable.
• The Phi Coefficient measures the association between two binary variables.
• The Kendall Rank Correlation Coefficient is a non-parametric measure of relationships between columns of ranked data.
• Lin’s concordance correlation coefficient measures bivariate pairs of observations relative to a “gold standard” test or measurement.
• Binomial coefficients tell us how many ways there are to choose k things out of larger set.
• The multinomial coefficients are used to find permutations when you have repeating values or duplicate items.
• The coefficient of dispersion, which actually has several different definitions; in general, it’s a statistic which measures dispersion.

Coefficient Definition in Mathematics

In mathematics, a coefficient is the number or multiplicative factor that goes before any variable in an equation or mathematical sentence.

If coefficients are numbers, they don’t change as the variables change, and we call them constants. They act upon the variables in a way that is always the same.

Examples of Coefficients

In the equation 4 x² + 3 x, both 4 and 3 are coefficients. The coefficient of x², 4, acts on the x² term and multiplies it by 4. The coefficient of x, 3, acts on the x term and multiplies it by 3.

In the equation 5 x⁴+ 567 x² + 24, the coefficients are 5, 567, and 24. 24 acts on the x⁴ term. 567 acts on the x². What about 24? It acts on a special, invisible term; the x⁰ term. Since any number to the 0th power is always 1, we normally condense this down to 1– or, when we write it with the coefficient, we skip it altogether. The coefficient of the x⁰ is called the constant coefficient.

In the equation x⁵ + 21 x ³ + 6 x ⁵ the coefficients are 1, 21, and 6. The fact that no number is written in front of x⁵ tells us immediately that the coefficient is the identity coefficient, the one number that leaves identical whatever it multiplies.

In the equation 24 x ⁸ + 56 ⁷ + 22 the coefficients are 24, 56, and 22. The leading coefficient is the coefficient of the highest-order term; the term in which our variable is raised to the highest power. In this case, that is x ⁸, so the leading coefficient is 8.

Nonconstant Coefficients

A coefficient can’t include the variables it acts upon, but it isn’t always a constant either. When it’s not a constant, the variables it includes are called parameters. In the equation y x⁴ + 4y x² + 3 x² + 4 x the coefficients are y, 4y+3, and 4.

Sources

Terms Factors and Coefficients

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