Statistics Definitions > Coefficient Definition

**Contents:**

## 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 x^{2} + 3 x, both 4 and 3 are coefficients. The coefficient of x^{2}, 4, acts on the x^{2} 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 x^{4}+ 567 x^{2} + 24, the coefficients are 5, 567, and 24. 24 acts on the x^{4} term. 567 acts on the x^{2}. What about 24? It acts on a special, invisible term; the x^{0} 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 x^{0} is called the **constant coefficient. **

In the equation x^{5} + 21 x ^{3} + 6 x ^{5} the coefficients are 1, 21, and 6. The fact that no number is written in front of x^{5} 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 x^{4} + 4y x^{2} + 3 x^{2} + 4 x the coefficients are y, 4y+3, and 4.

## Sources

Terms Factors and Coefficients

------------------------------------------------------------------------------**Need help with a homework or test question?** Chegg offers 30 minutes of free tutoring, so you can try them out before committing to a subscription. Click here for more details.

If you prefer an **online interactive environment** to learn R and statistics, this *free R Tutorial by Datacamp* is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try *this Statistics with R track*.

*Facebook page*.