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

**Coefficients **are numbers or letters that we use to multiply a variable. A **variable** is defined as a symbol (like *x *or *y*) that we use to describe any number. In a function the coefficient is located next to and in front of the variable. Single numbers, variables or the product of a number and a variable are known as **terms**.

## Coefficient Example

3*x* – 1*xy* + 2.3 + *y*

In the function above the first two coefficients are 3 and 1. Notice that 3 is next to and in front of variable *x*, while 1 is next to and in front of *xy*. The third coefficient is 2.3. This is known as a **constant coefficient** since its value will not change since it is not being multiplied by a variable. Simply defined, a **constant** is a term without a variable. Looking at the fourth term (*y*) we see that there is no coefficient. In such instances the coefficient is considered to be 1 since multiplying by 1 would not change the term.

## Like Terms

Looking at the four terms of the function above we can see that there are no like terms. **Like terms** are terms that have the same variable raised to the same power. Since the terms are 3*x*, 1*xy*, 2.3 and* y* and all have different variables there are no like terms in this function.

## Example of Like Terms

2*xy*^{2} + 3*xy*^{2} – 5*xy*^{2}

Notice that the coefficients (2, 3 and 5) are all different values. However, the function contains like terms since the variable (*xy*) for each term are raised to the second power.

Above we defined coefficients as being either numbers or letters. You may encounter a function with no numerical value in the coefficient spot. Simply treat the letter located in front of and next to the variable as the coefficient. For example:

a*x* + b*x* + *c*

In the function above *a* and *b* are coefficients while *x* is a variable. The third term (c) does not have a coefficient so the coefficient is considered to be 1.

## Sources

Terms Factors and Coefficients

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