Statistics Definitions > Coefficient Definition
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.
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.
3x – 1xy + 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.
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 3x, 1xy, 2.3 and y and all have different variables there are no like terms in this function.
Example of Like Terms
2xy2 + 3xy2 – 5xy2
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:
ax + bx + 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.
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