# What’s a Coefficient?

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This can be related to uniformity of velocity profile, temperature distribution, gas species (such as ammonia for an SCR, or activated carbon injection for mercury absorption), and other flow-related parameters. The Percent RMS also is used to assess flow uniformity in combustion systems, HVAC systems, ductwork, inlets to fans and filters, air handling units, etc. where performance of the equipment is influenced by the incoming flow distribution. A positive correlation indicates that the two move in the same direction, with a value of 1 denoting a perfect positive correlation. A value of -1 shows a perfect negative, or inverse, correlation, while zero means no linear correlation exists. Nor does the correlation coefficient show what proportion of the variation in the dependent variable is attributable to the independent variable. That’s shown by the coefficient of determination, also known as R-squared, which is simply the correlation coefficient squared.

The problem here is that you have divided by a relative value rather than an absolute. The correlation coefficient is particularly helpful in assessing and managing investment risks. For example, modern portfolio theory suggests diversification can reduce the volatility of a portfolio’s returns, curbing risk. The correlation coefficient between historical returns can indicate whether adding an investment to a portfolio will improve its diversification. The correlation coefficient does not describe the slope of the line of best fit; the slope can be determined with the least squares method in regression analysis. Measurements that are log-normally distributed exhibit stationary CV; in contrast, SD varies depending upon the expected value of measurements.

## Examples of coefficient in a Sentence

In the context of differential equations, an equation can often be written as equating to zero a polynomial in the unknown functions and their derivatives. In this case, the coefficients of the differential equation are the coefficients of this polynomial, and are generally non-constant functions. For avoiding confusion, the coefficient that is not attached to unknown functions and their derivative is generally called the constant term rather the constant coefficient. In particular, in a linear differential equation with constant coefficient, the constant term is generally not supposed to be a constant function. If, for example, the data sets are temperature readings from two different sensors (a Celsius sensor and a Fahrenheit sensor) and you want to know which sensor is better by picking the one with the least variance, then you will be misled if you use CV.

• Calculating the correlation coefficient for these variables based on market data reveals a moderate and inconsistent correlation over lengthy periods.
• Those relationships can be analyzed using nonparametric methods, such as Spearman’s correlation coefficient, the Kendall rank correlation coefficient, or a polychoric correlation coefficient.
• While a standard deviation (SD) can be measured in Kelvin, Celsius, or Fahrenheit, the value computed is only applicable to that scale.

In many scenarios, coefficients are numbers (as is the case for each term of the previous example), although they could be parameters of the problem—or any expression in these parameters. In such a case, one must clearly distinguish between symbols representing variables and symbols representing parameters. Following René Descartes, the variables are often denoted by x, y, …, and the parameters by a, b, c, …, but this is not always the case. For example, if y is considered a parameter in the above expression, then the coefficient of x would be −3y, and the constant coefficient (with respect to x) would be 1.5 + y. Correlation coefficients play a key role in portfolio risk assessments and quantitative trading strategies. For example, some portfolio managers will monitor the correlation coefficients of their holdings to limit a portfolio’s volatility and risk.

## Linear algebra

The values of -1 (for a negative correlation) and 1 (for a positive one) describe perfect fits in which all data points align in a straight line, indicating that the variables are perfectly correlated. In other words, the relationship is so predictable that the value of one variable can be determined from the matched value of the other. The closer the correlation coefficient is to zero the weaker the correlation, until at zero no linear relationship exists at all. The coefficient of variation is useful because the standard deviation of data must always be understood in the context of the mean of the data. In contrast, the actual value of the CV is independent of the unit in which the measurement has been taken, so it is a dimensionless number.

The coefficient of variation should be computed only for data measured on scales that have a meaningful zero (ratio scale) and hence allow relative comparison of two measurements (i.e., division of one measurement by the other). On the other hand, Kelvin temperature has a meaningful zero, the complete absence of thermal energy, and thus is a ratio scale. In plain language, it is meaningful to say that 20 Kelvin is twice as hot as 10 Kelvin, but only in this scale with a true absolute zero.

## How Do You Calculate the Correlation Coefficient?

A correlation coefficient of -1 describes a perfect negative, or inverse, correlation, with values in one series rising as those in the other decline, and vice versa. The correlation coefficient is calculated by determining the covariance of the variables and dividing that number by the product of those variables’ standard deviations. To calculate the Pearson correlation, start by determining each variable’s standard deviation as well as the covariance between them. The correlation coefficient is covariance divided by the product of the two variables’ standard deviations. The further the coefficient is from zero, whether it is positive or negative, the better the fit and the greater the correlation.

While a standard deviation (SD) can be measured in Kelvin, Celsius, or Fahrenheit, the value computed is only applicable to that scale. Different types of correlation coefficients are used to assess correlation based on the properties of the compared data. By far the most common is the Pearson coefficient, or Pearson’s r, which measures the strength and direction of a linear relationship between two variables. The Pearson coefficient cannot assess nonlinear associations between variables and cannot differentiate between dependent and independent variables. The correlation coefficient is a statistical measure of the strength of a linear relationship between two variables.

The constant coefficient, also known as constant term or simply constant is the quantity not attached to variables in an expression. For example, the constant coefficients of the expressions above are the number 3 and the parameter c, respectively. The coefficient attached to the highest degree of the variable in a polynomial is referred to as the leading coefficient. For example, in the expressions above, the leading coefficients are 2 and a, respectively. In Fluid Dynamics, the CV, also referred to as Percent RMS, %RMS, %RMS Uniformity, or Velocity RMS, is a useful determination of flow uniformity for industrial processes. The term is used widely in the design of pollution control equipment, such as electrostatic precipitators (ESPs),[16] selective catalytic reduction (SCR), scrubbers, and similar devices.

## Examples

It can also be distorted by outliers—data points far outside the scatterplot of a distribution. Those relationships can be analyzed using nonparametric methods, such as Spearman’s correlation coefficient, the Kendall rank correlation coefficient, or a polychoric correlation coefficient. The sample standard deviations are still 15.81 and 28.46, respectively, because the standard deviation is not affected by a constant offset.

Correlation coefficients are used in science and in finance to assess the degree of association between two variables, factors, or data sets. For example, since high oil prices are favorable for crude producers, one might assume the correlation between oil prices and forward returns on oil stocks is strongly positive. Calculating the correlation coefficient for these variables based on market data reveals a moderate and inconsistent correlation over lengthy periods. The Pearson correlation coefficient can’t be used to assess nonlinear associations or those arising from sampled data not subject to a normal distribution.

## Grammar Terms You Used to Know, But Forgot

If you want to create a correlation matrix across a range of data sets, Excel has a Data Analysis plugin on the Data tab, under Analyze. Correlation does not imply causation, as the saying goes, and the Pearson coefficient cannot determine whether one of the correlated variables is dependent on the other. Assessments of correlation strength based on the correlation coefficient value vary by application. In physics and chemistry, a correlation coefficient should be lower than -0.9 or higher than 0.9 for the correlation to be considered meaningful, while in social sciences the threshold could be as high as -0.5 and as low as 0.5. In the examples below, we will take the values given as randomly chosen from a larger population of values.