Technology & Engineering
Covariance and Correlation
Covariance and correlation are statistical measures used to quantify the relationship between two variables. Covariance measures how much two variables change together, while correlation standardizes this measure to a range of -1 to 1, indicating the strength and direction of the relationship. Both are important in analyzing data and understanding the associations between different technological and engineering factors.
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8 Key excerpts on "Covariance and Correlation"
- James E. De Muth(Author)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
Covariance is a measure of the strength of association between two continuous variables. It measures the linear relationship between two random variables. The term “linear dependence” is sometime used to refer to covariance which can serve as a measure of dependence between two variables. It provides the goodness of fit for the best possible linear function between two variables. Covariance is calculated as follows for population data: N ) y )( x ( ) Y , X cov( y i x i − − = μ μ Eq. 13.1 For sample data it will be slightly larger by dividing by n − 1. When there is a strong comparison, large positive deviations for x -values will match with large positive Chapter 13 314 deviations for y -values. At the same time large negatives will match with large negatives. A positive covariance indicates that values above the mean for one variable are associated with above the mean values for a second variable and below the mean values are similarly associated. Conversely, a negative covariance indicates that above the mean values of one variable are associated with below the mean values of the second variable. If two variables are completely unrelated to each other the covariance is zero. Values can range from negative infinity to positive infinity. It is difficult to compare the covariance between the x - and y -variables if they differ in magnitude (e.g., comparing patient weights in kilograms to heights in centimeters); therefore some type of standardization is required and this will be discussed below. This is accomplished by creating standardized values (subtracting the mean from each value and dividing by the standard deviation). This will result in a mean of 0 and standard deviation of 1. Covariances are useful when applied in the analysis of covariance (ANCOVA) for comparing two or more linear regression lines. The ANCOVA is used to compare two or more linear regression lines.
- Alan R. Jones(Author)
- 2018(Publication Date)
- Routledge(Publisher)
If two drivers are perfectly correlated with each other the one of them is redundant as it adds no further information to the behaviour of the entity we want to estimate. Definition 5.1 Correlation Correlation is a statistical relationship in which the values of two or more variables exhibit a tendency to change in relationship with one other.These variables are said to be positively (or directly) correlated if the values tend to move in the same direction, and negatively (or inversely) correlated if they tend to move in opposite directions. Pattern observed Properties Conclusion Data pattern goes from bot- tom left to top right Both val- ues increase together; both values decrease together The two variables are positively cor- related (not neces- sarily linearly) Linearity, Dependence and Correlation | 257 Usually estimators often like to keep things simple and given the chance will try to think in straight lines and look for the ‘Best Fit’ straight-line relationships between two variables. This is, of course, an oversimplification as the estimating GEEKs will tell us ( What’s an estimating GEEK? Someone who has ‘Got estimating experience & knowledge’, or, to put it another way, those of us who have suffered estimator’s war wounds and learnt the hard way from earlier mistakes.) Consequently, we will be looking at Correlation from the perspective of both linear and non-linear relationships, beginning with Linear Correlation. In order to understand what the Linear Correlation Coefficient is measuring, we are better considering another statistic first – Covariance, which measures how much of the variance in one variable can be explained by a variance in the other variable. 5.1 Covariance If two variables are related, then we would expect that they ‘ track together’ in a broad sense.
eBook - PDFQuantitative Techniques in Business, Management and Finance
A Case-Study Approach
- Umeshkumar Dubey, D P Kothari, G K Awari(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
The statistical tool with the help of which these relationships between two or more than two variables are studied is called correlation . The correlation analysis refers to the tech-niques used in measuring the closeness of the relationship between the variables. A relationship exists between two variables when one depends on the other. In some correlations, the two variables depend on each other and affect each other. A high degree of correlation between two variables may be due to a third variable acting on both. 276 Quantitative Techniques in Business, Management and Finance 12.2.1 Correlation Coefficient The measure of correlation called the correlation coefficient summarises in one number the direction and degree of correlation. 12.2.2 Correlation Analysis Correlation analysis is a technique that measures the nature, degree and extent of relation-ship existing between two or more variables. The correlation coefficient may also be said to be a measure of covariance between two series. 12.2.3 Bi-Variate Correlation The study of correlation between two variables is bi-variate correlation. The study of cor-relation between three or more variables is multivariate correlation. 12.2.3.1 Bi-Variate Data To study the correlation between two variables, we require pairs of observations. Such data is called bi-variate data . The detection and analysis of correlation between two statistical variables requires a relationship of some sort which associates the observations in pairs, one of each pair being a value of each of the two variables. 12.2.4 Correlation: Cause and Effect Relation 1. A high degree of correlation between two variables may be proved mathemati-cally even though they are not related to each other. Therefore, the correlation between the two may be due to chance or coincidence (Table 12.1). 2. A relationship exists between two variables when one depends on the other.
- Nicoleta Gaciu(Author)
- 2020(Publication Date)
- SAGE Publications Ltd(Publisher)
- The data should be arranged as a correlation table or plotted as a scatter graph. The table or scatterplot should be carefully examined to compare the variables and to see whether the paired data points follow a straight line which indicates that the value of one variable is linearly associated with the value of the other variable.
- If an association or a relationship exists between variables, the strength and direction of the relationship will be measured by a coefficient of correlation.
- To see if the relationship occurs by chance, a null hypothesis is formulated, and then the p-value is computed from the data.
- We cannot go directly from statistical correlation to causation, and further investigations are required.
13.1 Covariance and Correlation between two variables
Covariance and Correlation describe the association (relationship) between two variables, and they are closely related statistics to each other, but not the same. The covariance measures only the directional relationship between the two variables and reflects how they change together. A direct or positive covariance means that paired values of the two variables move in the same direction, while an indirect or negative covariance means they move in the opposite direction.The formula for covariance is:where xi is the ith x-value in the data set, is the mean of the x values, yi is the ith y-value in the data set, is the mean of the y-values and n is the number of data values in each data set.If cov(X, Y) > 0 there is a positive relationship between the dependent and independent variables, and if cov(X, Y) < 0 the relationship is negative.Example 13.1 Computing the covariance
Data file: Ex13_1.csv
Suppose that a physics teacher would like to convince her students that the amount of time they spend studying for a written test is related to their test score. She asks seven of her students to study for 0.5, 1, …, 3.5 hours and records their test scores, which are displayed in Table 13.1
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 7 Key Concepts in Statistics 1. Association In statistics, an association is any relationship between two measured quantities that renders them statistically dependent. The term association refers broadly to any such relationship, whereas the narrower term correlation refers to a linear relationship between two quantities. There are many statistical measures of association that can be used to infer the presence or absence of an association in a sample of data. Examples of such measures include the product moment correlation coefficient, used mainly for quantitative measurements, and the odds ratio, used for dichotomous measurements. Other measures of association are the distance correlation, tetrachoric correlation coefficient and Goodman and Kruskal's lambda In quantitative research, the term association is often used to emphasize that a relationship being discussed is not necessarily causal. 2. Distance correlation In statistics and in probability theory, distance correlation is a measure of dependence between two random variables. Its important property is that this measure of dependence is zero if and only if the random variables are statistically independent. This measure is derived from a number of other quantities that are used in its specification, specifically: distance variance , distance standard deviation and distance covariance . These take the same roles as the ordinary moments with corresponding names in the specification of the more standard correlation coefficient. These distance-based measures can be put into an indirect relationship to the ordinary moments by an alternative formulation (described below) using ideas related to Brownian ________________________ WORLD TECHNOLOGIES ________________________ motion, and this has led to the use of names such as Brownian covariance and Brownian distance covariance .
- Alan R. Jones, Alan Jones(Authors)
- 2018(Publication Date)
- Routledge(Publisher)
5 Measures of Linearity, Dependence and CorrelationWe have considered Measures of Central Tendency and Measures of Dispersion and Shape, but these are somewhat insular statistics or univariate statistics, meaning they are, in effect, one dimensional as they are expressing a view of a single value or range variable. Estimating is all about drawing relationships between variables that we hope will express insight into how the thing we are trying to estimate behaves in relation to something we already know, or at least, that we feel more confident in predicting.Ideally, we would like to be able to ascertain cause and effect between an independent variable or driver, and the dependent variable, or entity we are trying to estimate. However, the reality is that in many cases we cannot hope to understand the complex relationships of cause and effect and must suffice ourselves with drawing inference from relationships that suggest that things tend to move in the same direction or opposite directions, and therefore we can produce estimates by reading across changes in one variable (a driver) into changes in some other variable we want to estimate. In short, we want to have some bivariate or multivariate measures that can advise us when there appears to be a relationship between two or more variables. Correlation is a means of measuring the extent (if any) of any relationship.A word (or two) from the wise?'Statistician: A man who believes that figures don't lie, but admits that under analysis some of them won't stand up either'.Evan Esar (1899-1995) American humourist(I only say ‘appears to be a relationship’ because we are dealing with statistics here – heed the ‘A word [or two] from the wise’! We will always need to apply the sense check to the statistics.)Definition 5.1 CorrelationCorrelation is a statistical relationship in which the values of two or more variables exhibit a tendency to change in relationship with one other. These variables are said to be positively (or directly) correlated if the values tend to move in the same direction, and negatively (or inversely) correlated if they tend to move in opposite directions.
eBook - PDFStatistics with JMP
Graphs, Descriptive Statistics and Probability
- Peter Goos, David Meintrup(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
13 Covariance, correlation, and variance of linear functions Langdon frowned. Kohler was right. Holy wars were still making headlines. My God is better than your God. It seemed there was always close correlation between true believers and high body counts. (from Angels & Demons, Dan Brown, p. 57) In Section 3.9.2, we introduced the concepts of Covariance and Correlation between two quantitative variables as measures of (linear) relationship. Although these concepts were initially defined for sample data, we already briefly indicated that covariances and correlations can be calculated for populations. In this chapter, we focus on the calculation of population correlations and covariances. We also pay attention to the variance, which in fact is a special case of a covariance. The definitions and properties in this section are almost all valid for both continuous and discrete random variables. Only expressions with double summations are only valid for discrete random variables. 13.1 Covariance and Correlation We derive the terms Covariance and Correlation based on an example, and compute the population covariance for this example, as defined on page 91. Example 13.1.1 Assume that we want to investigate the covariance between two random variables X and Y, and that the joint and marginal probability distributions of these variables are given as in Table 13.1. In this example, X represents the size of Statistics with JMP: Graphs, Descriptive Statistics, and Probability, First Edition. Peter Goos and David Meintrup. © 2015 John Wiley & Sons, Ltd. Published 2015 by John Wiley & Sons, Ltd. Companion Website: wiley.com/go/goosandmeintrup COVARIANCE, CORRELATION, AND VARIANCE OF LINEAR FUNCTIONS 301 Table 13.1 Joint probability distribution p XY (x, y) and marginal probability distributions p X (x) and p Y (y) of X and Y for Example 13.1.1.
eBook - PDF- Mike Allen, Scott Titsworth, Stephen K. Hunt(Authors)
- 2008(Publication Date)
- SAGE Publications, Inc(Publisher)
Not surprisingly, the more accurate the prediction, the larger the correlation (i.e., closer to either 1 or –1); correspondingly, the smaller the correlation (i.e., closer to zero), the less accuracy in the prediction. A positive correlation indicates that as one value increases, the value for the other variable also increases (as height increases in persons, so does the average weight). A negative correlation indicates that as one value for a variable increases, the value of the other variable diminishes (as smoking tobacco increases, life expectancy diminishes). The size of the correlation value indicates the accuracy of the pre- diction in the direction indicated—larger correlations indicate greater accuracy. CALCULATING THE CORRELATION COEFFICIENT The correlation coefficient is a comparison (in a ratio form) of the covariance of the two variables to the total amount of variability. In other words, the correlation is a comparison, and when the covariance and variance are equal, the ratio is 1.00 (or –1.00). The formula below describes the relationship of covariance (the degree to which the two variables vary together) as it relates to the total amount of variability across both variables. Covariance means that as the values of one variable change, the value of the other variable changes in a predictable direction. As X increases, so does the corresponding value of Y, or as X increases, the value of Y decreases. For instance, consider the data reported in Table 9.1 comparing income (X) and level of communication apprehension (Y). Looking just at the X and Y columns, you can see that there is some degree of variability and that the two values share some degree of covariance. The purpose of the correlation procedure is to quantify the extent to which the covariance between X (income) and Y (communication apprehen- sion) is greater than the overall variance in the sample.
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