What's the difference between correlation and simple linear regression? In particular, I am referring to the Pearson product-moment correlation coefficient.
 A: Here is an answer I posted on the graphpad.com website:
Correlation and linear regression are not the same. Consider these differences:


*

*Correlation quantifies the degree to which two variables are related. Correlation does not fit a line through the data. 

*With correlation you don't have to think about cause and effect. You simply quantify how well two variables relate to each other. With regression, you do have to think about cause and effect as the regression line is determined as the best way to predict Y from X.

*With correlation, it doesn't matter which of the two variables you call "X" and which you call "Y". You'll get the same correlation coefficient if you swap the two. With linear regression, the decision of which variable you call "X" and which you call "Y" matters a lot, as you'll get a different best-fit line if you swap the two. The line that best predicts Y from X is not the same as the line that predicts X from Y (unless you have perfect data with no scatter.)

*Correlation is almost always used when you measure both variables. It rarely is appropriate when one variable is something you experimentally manipulate. With linear regression, the X variable is usually something you experimentally manipulate (time, concentration...) and the Y variable is something you measure.

A: From correlation we can only get an index describing the linear relationship between two variables; in regression we can predict the relationship between more than two variables and can use it to identify which variables x can predict the outcome variable y.
A: Quoting Altman DG, "Practical statistics for medical research" Chapman & Hall, 1991, page 321:
"Correlation reduces a set of data to a single number that bears no direct relation to the actual data. Regression is a much more useful method, with results which are clearly related to the measurement obtained. The strength of the relation is explicit, and uncertainty can be seen clearly from confidence intervals or prediction intervals"
A: In the single predictor case of linear regression, the standardized slope has the same value as the correlation coefficient.  The advantage of the linear regression is that the relationship can be described in such a way that you can predict (based on the relationship between the two variables) the score on the predicted variable given any particular value of the predictor variable.  In particular one piece of information a linear regression gives you that a correlation does not is the intercept, the value on the predicted variable when the predictor is 0.
In short - they produce identical results computationally, but there are more elements which are capable of interpretation in the simple linear regression.  If you are interested in simply characterizing the magnitude of the relationship between two variables, use correlation - if you are interested in predicting or explaining your results in terms of particular values you probably want regression.
A: All of the given answers so far provide important insights but it should not be forgotten that you can transform the parameters of one into the other:
Regression: $y = mx + b$
Connection between regression parameters and correlation, covariance, variance, standard deviation and means:
$$m = \frac{Cov(y, x)}{Var(x)} = \frac{Cor(y, x) \cdot Sd(y)}{Sd(x)}$$
$$b = \bar{y}-m\bar{x}$$
So you can transform both into each other by scaling and shifting their parameters.
An example in R:
y <- c(4.17, 5.58, 5.18, 6.11, 4.50, 4.61, 5.17, 4.53, 5.33, 5.14)
x <- c(4.81, 4.17, 4.41, 3.59, 5.87, 3.83, 6.03, 4.89, 4.32, 4.69)
lm(y ~ x)
## 
## Call:
## lm(formula = y ~ x)
## 
## Coefficients:
## (Intercept)            x  
##      6.5992      -0.3362
(m <- cov(y, x) / var(x)) # slope of regression
## [1] -0.3362361
cor(y, x) * sd(y) / sd(x) # the same with correlation
## [1] -0.3362361
mean(y) - m*mean(x)       # intercept
## [1] 6.599196

A: What's the difference between the correlation between $X$ and $Y$ and a linear regression predicting $Y$ from $X$?
First, some similarities:

*

*the standardised regression coefficient is the same as Pearson's correlation coefficient

*The square of Pearson's correlation coefficient is the same as the $R^2$ in  simple linear regression

*The sign of the unstandardized coefficient (i.e., whether it is positive or negative) will the same as the sign of the correlation coefficient.

*Neither simple linear regression nor correlation answer questions of causality directly. This point is important, because I've met people that think that simple regression can magically allow an inference that $X$ causes $Y$.

*Standard tests of the null hypothesis (i.e., "correlation = 0" or, equivalently, "slope = 0" for the regression in either order), such as carried out by lm and cor.test in R, will yield identical p-values.

Second, some differences:

*

*The regression equation (i.e., $a + bX$) can be used to make predictions on $Y$ based on values of $X$

*While correlation typically refers to the linear relationship, it can refer to other forms of dependence, such as polynomial or truly nonlinear relationships

*While correlation typically refers to Pearson's correlation coefficient, there are other types of correlation, such as Spearman's.

*The correlation between X and Y is the same as the correlation between Y and X. In contrast, the unstandardized coefficient typically changes when moving from a model predicting Y from X to a model predicting X from Y.

A: Correlation analysis only quantifies the relation between two variables ignoring which is dependent variable and which is independent. But before appliyng regression you have to calrify that impact of which variable you want to check on the other variable.
A: The regression analysis is a technique to study the cause of effect of a relation between two variables.
whereas, The correlation analysis is a technique to study the quantifies the relation between two variables.
A: Correlation is an index (just one number) of the strength of a relationship. Regression is an analysis (estimation of parameters of a model and statistical test of their significance) of the adequacy of a particular functional relationship. The size of the correlation is related to how accurate the predictions of the regression will be.
