Simple linear regression describes the linear relationship between a response variable denoted by y and an explanatory variable denoted by x using a statistical model. Statistical models are used to make predictions. In finance, for example, correlation is used in several analyses including the calculation of portfolio standard deviation. Because it is so time-consuming, correlation is best calculated using software like Excel.
Correlation combines statistical concepts, namely, variance and standard deviation. Variance is the dispersion of a variable around the mean, and standard deviation is the square root of variance.
There are several methods to calculate correlation in Excel. The simplest is to get two data sets side-by-side and use the built-in correlation formula:. If you want to create a correlation matrix across a range of data sets, Excel has a Data Analysis plugin that is found on the Data tab, under Analyze. Select the table of returns.
In this case, our columns are titled, so we want to check the box "Labels in first row," so Excel knows to treat these as titles. Then you can choose to output on the same sheet or on a new sheet. Once you hit enter, the data is automatically created. You can add some text and conditional formatting to clean up the result. The linear correlation coefficient is a number calculated from given data that measures the strength of the linear relationship between two variables, x and y.
Correlation combines several important and related statistical concepts, namely, variance and standard deviation. The formula is:. The computing is too long to do manually, and sofware, such as Excel, or a statistics program, are tools used to calculate the coefficient. As variable x increases, variable y increases. As variable x decreases, variable y decreases.
A correlation coefficient of -1 indicates a perfect negative correlation. As variable x increases, variable z decreases. As variable x decreases, variable z increases. A graphing calculator is required to calculate the correlation coefficient. The following instructions are provided by Statology.
Step 1: Turn on Diagnostics. You will only need to do this step once on your calculator. After that, you can always start at step 2 below. This is important to repeat: You never have to do this again unless you reset your calculator. Step 2: Enter Data.
Step 3: Calculate! Finally, select 4:LinReg and press enter. Now you can simply read off the correlation coefficient right from the screen its r. This is also the same place on the calculator where you will find the linear regression equation and the coefficient of determination. The linear correlation coefficient can be helpful in determining the relationship between an investment and the overall market or other securities.
It is often used to predict stock market returns. This statistical measurement is useful in many ways, particularly in the finance industry. For example, it can be helpful in determining how well a mutual fund is behaving compared to its benchmark index, or it can be used to determine how a mutual fund behaves in relation to another fund or asset class. By adding a low, or negatively correlated, mutual fund to an existing portfolio, diversification benefits are gained.
The result is shown below. Our scatterplot shows a strong relation between income over and freelancers who had a low income over leftmost dots typically had a low income over as well lower dots and vice versa. Furthermore, this relation is roughly linear ; the main pattern in the dots is a straight line. The extent to which our dots lie on a straight line indicates the strength of the relation. The Pearson correlation is a number that indicates the exact strength of this relation.
A correlation coefficient indicates the extent to which dots in a scatterplot lie on a straight line. This implies that we can usually estimate correlations pretty accurately from nothing more than scatterplots. The figure below nicely illustrates this point. Some basic points regarding correlation coefficients are nicely illustrated by the previous figure. The least you should know is that. When interpreting correlations, you should keep some things in mind. An elaborate discussion deserves a separate tutorial but we'll briefly mention two main points.
A Variables : The variables to be used in the bivariate Pearson Correlation. You must select at least two continuous variables, but may select more than two. The test will produce correlation coefficients for each pair of variables in this list. B Correlation Coefficients: There are multiple types of correlation coefficients.
By default, Pearson is selected. Selecting Pearson will produce the test statistics for a bivariate Pearson Correlation. C Test of Significance: Click Two-tailed or One-tailed , depending on your desired significance test. SPSS uses a two-tailed test by default. E Options : Clicking Options will open a window where you can specify which Statistics to include i.
Perhaps you would like to test whether there is a statistically significant linear relationship between two continuous variables, weight and height and by extension, infer whether the association is significant in the population. You can use a bivariate Pearson Correlation to test whether there is a statistically significant linear relationship between height and weight, and to determine the strength and direction of the association.
Before we look at the Pearson correlations, we should look at the scatterplots of our variables to get an idea of what to expect. In particular, we need to determine if it's reasonable to assume that our variables have linear relationships. When finished, click OK. To add a linear fit like the one depicted, double-click on the plot in the Output Viewer to open the Chart Editor.
Notice that adding the linear regression trend line will also add the R-squared value in the margin of the plot. If we take the square root of this number, it should match the value of the Pearson correlation we obtain. From the scatterplot, we can see that as height increases, weight also tends to increase. There does appear to be some linear relationship. Select the variables Height and Weight and move them to the Variables box. Indeed, the calculations for Pearson's correlation coefficient were designed such that the units of measurement do not affect the calculation.
This allows the correlation coefficient to be comparable and not influenced by the units of the variables used. The Pearson product-moment correlation does not take into consideration whether a variable has been classified as a dependent or independent variable.
It treats all variables equally. For example, you might want to find out whether basketball performance is correlated to a person's height. You might, therefore, plot a graph of performance against height and calculate the Pearson correlation coefficient.
That is, as height increases so does basketball performance. This makes sense. This is because the Pearson correlation coefficient makes no account of any theory behind why you chose the two variables to compare. This is illustrated below:. It is important to realize that the Pearson correlation coefficient, r , does not represent the slope of the line of best fit. It simply means that there is no variation between the data points and the line of best fit. This is not uncommon when working with real-world data, which is often "messy", as opposed to textbook examples.
We briefly set out the seven assumptions below, three of which relate to your study design and how you measured your variables i. Note: We list seven assumptions below, but there is disagreement in the statistics literature whether the term "assumptions" should be used to describe all of these e. We highlight this point for transparency. Note: The independence of cases assumption is also known as the independence of observations assumption.
Since assumptions 1, 2 and 3 relate to your study design and how you measured your variables , if any of these three assumptions are not met i.
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