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Difference Between Linear and Curvilinear Correlation

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Studying in Pairs – Correlation

In statistical studies, we measure more than one variable for each individual. Like we measure precipitation and plant growth, or we measure the number of young birds with their nesting habitat. So, we collect pairs of data and study them together instead of studying the pair or variable individually. This gives rise to correlation.

Studying Correlation, We will Find Two Terms – Linear and Curvilinear Correlation. In this section, our motive is to explain the differences between these two terms, we will also define the terms under the heading ‘meaning of linear correlation’ and ‘what is curvilinear correlation?’ Hence, let us get started. 


Difference between Linear and Curvilinear Correlation

As already mentioned, we will prioritize our study on the difference between Linear and Curvilinear Correlation. Apart from this, it is important for us to know the meaning of correlation, linear correlation, and curvilinear correlation in depth. 

Well, in simple words we can say – Linear Correlation means the ratio of change that is constant. While Curvilinear Correlation or Non-Linear Correlation means the ratio of change is not constant. 

This is only a brief discussion of the difference between Linear and Curvilinear Correlation, we will represent the same in detail in our forthcoming section. 


The Bivariate Data

If we examine each variable separately, it means we are examining the univariate data, but in correlation, we find ways to describe and analyze the bivariate data, in which there are two variables. If you take a sample, we can describe the relationship between the two variables taken in the sample graphically and numerically. So, we can say correlation is defined as the statistical correlation between two variables.


Meaning of Linear Correlation

Linear correlation is referred to as the measure of relationship which is between two random variables with their values which range from -1 and 1. This is proportional to the covariance which can be interpreted in the same way as the covariance.

Linear correlation can also be said to be based on the straight-line relationship which occurs between two random variables given in the sample.

Correlation is known as linear if the ratio of change remains constant. Suppose, the amount of output in a factory gets doubled by doubling the number of workers, if the data is graphed it will be an example of linear correlation.

Graphically Represented - when all the points on the scatter diagram incline to lie near a line that looks like a straight line, then the correlation is said to be a linear equation. 

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What is Curvilinear Correlation?

Non-linear, also known as the curvilinear correlation is said to occur when the ratio of change happens between two variables that are not constant. This can happen when the value of one variable increases, with the value of another variable also increasing side by side. This will happen to an extent, after which the increase in that value of one variable will start resulting in the decrease in the value of the other variable.

To represent the curvilinear equation graphically – It is an inverted U 

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Difference Between Linear and Curvilinear Correlation

The difference between linear and curvilinear correlation is depicted below with the help of a graph.

Linear Correlation 

Curvilinear Correlation

There exists a linear correlation if the ratio of change in the two variables is constant here. 


Suppose we plot the coordinates in a graph-

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There exists a curvilinear correlation if the change in the variable is not constant here. 


Suppose we plot the coordinates in a graph- 

(Image)


A Scatter Diagram Depicts many Relations between Two Variables 

A correlation happens to be between two variables only when one of them is related to the other in some or another way. We can take an example of a scatter diagram, which is a graph of the paired (x, y) sample data with the horizontal x-axis and with its vertical y-axis. Each individual (x, y) pair is plotted in the form of a single point.

Here, we plot the bear chest girth (y) against the bear length (x). When examining the scatter diagram, we need to study the overall pattern of the plotted points. We will find that the value for the chest girth tends to increase as the value of the length increases. Also, we will observe the upward slope and a straight-line pattern in the plotted data points illustrated in the graph.

Thus, we can say that a scatter diagram can identify several different types of relationships between any two related variables. 

The Relationships are as Follows:

  • A relationship will have no correlation when the points on a scatter diagram do not show any specific pattern.

  • A relationship is called non-linear when the points on the scatter diagram follow a pattern but the pattern is not a straight line.

  • A relationship is called linear when the points on the scatter diagram give rise to a straight-line pattern.

  • Linear relationships can be any - positive or negative. Positive relations have their points which are inclined upwards to the right side. If the value increases, the value of y increases. If the value of x decreases, the value of y decrease. 

  • If there is a negative relationship then their points will decline downwards to the right. If the value of x increases, the value of y decreases. While, if the value of x decreases, the value of y increases. 

  • Non-linear relationships will have a clear pattern, they are not only linear. 

  • Also, to note – if two variables have no relationship, then there will be neither a straight-line relationship nor a non-linear relationship. 

FAQs on Difference Between Linear and Curvilinear Correlation

1. What is the primary difference between linear and curvilinear correlation?

The primary difference lies in the nature of the relationship between two variables. In linear correlation, the ratio of change between the variables is constant, meaning the relationship can be represented by a straight line on a graph. In curvilinear (or non-linear) correlation, the ratio of change is not constant, causing the relationship to form a curve when plotted. For more details on correlation types, you can explore the Types of Correlation Explained.

2. How can a scatter plot help identify if a correlation is linear or curvilinear?

A scatter plot is a powerful visual tool for this purpose. If the points on the plot tend to cluster around a straight line (either rising or falling), it indicates a linear correlation. If the points form a distinct curve or pattern (like a 'U' shape or an inverted 'U'), it signifies a curvilinear correlation. A random scattering of points with no clear pattern suggests no correlation. You can practice identifying these patterns in our MCQs on Correlation and Regression.

3. What is a real-world example of linear correlation in commerce?

A classic example of positive linear correlation in commerce is the relationship between advertising expenditure and sales revenue. Generally, as a company increases its spending on advertising, its sales revenue tends to increase at a relatively consistent rate. Another example is the negative linear correlation between the price of a product and its quantity demanded, as per the law of demand.

4. Can you provide a common example of curvilinear correlation?

A common example is the relationship between the age of a worker and their income. Initially, as a person gains experience, their income increases. However, after a certain peak age (e.g., late 40s or 50s), income may start to plateau or even decrease as they approach retirement. This relationship, when plotted, forms a curve, not a straight line, representing a curvilinear correlation.

5. Why is it important for a student to distinguish between linear and curvilinear correlation?

Distinguishing between them is crucial for accurate statistical analysis. Applying a linear model (like Pearson's correlation coefficient or linear regression) to a strong curvilinear relationship can be misleading. It might show a weak or zero correlation, causing you to wrongly conclude that no relationship exists. Recognising the type of correlation ensures you select the appropriate statistical method to measure the relationship's true strength and nature.

6. If the correlation coefficient (r) is zero, does it always mean there is no relationship between variables?

No, this is a common misconception. A correlation coefficient (r) of zero specifically indicates the absence of a linear relationship. It is entirely possible for two variables to have a perfect curvilinear relationship (e.g., shaped like a parabola) and still have a linear correlation coefficient of zero. Therefore, it is essential to visualise the data with a scatter plot before concluding that no relationship exists.

7. Is 'correlation' the same as 'linear correlation'?

Not exactly. 'Correlation' is a broad term that refers to any statistical relationship or association between two variables. This can be linear or non-linear. 'Linear correlation' is a specific type of correlation where the relationship can be graphically represented as a straight line. While linear correlation is the most commonly studied type in introductory statistics, it is just one of several ways variables can be related.