Measuring Correlation: Visual and Numerical Methods

As explained in the previous blog, relationships can be found everywhere, and they need to be measured to determine whether they actually exist.

Both visual and numerical methods can be used to measure correlation.

Visual and Numerical Methods to Measure Correlation

1. Scatter plot

When the values of one quantitative variable are plotted against the corresponding values of the other quantitative variable on a coordinate plane, the result is a scatter plot.

Quantitative variables are the variables that represent numerical values that can be measured or counted (Age, Height, Weight, Temperature, etc).

In simple terms, it visualizes the data points of one variable against those of another variable using dots.

Normally, the independent variable is presented on the horizontal axis (X-axis) while the dependent variable is presented on the vertical axis (Y-axis).

The example scatter plot below shows the relationship between sales and advertising expenses.

Advertising Expenses Vs Sales Revenue

The above scatter plot shows that sales tend to increase with the increase in advertising expenses.

A thorough examination of a scatter plot can answer the following questions.

• Is the relationship linear or non-linear?
• Is the linear relationship positive or negative?
• Is the linear relationship perfect, strong, or weak?

1. Answering the first question

A linear relationship can be identified when the scatter (dots) gather around an imaginary straight line. In other words, if those points follow a straight-line pattern, either upward or downward, it indicates a linear relationship between the two quantitative variables.

The scatter plot for the above example of sales and advertising expenses shows a linear relationship, since the data points follow an upward straight line.

Linear Relationship

A non-linear relationship can be identified when the points in the scatter plot do not follow a straight line.

Non-linear Relationship

2. Answering the second question

Even though the scatter plot shows a linear relationship, it is important to identify whether it is positive or negative.

A positive relationship can be identified when the points move in the same direction (bottom left to the top right). In simple terms, a positive relationship exists when both variables increase or decrease together.

Positive Correlation

A negative relationship can be identified when the points move in the opposite direction (top left to the bottom right).

Negative Correlation

3. Answering the third question

A perfect positive or negative relationship can be identified when all the points lie exactly on a straight line.

Perfect Positive Correlation

Perfect Negative Correlation

A strong relationship can be determined when the points lie very closely around the imaginary straight line.

Strong Positive Correlation

Strong Negative Correlation

A weak positive or negative relationship can be recognized when the points are widely scattered from a straight line. However, it still shows a positive or negative relationship.

Weak Positive Correlation

Weak Negative Correlation

Advantages of Scatter Plots

1. Helps Identify Relationships

Scatter plots are useful for identifying the nature (direction and strength) of the relationship between two variables. They give a rough idea of how the two variables are related.

2. Detects Linear and Non-Linear Patterns

Scatter plots help identify whether the relationship follows a linear or a non-linear pattern. It is important to select appropriate statistical methods.

3. Identifies Outliers

Points that are far from the general pattern are known as outliers or extreme values. Scatter plots make it easy to detect outliers, which are farther away than the other data points.

4. Simple and Easy to Interpret

Reading a scatter plot is not a complex task. It provides a quick visual understanding of data patterns. It is useful for both statistical analysis and decision-making.

5. Helps in Predictive Analysis

Since scatter plots provide an initial idea of the nature of the association between variables, they help to predict the changes in one variable when the other variable changes. For example, a positive relationship between advertising expenses and sales may suggest that higher advertising spending is associated with increased sales.

6. Useful Before Conducting Correlation Analysis

Since Pearson’s correlation coefficient is only used to measure linear relationships, a scatter plot helps determine whether a linear relationship exists before calculating it.

Limitations of Scatter Plots

1. Cannot Measure the Exact Strength of a Relationship

Although the scatter plot provides a brief idea of the relationship, it does not give an exact numerical value for the correlation. Another numerical measurement should be used to measure it precisely.

2. Subjective Interpretation

Interpretations given by different people can vary for relationships using the scatter plot.

3. Limited to Two Variables at a Time

A scatter plot shows the relationship between only two quantitative variables. It cannot be used to study more than two variables simultaneously.

4. Difficult to Interpret with Large Datasets

When there are many points, they may overlap, creating a packed plot. It would be difficult to identify the patterns.

5. May Not Reveal Complex Relationships

Scatter plots can be used only to analyze simple patterns, not the complex patterns between multiple variables.

Scatter plots are an excellent first step for studying relationships between variables. However, numerical measures and statistical analysis would be required for accurate conclusions.

Free-hand Method

The free-hand method is a graphical technique in which a straight line is drawn by eye on the scatter plot so that the data points are distributed as evenly as possible on both sides of the straight line.

The free-hand method gives the approximate line of best fit that represents the overall trend of the relationship between two variables.

Free-hand Method

Advantages of the Free-hand Method

1. Simple and Easy Method

This is a very simple and straightforward method to get a trend line that doesn’t require complex calculations or statistical software. This trend line can be easily explained and interpreted.

2. Shows the Overall Trend

This trend line provides a clear visual representation of the overall direction (positive or negative) and strength (perfect, strong, or weak) of the relationship between two variables.

3. Helps in Initial Data Analysis

This trend line provides a basic understanding of the relationship before using any numerical measurements such as the correlation coefficient.

4. Useful for Estimation

The free-hand line can be used to make rough estimates of one variable based on the changes in the other variable. But these predictions must be confirmed using a numerical method.

Limitations of the Free-Hand Method

1. Subjective in Nature

Since this line is drawn by eye, different people may draw different lines for the same scatter plot based on their personal judgment. Therefore, the interpretations can vary.

2. Does Not Provide an Exact Measurement

The free-hand method fails to provide an exact numerical measure of correlation. It is less accurate than the numerical methods such as the least-squares regression line.

3. Not Suitable for Complex Relationships

When there is a complex or non-linear relationship between two variables, the free-hand method is not very useful for making further statistical analysis.

4. Influenced by Outliers

Extreme values may affect how the line is drawn, resulting in a misleading representation of the overall trend.

In modern statistical analysis, the free-hand method is rarely used due to its lower accuracy and subjectivity. But it is a simple and useful graphical technique for beginners to understand the overall trends of relationships.

Coefficient of Correlation

Both scatter plots and the free-hand method provide a simple visual illustration of the relationship between two variables.

However, since these methods do not provide an exact numerical value for correlation, the coefficient of correlation can be used as a statistical measurement to evaluate the direction and the strength of the linear relationship between two quantitative variables.

The coefficient of correlation is a numerical value that ranges from -1 to +1 to quantify the direction and strength of the linear relationship between two quantitative variables.

The most commonly used coefficient of correlation for linear relationships is Pearson’s Correlation Coefficient. It is represented by r.

If the coefficient of correlation (r) is closer to -1 or +1, it is a strong negative or positive linear relationship, while if it is closer to 0, it is a weak negative or positive linear relationship.

The formula to obtain the coefficient of correlation can be shown as follows.

Coefficient of Correlation

Properties of the Coefficient of Correlation

1. The value of r falls between -1 and +1, including -1 and +1.
2. The coefficient of correlation doesn’t have a unit of measurement.
3. The coefficient of correlation has a magnitude and a direction, where the sign of the value indicates the direction while the absolute value shows the magnitude of the relationship.
4. The value of the coefficient of correlation doesn’t change when adding, subtracting, multiplying, or dividing both dependent and independent variables by the same amount.
5. Symmetric property — The coefficient of correlation between x and y is the same as between y and x.
6. Zero coefficient of correlation means there is no linear relationship between two variables. But there might be a non-linear relationship; that means a zero coefficient of correlation doesn’t always imply independence between variables. However, independent variables do not have any relationship. Therefore, independence implies zero correlation.

Limitations of the Coefficient of Correlation

1. The correlation coefficient only measures whether there is a linear relationship. It fails to determine the nature of non-linear relationships. Extreme values can increase or decrease the coefficient.
2. The correlation coefficient value depends on the range of observations used to calculate it.
3. It only measures the correlation between two variables.

It can be concluded that the coefficient of correlation is statistically useful rather than scatter plots and the free-hand method.