How to Calculate and Interpret Karl Pearson Coefficient of Correlation
Karl Pearson's Correlation Coefficient is a fundamental concept in Commerce used to measure the strength and direction of a linear relationship between two variables. It plays a significant role in fields like Accounting and Economics, where understanding relationships between data points is essential for informed business decisions.
The coefficient, denoted as r, quantifies how two numerical variables, such as sales and advertising spend or price and supply, move together. The value of r ranges between -1 and +1. A value of +1 indicates a perfect positive linear correlation, -1 indicates a perfect negative linear correlation, and 0 means no linear relationship.
Karl Pearson's Correlation: Definition and Importance
Karl Pearson’s correlation method helps to identify and express the direction and magnitude of the association between two variables in Commerce studies. For example, in Economics, it is often applied to analyze how a price change may influence product demand or supply.
Understanding this method is important for interpreting business trends, predicting outcomes, and optimizing decision-making processes.
Formula and Calculation Approach
The calculation of Karl Pearson’s coefficient uses the following formula:
r = [ Σ(X – X̄)(Y – Ȳ) ] / [ n × σX × σY ]
Where:
X, Y = Individual values of variables
X̄, Ȳ = Means of X and Y respectively
σX, σY = Standard deviations of X and Y
n = Number of pairs of observations
r = Karl Pearson correlation coefficient
It is crucial to ensure a linear relationship exists between the variables before applying this formula to draw meaningful conclusions.
Worked Example: Step-by-Step Solution
Suppose you want to analyze if there is a correlation between the price of a product and its supply. Consider this data:
| Price (X) | 15 | 25 | 35 | 40 | 50 | 65 | 75 |
|---|---|---|---|---|---|---|---|
| Supply (Y) | 2 | 5 | 6 | 8 | 9 | 10 | 14 |
- Find the mean values of X and Y.
- Compute the deviations of each X and Y from their mean (X–X̄, Y–Ȳ).
- Calculate the products (X–X̄)(Y–Ȳ), square the deviations, and sum up all values as required in the formula.
- Determine σX and σY (standard deviations).
- Substitute all values into the formula to find r.
The final value of r will indicate whether price and supply move in the same (positive) or opposite (negative) direction, and how strong their linear relationship is.
Key Principles and Interpretation
Interpretation of Karl Pearson’s r is straightforward:
- If r is close to +1, there is a strong positive linear relationship.
- If r is close to -1, there is a strong negative linear relationship.
- If r is near 0, there is little to no linear relationship between the variables.
In Commerce, this helps in forecasting, identifying trends, and making evidence-based business recommendations.
Assumptions and Limitations
To rely on results from Karl Pearson’s correlation, certain assumptions must be followed:
- There must be a linear relationship between variables.
- Data should not contain extreme outliers, as these can distort the coefficient.
Remember, correlation does not mean causation. Even if r is high, one variable may not be causing changes in the other.
Interpretation Scale for r
| Value of r | Relationship Type |
|---|---|
| +1 | Perfect positive correlation |
| 0 | No linear correlation |
| -1 | Perfect negative correlation |
Values between 0 and ±1 reflect the degree of strength and direction as described above.
Practical Application in Commerce Studies
Karl Pearson’s coefficient is widely used in Commerce subjects to:
- Analyze the sensitivity between costs and revenues.
- Assess supply and demand relationships in Economics.
- Understand the impact of marketing efforts on sales.
Next Steps for Mastery
To strengthen your understanding, practice problems on supply-demand data, financial statistics, or any dataset with two numeric variables. Using the step-by-step method above ensures clarity and helps avoid calculation errors.
For more Commerce practice sheets and focused concept explanations, visit Vedantu Commerce Practice Questions.
Continue Learning with Vedantu
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FAQs on Karl Pearson Coefficient of Correlation – Formula, Steps & Example
1. What is Karl Pearson's coefficient of correlation?
Karl Pearson's coefficient of correlation (r) is a statistical measure that determines the strength and direction of a linear relationship between two quantitative variables.
Key points:
• Value of r ranges from -1 to +1
• +1 indicates perfect positive correlation
• -1 indicates perfect negative correlation
• 0 means no linear correlation exists between variables
2. What is the formula for Karl Pearson's coefficient of correlation?
The formula for Karl Pearson's coefficient of correlation (r) is:
r = Σ[(X – X̄)(Y – Ȳ)] / [n × σX × σY]
Where:
• X and Y = individual data values
• X̄ & Ȳ = mean of X and Y
• σX & σY = standard deviations of X and Y
• n = number of observations
3. How do you calculate Karl Pearson's coefficient of correlation step by step?
Follow these step-wise procedures to calculate Karl Pearson's r:
1. Find means of X and Y
2. Calculate deviations (X – X̄) and (Y – Ȳ)
3. Compute products (X – X̄)(Y – Ȳ) and squares (X – X̄)2, (Y – Ȳ)2
4. Add all columns
5. Find standard deviations of X and Y
6. Apply values in the formula:
r = Σ[(X–X̄)(Y–Ȳ)] / [n × σX × σY]
4. What do the values of the correlation coefficient mean?
Karl Pearson's r interpretation:
• +1: Perfect positive correlation
• 0.7 to 0.99: Strong positive
• 0.3 to 0.69: Moderate positive
• 0.01 to 0.29: Weak positive
• 0: No correlation
• -0.01 to -0.29: Weak negative
• -0.3 to -0.69: Moderate negative
• -0.7 to -0.99: Strong negative
• -1: Perfect negative correlation
Interpretation helps in understanding the strength and direction of the relationship between the two variables.
5. What are the assumptions of Karl Pearson's coefficient of correlation?
Karl Pearson's coefficient assumes:
• The relationship between variables is linear
• Both variables are measured on interval or ratio scales
• Data should be free from significant outliers
• Both variables are quantitative and continuous
6. What is the difference between Karl Pearson and Spearman’s correlation coefficient?
Main differences between Karl Pearson and Spearman’s correlation:
• Karl Pearson: Used for quantitative, continuous data; assumes linearity; based on means and standard deviations
• Spearman’s rank: Used for ordinal or ranked data; measures monotonic relationships; based on ranks not original values
• Pearson's coefficient is sensitive to outliers, Spearman’s is more robust
7. Can Karl Pearson's correlation coefficient be negative?
Yes, Karl Pearson's coefficient (r) can be negative. A negative value indicates a negative linear relationship – as one variable increases, the other decreases. The value ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation).
8. In which situations is Pearson's correlation not appropriate?
Pearson's correlation should not be used when:
• The relationship between variables is non-linear
• One or both variables are ordinal, nominal, or not continuous
• There are major outliers present
• Sample size is very small
In such cases, consider Spearman’s rank correlation as an alternative.
9. How is the Karl Pearson coefficient used in commerce and business?
Karl Pearson's coefficient is widely used in commerce and business to:
• Measure relationship between sales and advertising expenditure
• Analyze correlation between price and demand
• Study association between production volume and profit
• Support data-driven decision-making in economics, accountancy, and business analytics
10. What are the properties of Karl Pearson's coefficient of correlation?
The properties of Karl Pearson’s correlation coefficient (r) are:
• Ranges from -1 to +1
• Unit-free (dimensionless measure)
• Affected by extreme values (outliers)
• Shows only linear relationships
• Interchanging X and Y does not change the value of r
11. Is a high correlation coefficient always evidence of causation?
No, a high correlation coefficient shows only a strong relationship, not causation.
Remember:
• Correlation ≠ Causation
• Two variables may move together due to coincidence, hidden factors, or mutual influence
• Always examine context and use additional analysis to infer causality
12. How do you interpret an r value close to zero?
An r value close to zero (e.g. 0.05, -0.03) indicates no linear correlation between the two variables.
• The variables do not move together in any consistent linear pattern
• There might still be a non-linear or other form of relationship
