Karl Pearson Coefficient of Correlation
Ca Narayan3 Aug, 2023
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Karl Pearson’s Coefficient of Correlation is a mathematical method in which the numerical expression is used to estimate or determine the range or magnitude and the direction of the link between two linearly related variables.
It is a well-known and widely used method called the Pearsonian Coefficient of Correlation. Karl Pearson's Coefficient of Correlation formula is used to calculate the evaluation of the relationship between two variables.What is the Coefficient of Correlation?
The coefficient of correlation, which is generally represented as "r," is a statistical metric that is used to assess the intensity and direction of the association between two variables. The coefficient of correlation, or simply correlation coefficient, ranges between -1 and 1. A positive value of "r" shows a positive correlation, suggesting that when one variable develops, the other tends to increase as well. Conversely, a negative value of "r" suggests a negative correlation, indicating that when one variable increases, the other tends to decrease. An "r" value of 0 denotes no correlation between the variables, suggesting that changes in one variable do not impact the other.Types of Correlation
Correlation, the measure of association between two variables, comes in various forms, each offering valuable insights into relationships within data. Here, we explore the types of correlation in simple language to help ease comprehension.Positive Correlation:
When two variables demonstrate a positive correlation, their values tend to grow or drop together. A positive correlation coefficient (r) sits between 0 and 1, suggesting a direct association.Negative Correlation:
Conversely, with a negative correlation, the variables display an inverse connection. As one variable grows, the other lowers. Represented by a negative "r" value between 0 and -1, a negative correlation implies that changes in one variable correlate to contradictory changes in the other.Zero Correlation:
A zero correlation signifies no discernible relationship between the two variables. The correlation coefficient "r" is 0, implying that changes in one variable have no impact on the other.Perfect Correlation:
When two variables have a correlation coefficient of either 1 or -1, it indicates a perfect correlation. Perfect correlation represents a strong, predictable connection between the variables.Strong Correlation:
A strong correlation, whether positive or negative, is indicated by an "r" value close to 1 or -1, respectively. It suggests a robust relationship between the variables, implying that changes in one variable significantly influence the other.Weak Correlation:
When the correlation coefficient approaches 0, it points to a weak correlation. Weak correlation denotes limited association between variables, and changes in one variable have little effect on the other.Methods to Calculate Karl Pearson's Coefficient of Correlation
Karl Pearson's Coefficient of Correlation, typically written as "r," is a frequently used statistical metric that assesses the strength and direction of the association between two variables in a dataset. Various methods are available to determine this correlation coefficient, each adjusted to fit various data kinds and needs. Here, we present a clear and concise explanation of three popular methods to calculate "r" and define the key terms involved.Method of Direct Calculation:
The formula for Karl Pearson's coefficient using the direct calculation method is: r = (nΣxy - ΣxΣy) / √((nΣx² - (Σx)²) * (nΣy² - (Σy)²)) Where, Σxy: The summation of the product of corresponding values of the two variables. Σx: The sum of all values of the first variable. Σy: The sum of all values of the second variable. Σx²: The sum of squares of all values of the first variable. Σy²: The sum of squares of all values of the second variable. n: The number of data points in the dataset.Method of Deviations:
The formula for Karl Pearson's coefficient using the method of deviations is: r = (∑(xᵢ - x̄)(ȳᵢ - ȳ)) / √(∑(xᵢ - x̄)² * ∑(ȳᵢ - ȳ)²) Where, xᵢ: The value of the first variable at the "i-th" data point. ȳ: The mean of the second variable. ȳᵢ: The value of the second variable at the "i-th" data point.Method of Covariance and Standard Deviations:
The formula for Karl Pearson's coefficient using the method of covariance and standard deviations is: r = Cov(x, y) / (σₓ * σᵧ) Cov(x, y): The covariance between the first and second variables. σₓ: The standard deviation of the first variable. σᵧ: The standard deviation of the second variable.Method of Rank Correlation:
The formula for Pearson's coefficient using the Method of Rank Correlation is: r = 1 - ((6 * ∑dᵢ²) / (n * (n² - 1))) Where, dᵢ: The difference in ranks of corresponding data points for both variables (x and y). n: The number of data points in the dataset.Karl Pearson's Coefficient of Correlation Numericals
Numerical Question 1: An ice cream parlor owner is analyzing the relationship between daily temperature and the number of ice cream units sold. For a period of 10 days, the recorded data is as follows:| Day | Temperature (°C) | Ice Cream Sales (units) |
| 1 | 25 | 85 |
| 2 | 30 | 92 |
| 3 | 27 | 78 |
| 4 | 22 | 65 |
| 5 | 32 | 100 |
| 6 | 29 | 90 |
| 7 | 28 | 85 |
| 8 | 24 | 70 |
| 9 | 26 | 80 |
| 10 | 31 | 95 |
| Individual | Exercise Duration (minutes) | Heart Rate (bpm) |
| 1 | 20 | 110 |
| 2 | 30 | 125 |
| 3 | 15 | 105 |
| 4 | 25 | 120 |
| 5 | 40 | 135 |
| 6 | 35 | 130 |
| 7 | 10 | 100 |
| 8 | 45 | 140 |
| 9 | 50 | 150 |
| 10 | 55 | 160 |
| 11 | 5 | 90 |
| 12 | 60 | 170 |
Uses of Karl Pearson's Coefficient of Correlation
Karl Pearson's Coefficient of Correlation is a powerful statistical tool widely employed in various fields to measure the relationship between two variables. It plays a crucial role in drawing valuable insights and making informed decisions based on data analysis. It is used to:- Assess association and measure the strength of relationships between variables.
- Validate hypotheses and support data-driven decision-making.
- Enable predictive analysis by identifying patterns and trends.
- Aid in risk assessment and optimization strategies.
- Facilitate social sciences research, economic analysis, and quality control.
- Contribute to medical research and market studies.
- Enhance educational analysis and improve teaching methods.
- Provide valuable insights across diverse disciplines.
| Evolution And Definition Of Economics | Emergence Of Macroeconomics |
| Education Sector In India | Intermediate Good |
Karl Pearson's Coefficient of Correlation FAQs
Ques1:- How is Karl Pearson's coefficient used in prediction? Ans:- It helps make predictions about one variable based on the known value of the other. Ques2:- In which fields is the coefficient widely used? Ans:- It is extensively used in social sciences, economics, medical research, and market analysis. Ques3:- How is a positive correlation interpreted? Ans:- A positive correlation suggests that as one variable increases, the other tends to increase as well. Ques4:- What does a negative correlation indicate? Ans:- A negative correlation indicates that as one variable increases, the other tends to decrease. Ques5:- How is zero correlation interpreted? Ans:- Zero correlation means there is no meaningful relationship between the two variables.
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