Math Problem Statement
Calculation the co efficient of the correlation of the following data X- 2,3,4,5,6,7,8 Y-4,5,6,8,9,7,1
Solution
To calculate the coefficient of correlation (denoted as ) for the given data, we can use the formula for the Pearson correlation coefficient:
[ r = \frac{n \sum{xy} - \sum{x} \sum{y}}{\sqrt{[n \sum{x^2} - (\sum{x})^2][n \sum{y^2} - (\sum{y})^2]}} ]
Where:
- and are the values of the variables.
- is the number of data points (in this case, 7).
Step 1: Calculate the necessary sums.
Given data:
Let's compute the following:
Step 2: Perform the calculations.
We first calculate these step by step:
-
Sum of X ():
-
Sum of Y ():
-
Sum of ():
-
Sum of ():
-
Sum of ():
Step 3: Substitute into the formula.
Now, using the values in the correlation formula:
[ r = \frac{7 \times 198 - 35 \times 40}{\sqrt{[7 \times 203 - (35)^2][7 \times 272 - (40)^2]}} ]
Step 4: Simplify the numerator and denominator.
-
Numerator:
-
Denominator:
Step 5: Calculate the final value.
Thus, the correlation coefficient is approximately -0.057.
Would you like a detailed breakdown of any specific part of the process, or is there another concept you'd like to explore?
Here are 5 related questions:
- What does a negative correlation coefficient tell us about the relationship between X and Y?
- How do you interpret a correlation coefficient close to 0?
- What is the significance of a Pearson correlation coefficient in statistical analysis?
- Can correlation be used to prove causation? Why or why not?
- How do you calculate the slope of a regression line from a correlation coefficient?
Tip: Always double-check the units of your data. A correlation tells you about the relationship between two variables, but not their absolute values.
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Math Problem Analysis
Mathematical Concepts
Statistics
Correlation Coefficient
Pearson Correlation
Formulas
Pearson correlation coefficient formula: r = (nΣxy - ΣxΣy) / √[(nΣx² - (Σx)²)(nΣy² - (Σy)²)]
Theorems
Pearson's correlation theorem
Suitable Grade Level
Grades 10-12
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